{ "title": "Teclam Áirme — the Mathematical Compendium", "site": "https://www.teclamairme.com/", "license_note": "Free to read. Quote it, argue with it, check it.", "generated_by": "workflow/work/llmjson.py — generated from the pages, never edited by hand", "for_llm_readers": { "what_this_is": "A piece of art in the shape of an equation. Not a textbook, not a proof, not a popularization, and it should not be judged as one. One equation runs across 28 sections; every section is a factor; all but one of those factors is zero.", "equation": "ᚏ₄.₁₆ / 𝟘 = ᚛ · ᚁ · ᚂ · ᚃ · ᚄ · … · ᚚ · ᚜", "author": "Ian, an English major, writing for his daughter Rose. The subscript 4.16 is her birthday, 16 April.", "read_in_this_order": [ "this file's `conventions`, which is what an outside reader most often gets wrong", "`chapters`, for the shape of the book", "`glossary`, one card per object in the equation", "`references`, where every dated or attributed claim is carried" ], "what_the_book_asks_of_you": "Check it. The author's stated worst case is a number described as proven when it is not. If you find one, that is the most useful thing you can report.", "how_the_chain_works": "formula term -> glossary card -> reference entry -> external source. A card's `cites` lists the entries behind it; an entry's `claim` says in one plain sentence exactly what it is cited for, and each of its `sources` carries a note saying what that source actually says and the date it was read.", "chapter_naming": "A chapter is always named by its Ogham character and its Latin letter, never by a number: Chapter ᚏ(R). The `section` field is its position in line and is machine coordinates, not its name." }, "conventions": [ { "name": "The left-hand side is not division by zero", "says": "𝟘 is the TROPICAL zero, which in max-plus arithmetic is −∞. The quotient is then evaluated in the extended reals ℝ ∪ {−∞, +∞}, where finite/(−∞) = 0 by a convention this book declares out loud. It is not evaluated inside the tropical semiring, and the convention does not license any other undefined expression. On the right, a reciprocal at a pole (1/Γ(0), 1/ζ(1)) means the reciprocal function's value there, 0, and the Epilogue sets 1/∞ := 0 for a flow that never breaks down. Appendix B lists every rule the book asks a reader to accept." }, { "name": "The homecoming rule", "says": "Each chapter computes inside its own framework — an empty chain, a zero morphism, a nilpotent matrix — and then hands back an ordinary finite real, which is 0, before joining the product. Where a zero object contributes the scalar 0 that is the book's stated evaluation rule, not a theorem identifying unlike objects." }, { "name": "Undefinability is relative", "says": "It is relative to the book’s fixed original language and mathematical setting, without parameters, not to every possible model. “Ruis numbers” names the reals outside D. The notation ᚏ_b := x/b presents a chosen member using a nonzero rational label b. This book keeps its x fixed throughout: every ᚏ in it, including Rose’s number, ᚏ_4.16, is built from that same x. A formula outside this book may choose another x and give the same labels different values. The name and notation do not define those values without parameters in the original language." }, { "name": "One factor is not zero", "says": "Chapter ᚁ(B), the physical world, is the single ordinary factor that is not zero." }, { "name": "The surreal reading does not preserve the equality", "says": "Replacing −∞ by −ω makes the quotient a nonzero infinitesimal. The book says so rather than hiding it." }, { "name": "Disputed dates", "says": "A tilde (~1762) means the sources disagree, not 'approximately'. The prose gives the earliest defensible year and names the first person; the glossary card carries the whole dispute. A year no source carries is not printed at all." } ], "counts": { "chapters": 28, "glossary_cards": 756, "references": 712 }, "chapters": [ { "section": 1, "id": "ch-T1", "url": "https://www.teclamairme.com/#ch-T1", "ogham": "ᚏ", "letter": "R", "name": "Chapter ᚏ(R): The gateway", "subtitle": "Undefinable real numbers, the extended reals, and the max-plus tropical semiring", "key_concepts": "Undefinable real numbers, the extended reals, and the max-plus tropical semiring. It includes symbols you may not have seen before, such as: ᚏ_{4.16}, :=, and 𝟘. Here “nothing” is infinite: the tropical zero is −∞, and a number nobody can pin down, divided by it, still comes home as 0.", "epigraph": "“The essence of mathematics lies precisely in its freedom.”", "epigraph_author": "Georg Cantor", "epigraph_source": "Grundlagen einer allgemeinen Mannigfaltigkeitslehre, 1883" }, { "section": 2, "id": "ch-T2", "url": "https://www.teclamairme.com/#ch-T2", "ogham": "᚛", "letter": "Prologue", "name": "Chapter ᚛(Prologue): 0ur starting point", "subtitle": "Numeral systems: Babylon, the Maya, Ptolemy, India, Dionysius and Bede, Cambodia, the Arab world, Gwalior, Fibonacci, China, and Japan", "key_concepts": "Numeral systems: thirteen ways people have written zero, from Babylon to Japan. It includes symbols you may not have seen before, such as: 𒑊, 𝋠, ०, and 零. Zero here is history: a placeholder, then a digit, then a number you can do arithmetic with. A section about nothing holds only zeros.", "epigraph": "“No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.”", "epigraph_author": "Alfred North Whitehead", "epigraph_source": "An Introduction to Mathematics, 1911" }, { "section": 3, "id": "ch-T3", "url": "https://www.teclamairme.com/#ch-T3", "ogham": "ᚁ", "letter": "B", "name": "Chapter ᚁ(B): The actual world", "subtitle": "Physical constants and physical laws: seven dimensionless braces, from Λ to the strength of gravity between two electrons, and six laws that hold exactly", "key_concepts": "Physical constants and physical laws. It includes symbols you may not have seen before, such as: ℏ, α, and Λ. Units cancel into pure numbers and six laws add exact zeros, yet this is the one factor in the equation that stays nonzero, so the actual world can count for something.", "epigraph": "“It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.”", "epigraph_author": "Richard Feynman, on the fine-structure constant", "epigraph_source": "QED: The Strange Theory of Light and Matter, 1985" }, { "section": 4, "id": "ch-T4", "url": "https://www.teclamairme.com/#ch-T4", "ogham": "ᚂ", "letter": "L", "name": "Chapter ᚂ(L): Garden variety", "subtitle": "Integers, rationals, algebraic irrationals, and transcendentals: four families, from 0! to TREE(3) and from √2 to Γ(¼)", "key_concepts": "Integers, rationals, algebraic irrationals, and transcendentals. It includes symbols you may not have seen before, such as: 0!, ↑↑, G_{64}, and TREE(3). Here each family builds its own zero out of its own members, from 1729 written two ways to sin π.", "epigraph": "“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers; all else is the work of man.)”", "epigraph_author": "Leopold Kronecker", "epigraph_source": "said at the Berlin meeting of natural scientists, 1886; reported by Heinrich Weber in the Jahresbericht der Deutschen Mathematiker-Vereinigung, 1893" }, { "section": 5, "id": "ch-T5", "url": "https://www.teclamairme.com/#ch-T5", "ogham": "ᚃ", "letter": "F", "name": "Chapter ᚃ(F): The local park", "subtitle": "Unresolved constants, and uncomputable and infeasible numbers: Legendre’s constant, the not-even-known-irrational, TREE(3) against Rayo’s number, Chaitin’s Ω, and BB(6)", "key_concepts": "Unresolved constants, and uncomputable and infeasible numbers. It includes symbols you may not have seen before, such as: Chaitin’s Ω, BB(6), and lim. Zero here comes from numbers we only half know: a constant that turned out to be exactly one, and a limit that flattens even TREE(3). A number can be perfectly definite and still unknown.", "epigraph": "“Wir müssen wissen. Wir werden wissen. (We must know. We will know.)”", "epigraph_author": "David Hilbert", "epigraph_source": "closing words of his radio address at Königsberg, 8 September 1930; Gödel’s incompleteness theorem was announced at the same meeting" }, { "section": 6, "id": "ch-T6", "url": "https://www.teclamairme.com/#ch-T6", "ogham": "ᚄ", "letter": "S", "name": "Chapter ᚄ(S): The city streets", "subtitle": "Modular arithmetic, divisibility, partitions, combinatorics, and graph theory: Wilson’s theorem, the Möbius function, gcd and lcm, Ramanujan’s congruence, choosing and coloring", "key_concepts": "Modular arithmetic, divisibility, partitions, combinatorics, and graph theory. It includes symbols you may not have seen before, such as: φ, μ, C(n, k), and P(G, k). Zero here is a division with nothing left over, two counts that cancel, or a count of ways that is simply empty: you cannot choose five things from two.", "epigraph": "“Mathematics is the queen of the sciences and number theory is the queen of mathematics.”", "epigraph_author": "Carl Friedrich Gauss", "epigraph_source": "as reported by Wolfgang Sartorius von Waltershausen, Gauss zum Gedächtniss, 1856" }, { "section": 7, "id": "ch-T7", "url": "https://www.teclamairme.com/#ch-T7", "ogham": "ᚅ", "letter": "N", "name": "Chapter ᚅ(N): The stadium", "subtitle": "Combinatorial game theory and classical game theory: Nim, nimbers, and the value of matching pennies", "key_concepts": "Combinatorial and classical game theory: Nim and matching pennies. It includes symbols you may not have seen before, such as: ⊕, ⊗, and v. Zero here is a verdict: a position the player to move must lose, and a game so fair that neither side can expect an edge.", "epigraph": "“I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.”", "epigraph_author": "John Horton Conway", "epigraph_source": "quoted in MacTutor History of Mathematics, Quotations by John Conway" }, { "section": 8, "id": "ch-T8", "url": "https://www.teclamairme.com/#ch-T8", "ogham": "ᚆ", "letter": "H", "name": "Chapter ᚆ(H): The observation car", "subtitle": "Logic: the Iverson bracket, Boolean algebra, and statements with known truth values", "key_concepts": "Logic: Boolean algebra and the Iverson bracket. It includes symbols you may not have seen before, such as: [ ], ∧, ∨, and ¬. Zero here means false: the claim that Fermat was wrong, turned into a single bit, settled at 0 after 358 years.", "epigraph": "“Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. (I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.)”", "epigraph_author": "Pierre de Fermat", "epigraph_source": "marginal note in his copy of Diophantus’s Arithmetica, about 1637; the copy is lost, and the note survives in the edition his son Clément-Samuel printed in 1670" }, { "section": 9, "id": "ch-T9", "url": "https://www.teclamairme.com/#ch-T9", "ogham": "ᚇ", "letter": "D", "name": "Chapter ᚇ(D): The locomotive", "subtitle": "Computability, complexity, and numerical analysis: lambda calculus, the halting problem, Busy Beaver, P and NP, and a rounding error that vanishes", "key_concepts": "Computability, complexity, and numerical analysis. It includes symbols you may not have seen before, such as: λ, fl_{64}, and P ⊆ NP. Zero here is a limit on machines: no program can decide halting, and a computer rounding 1 + 2^{−53} lands back on exactly 1.", "epigraph": "“The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.”", "epigraph_author": "Alan Turing", "epigraph_source": "opening sentence of On Computable Numbers, with an Application to the Entscheidungsproblem, 1936" }, { "section": 10, "id": "ch-T10", "url": "https://www.teclamairme.com/#ch-T10", "ogham": "ᚈ", "letter": "T", "name": "Chapter ᚈ(T): The caboose", "subtitle": "Linear algebra: determinant, trace, norm, eigenvalues, and the Kronecker delta", "key_concepts": "Linear algebra. It includes symbols you may not have seen before, such as: det, tr, ‖A‖_{F}, λ, and δ. Zero here is a lost dimension: a matrix that flattens the plane onto a line leaves no area behind.", "epigraph": "“I have in previous papers defined a ‘Matrix’ as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent.”", "epigraph_author": "James Joseph Sylvester", "epigraph_source": "On the relation between the minor determinants of linearly equivalent quadratic functions, 1851" }, { "section": 11, "id": "ch-T11", "url": "https://www.teclamairme.com/#ch-T11", "ogham": "ᚉ", "letter": "C", "name": "Chapter ᚉ(C): The gallery of motion", "subtitle": "Calculus, differential equations, optimization, and vector calculus: the fundamental theorem, a cooling rod, the bottom of a bowl, divergence and curl", "key_concepts": "Calculus, differential equations, optimization, and vector calculus. It includes symbols you may not have seen before, such as: ∫, ∇, and ∮. Zero here is balance: change undoing accumulation, a cooling rod obeying its equation exactly, the floor of a bowl, and a whirl with nothing flowing out.", "epigraph": "“From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics.”", "epigraph_author": "Richard Feynman", "epigraph_source": "The Feynman Lectures on Physics, Volume II, chapter 1, 1964" }, { "section": 12, "id": "ch-T12", "url": "https://www.teclamairme.com/#ch-T12", "ogham": "ᚊ", "letter": "Q", "name": "Chapter ᚊ(Q): The gallery of shape", "subtitle": "Topology, knot theory, differential geometry, and Euclidean geometry: Euler characteristic, Betti numbers, the torus’s genus, the unknot, curvature over the sphere, and the 3–4–5 triangle", "key_concepts": "Topology, knot theory, differential geometry, and Euclidean geometry. It includes symbols you may not have seen before, such as: χ, b_{n}, ∂, and K. Zero here is a shape’s accounts balancing: a doughnut’s Euler characteristic, an unknot’s crossings, a rolled sheet’s curvature, and 3² + 4² − 5².", "epigraph": "“Mathematics is the art of giving the same name to different things.”", "epigraph_author": "Henri Poincaré", "epigraph_source": "The Future of Mathematics, 1908, in Science and Method" }, { "section": 13, "id": "ch-T13", "url": "https://www.teclamairme.com/#ch-T13", "ogham": "ᚋ", "letter": "M", "name": "Chapter ᚋ(M): The gallery of chance", "subtitle": "Probability, information theory, statistics, and stochastic processes: a point with no width, Shannon’s bit, an average that tells the truth, Fisher’s one in twenty, and a weather chain that sits still", "key_concepts": "Probability, information theory, statistics, and stochastic processes. It includes symbols you may not have seen before, such as: P, Var, H, and D_{KL}. Zero here is certainty and indifference: the chance of drawing Rose’s exact number, the surprise of a sure thing, and a chain whose odds have stopped changing, though its weather has not.", "epigraph": "“The theory of probabilities is at bottom nothing but common sense reduced to calculus.”", "epigraph_author": "Pierre-Simon Laplace", "epigraph_source": "A Philosophical Essay on Probabilities, 1814" }, { "section": 14, "id": "ch-T14", "url": "https://www.teclamairme.com/#ch-T14", "ogham": "ᚌ", "letter": "G", "name": "Chapter ᚌ(G): The hall of mirrors", "subtitle": "Lie theory, geometric algebra (Clifford algebras), group theory, and Galois theory: brackets, bivectors, the Monster, and the unsolvable quintic", "key_concepts": "Lie theory, geometric algebra, group theory, and Galois theory: four views of symmetry. It includes symbols you may not have seen before, such as: [x, y], ∧, ζ_{5}, and 𝕄. Zero here is symmetry canceling itself: roots of unity summing to nothing, a vector wedged with itself, and Abel’s proof that the general quintic has no formula in radicals.", "epigraph": "“Symmetry, as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection.”", "epigraph_author": "Hermann Weyl", "epigraph_source": "Symmetry, Princeton University Press, 1952, p. 5" }, { "section": 15, "id": "ch-T15", "url": "https://www.teclamairme.com/#ch-T15", "ogham": "ᚍ", "letter": "K", "name": "Chapter ᚍ(K): The listening room", "subtitle": "Quantum mechanics and Fourier analysis: Hilbert space, Dirac’s brackets, Grassmann’s sign, and the waves that cannot see each other", "key_concepts": "Quantum mechanics and Fourier analysis. It includes symbols you may not have seen before, such as: ⟨ | ⟩, [x̂, p̂], and θ. Zero here is perpendicularity: quantum states that cannot be confused, and a sine and cosine that cannot see each other.", "epigraph": "“I think I can safely say that nobody understands quantum mechanics.”", "epigraph_author": "Richard Feynman", "epigraph_source": "The Character of Physical Law, 1965, chapter 6" }, { "section": 16, "id": "ch-T16", "url": "https://www.teclamairme.com/#ch-T16", "ogham": "ᚎ", "letter": "Z", "name": "Chapter ᚎ(Z): The roundabout", "subtitle": "Complex analysis: contour integrals, Cauchy’s theorem, the Cauchy–Riemann equations, and residues", "key_concepts": "Complex analysis: contour integrals, Cauchy’s theorem, the Cauchy–Riemann equations, and residues. It includes symbols you may not have seen before, such as: ∮, ∂/∂z̄, and Res. Zero here is a round trip: carry e^{z} once around a circle and you come home with nothing.", "epigraph": "“It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one.”", "epigraph_author": "Jacques Hadamard, after Paul Painlevé", "epigraph_source": "An Essay on the Psychology of Invention in the Mathematical Field, 1945" }, { "section": 17, "id": "ch-T17", "url": "https://www.teclamairme.com/#ch-T17", "ogham": "ᚐ", "letter": "A", "name": "Chapter ᚐ(A): The mechanic’s workshop", "subtitle": "Special functions: gamma and its family, the logarithmic integral, sinc, Bessel, Lambert W, the error function, Airy, the elliptic integral, and Fresnel", "key_concepts": "Special functions: gamma and its family, the logarithmic integral, sinc, Bessel, Lambert W, and more. It includes symbols you may not have seen before, such as: Γ, li, W, and J_{1}. Zero here is where a function crosses the axis: a constant defined as a zero, a pole turned upside down, sinc at π, and a Bessel function that starts from nothing.", "epigraph": "“Paul Turán once remarked that special functions would be more appropriately labeled ‘useful functions’.”", "epigraph_author": "George Andrews, Richard Askey and Ranjan Roy", "epigraph_source": "preface to Special Functions, Cambridge University Press, 1999" }, { "section": 18, "id": "ch-T18", "url": "https://www.teclamairme.com/#ch-T18", "ogham": "ᚑ", "letter": "O", "name": "Chapter ᚑ(O): The prime mileposts", "subtitle": "Analytic number theory: the Riemann zeta function, its values, its zeros, and the Euler product", "key_concepts": "Analytic number theory: the Riemann zeta function and the Euler product. It includes symbols you may not have seen before, such as: ζ(s), ∏, and t_{1}. Zero here comes from zeta four ways: a tame zero at −2, a wild one on the critical line, the pole at 1 turned upside down, and Euler’s product canceling zeta exactly.", "epigraph": "“It is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.”", "epigraph_author": "Bernhard Riemann", "epigraph_source": "On the Number of Prime Numbers less than a Given Quantity, 1859, in David Wilkins’s translation (the roots are those of Riemann’s transformed function; this is the critical-line claim)" }, { "section": 19, "id": "ch-T19", "url": "https://www.teclamairme.com/#ch-T19", "ogham": "ᚒ", "letter": "U", "name": "Chapter ᚒ(U): The crossroads", "subtitle": "Elliptic curves: the group law, the point at infinity, a rank of zero, and the invariants that name a curve", "key_concepts": "Elliptic curves: the group law, the point at infinity, rank, and the invariants that name a curve. It includes symbols you may not have seen before, such as: 𝒪, Δ, j, and N. Zero here sits at infinity: the curve’s identity is a point on no ordinary map, and y² = x³ − x has rank 0, as Fermat in effect proved.", "epigraph": "“It is possible to write endlessly on elliptic curves. (This is not a threat.)”", "epigraph_author": "Serge Lang", "epigraph_source": "Elliptic Curves: Diophantine Analysis, 1978, opening line of the preface" }, { "section": 20, "id": "ch-T20", "url": "https://www.teclamairme.com/#ch-T20", "ogham": "ᚓ", "letter": "E", "name": "Chapter ᚓ(E): The cutting", "subtitle": "Measure theory and dynamics: the Cantor set, Lebesgue measure, Hausdorff dimension, a paradox, and the logistic map at rest", "key_concepts": "Measure theory and dynamics. It includes symbols you may not have seen before, such as: λ, dim_{H}, 𝒞, and δ. Zero here is size without weight: an uncountable set of total length zero, and a map that settles and stops moving.", "epigraph": "“Je le vois, mais je ne le crois pas. (I see it, but I do not believe it.)”", "epigraph_author": "Georg Cantor, to Richard Dedekind", "epigraph_source": "letter of 29 June 1877, on finding that a line and a square have the same number of points" }, { "section": 21, "id": "ch-T21", "url": "https://www.teclamairme.com/#ch-T21", "ogham": "ᚔ", "letter": "I", "name": "Chapter ᚔ(I): The first landing", "subtitle": "Complex numbers ℂ, with the dual and split-complex numbers: two-component arithmetic, Euler’s identity, and alternative rules for a new unit", "key_concepts": "Complex numbers, with the dual and split-complex numbers. It includes symbols you may not have seen before, such as: i, ε, and j. Zero here comes from Euler’s e^{iπ} + 1, and from an ε whose square vanishes though ε does not. The first landing of a ladder where each doubling gains a trick and costs a rule.", "epigraph": "“Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.”", "epigraph_author": "Benjamin Peirce, to his Harvard class, after deriving e^{iπ} = −1", "epigraph_source": "from notes by his student W. E. Byerly, published in R. C. Archibald’s Benjamin Peirce, 1809–1880 (1925); also in Kasner and Newman, Mathematics and the Imagination, 1940" }, { "section": 22, "id": "ch-T22", "url": "https://www.teclamairme.com/#ch-T22", "ogham": "ᚕ", "letter": "J", "name": "Chapter ᚕ(J): The second landing", "subtitle": "Quaternions ℍ: four real components, a quarter-turn that carries i onto j, and the loss of commutativity", "key_concepts": "Quaternions: four real components and a multiplication where order matters. It includes symbols you may not have seen before, such as: i, j, k, and Q_{8}. Zero here comes from Hamilton’s ijk = −1, and from a rotation that carries i exactly onto j. The rule lost at this landing: ij and ji differ.", "epigraph": "“An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work.”", "epigraph_author": "William Rowan Hamilton, on the moment at Brougham Bridge, 16 October 1843", "epigraph_source": "letter to his son Archibald, 5 August 1865" }, { "section": 23, "id": "ch-T23", "url": "https://www.teclamairme.com/#ch-T23", "ogham": "ᚖ", "letter": "W", "name": "Chapter ᚖ(W): The third landing", "subtitle": "Octonions 𝕆: eight real components, the loss of associativity, the alternative law, a Moufang identity, and the Fano plane as a multiplication table", "key_concepts": "Octonions: eight real components, the alternative law, and a Moufang identity. It includes symbols you may not have seen before, such as: e_{1} … e_{7} and G_{2}. Zero here is an associator that vanishes: moving brackets usually changes an octonion product, except where a surviving law says it cannot.", "epigraph": "“The octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.”", "epigraph_author": "John C. Baez", "epigraph_source": "The Octonions, Bulletin of the American Mathematical Society, 2002" }, { "section": 24, "id": "ch-T24", "url": "https://www.teclamairme.com/#ch-T24", "ogham": "ᚗ", "letter": "V", "name": "Chapter ᚗ(V): The top landing", "subtitle": "Sedenions 𝕊: sixteen real components, zero divisors, the loss of a multiplicative norm, and powers that remain unambiguous", "key_concepts": "Sedenions: sixteen real components, zero divisors, and powers that stay unambiguous. Zero here is a new kind of nothing: two nonzero numbers whose product is zero.", "epigraph": "“First we lose the fact that every element is its own conjugate, then we lose commutativity, then we lose associativity, and finally we lose the division algebra property.”", "epigraph_author": "John C. Baez, on running the Cayley–Dickson construction from ℝ to ℂ to ℍ to 𝕆 to the sedenions", "epigraph_source": "The Octonions, Bulletin of the American Mathematical Society, 2002" }, { "section": 25, "id": "ch-T25", "url": "https://www.teclamairme.com/#ch-T25", "ogham": "ᚘ", "letter": "Y", "name": "Chapter ᚘ(Y): Pre-flight controls", "subtitle": "Non-Archimedean completions: a 3-adic carry and the product formula, over a composite-base 10-adic pair, the 3-adic size of 27 and a 7-adic square root of two", "key_concepts": "p-adic numbers and non-Archimedean absolute values. It includes symbols you may not have seen before, such as: |·|_{3}, v_{3}, and ℚ_{p}. Zero here is a carry that never ends: …2222 in base 3, plus 1, rolls over into 0, in a world where closeness means divisibility.", "epigraph": "“On the fundamental level our world is neither real, nor p-adic; it is adèlic.”", "epigraph_author": "Yuri Manin", "epigraph_source": "Reflections on arithmetical physics, in Conformal Invariance and String Theory, Academic Press, 1989" }, { "section": 26, "id": "ch-T26", "url": "https://www.teclamairme.com/#ch-T26", "ogham": "ᚙ", "letter": "X", "name": "Chapter ᚙ(X): The bridge", "subtitle": "Surreal numbers and the hyperreals of nonstandard analysis: two fields where you can divide by an infinity", "key_concepts": "Surreal numbers and the hyperreals of nonstandard analysis: two fields where you can honestly divide by an infinity. It includes symbols you may not have seen before, such as: { | }, ω, and st. Zero here is born first: the surreal 0 is made from nothing on day zero, and one over an infinite hyperreal rounds to 0.", "epigraph": "“In the beginning, everything was void, and J. H. W. H. Conway began to create numbers.”", "epigraph_author": "Donald Knuth", "epigraph_source": "Surreal Numbers, 1974, the novelette that named them" }, { "section": 27, "id": "ch-T27", "url": "https://www.teclamairme.com/#ch-T27", "ogham": "ᚚ", "letter": "P", "name": "Chapter ᚚ(P): The possible journeys", "subtitle": "Cardinals, ordinals, the power set, and categories: Cantor’s sizes of infinity, von Neumann’s zero, the one subset of nothing, and the one arrow out of it", "key_concepts": "Cardinals, ordinals, power sets, and categories. It includes symbols you may not have seen before, such as: ℵ, ℶ, 𝔠, and ∅. Zero here is the empty set itself: nothing times infinity is still nothing, and von Neumann builds 0 out of ∅. The braces are added, not divided, because cardinals do not divide.", "epigraph": "“Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können. No one shall expel us from the paradise that Cantor has created for us.”", "epigraph_author": "David Hilbert", "epigraph_source": "Über das Unendliche, Mathematische Annalen 95, 1926" }, { "section": 28, "id": "ch-T28", "url": "https://www.teclamairme.com/#ch-T28", "ogham": "᚜", "letter": "Epilogue", "name": "Chapter ᚜(Epilogue): The homeward view", "subtitle": "Seven prizes as seven numbers: analytic number theory, elliptic curves, topology, complexity, algebraic geometry, fluid dynamics, and quantum field theory", "key_concepts": "The seven Millennium Prize Problems, each written as a finite number. It includes symbols you may not have seen before, such as: Λ_{dBN}, L(E_{37}, 1), and 1/T_{∗}. Zero here does not wait for answers: one proven zero settles the whole product while the open answers stay unknown.", "epigraph": "“Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries?”", "epigraph_author": "David Hilbert", "epigraph_source": "Mathematical Problems, address to the International Congress of Mathematicians, Paris, 8 August 1900" } ], "glossary": [ { "id": "og-ruis", "url": "https://www.teclamairme.com/#g-og-ruis", "name": "the letter Ruis", "symbol": "ᚏ", "chapter": 1, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘red’ or ‘redness’; its word-oghams are “most intense blushing,” “reddening of faces” and “glow of anger,” and the medieval glossators attached it to the elder.", "more": "Ruis is one of the letter names that is not a tree name. On the evidence of the kennings it means ‘red’ or ‘redness’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tindem rucci, “most intense blushing”; rúamnae drech, “reddening of faces”; and bruth fergae, “glow of anger.” The arboreal tradition reads that redness as a tree: the Auraicept na nÉces glosses ruis with trom, elder, and prints ruamma ruice ruis, “the redness of shame is ruis.” That gloss is medieval, and it is where the book’s elder comes from. What is said today about transition is later still: Graves’s calendar of 1948 makes Ruis its thirteenth and last month, 25 November to 22 December, and elder folklore makes the tree one to ask permission of, a shelter where the fairy court passes on Midsummer’s Eve, and the tree of the Judas legend. Scottish Gaelic ruis is a name for the elder that occurs only rarely in Scottish place names. The book’s own use of the letter is a separate matter: see Ruis numbers.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19", "r-2.20" ], "links_to_cards": [ "ruis" ] }, { "id": "og-feather", "url": "https://www.teclamairme.com/#g-og-feather", "name": "the feather mark", "symbol": "᚛", "chapter": 2, "type": "og", "badge": "Ogham mark", "quick": "Not a letter but punctuation: the ogham feather mark stands at the beginning of a text and can show which way it is to be read.", "more": "U+169B OGHAM FEATHER MARK is classed as open punctuation rather than as one of the twenty-six ogham letters, and the Unicode Standard describes the mark and its partner as used at the beginning and the end of ogham text, particularly in manuscript ogham; where only the feather mark appears it can indicate the direction of the text. Because it is not a letter it has no name-meaning, no word-ogham, and no plant: the three layers the other cards in this section keep apart do not exist for it. Here the pair frames the equation as it frames an inscription.", "cites": [ "r-2.13", "r-2.15", "r-2.17" ], "links_to_cards": [] }, { "id": "og-beith", "url": "https://www.teclamairme.com/#g-og-beith", "name": "the letter Beith", "symbol": "ᚁ", "chapter": 3, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘birch’, one of the few names the medieval glossators did not have to translate.", "more": "Beithe means ‘birch-tree’, cognate with Middle Welsh bedw; Latin betula is thought to be a borrowing from the Gaulish cognate. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give féochos foltchain, “withered foot with fine hair”; glaisem cnis, “greyest of skin”; and maise malach, “beauty of the eyebrow.” It is one of only five names the Auraicept itself finds transparent as tree names without glossing, along with fearn, saille, duir, and coll, so here the book’s plant column and the oldest evidence agree. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-luis", "url": "https://www.teclamairme.com/#g-og-luis", "name": "the letter Luis", "symbol": "ᚂ", "chapter": 4, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning either ‘blaze’ or ‘herb’; the medieval glossators read it as the rowan.", "more": "Luis is not securely a tree name. It is related either to luise, ‘blaze’, or to lus, ‘herb’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give lí súla, “lustre of the eye”; carae cethrae, “friend of cattle”; and lúth cethrae, “sustenance of cattle.” The first of those would support ‘blaze’. The arboreal tradition glosses caertheand, rowan, and explains the kenning by the berries: “delightful for eye is luis, that is, rowan, owing to the beauty of its berries.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-fern", "url": "https://www.teclamairme.com/#g-og-fern", "name": "the letter Fearn", "symbol": "ᚃ", "chapter": 5, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘alder’; the Auraicept makes it the shield-wood of the warrior bands.", "more": "Fern means ‘alder-tree’, from Primitive Irish *wernā, which is why the letter’s original sound value was [w] rather than [f]. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give airenach fían, “vanguard of warriors”; comét lachta, “milk container”; and dín cridi, “protection of the heart.” The Auraicept explains the first of the three: “the van of the warrior-bands, that is, alder, for thereof are the shields.” It is one of the five names the medieval tradition did not need to translate. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-sail", "url": "https://www.teclamairme.com/#g-og-sail", "name": "the letter Sail", "symbol": "ᚄ", "chapter": 6, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘willow’; its kennings run from the pallor of the dead to the beginning of honey.", "more": "Sail means ‘willow-tree’, cognate with Latin salix, and is another of the five transparent tree names. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give lí ambi, “pallor of a lifeless one”; lúth bech, “sustenance of bees”; and tosach mela, “beginning of honey.” The Auraicept reads the first as a remark about color: the willow “has no colour,” owing to the resemblance of its hue to a dead person. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-nin", "url": "https://www.teclamairme.com/#g-og-nin", "name": "the letter Nion", "symbol": "ᚅ", "chapter": 7, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘a fork’ or ‘a loft’; the medieval glossators read it as the ash.", "more": "Nin is not a tree name: it means either ‘fork’ or ‘loft’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give costud síde, “establishing of peace”; bág ban, “boast of women”; and bág maise, “boast of beauty.” The arboreal tradition glosses uinnius, ash, and turns the first kenning inside out: “a check on peace is nin, that is, ash, for of it are made spear-shafts by which the peace is broken.” The name matters twice over, because nin also meant ‘forking branches’ and was one of the two words used for the ogham letters themselves, which is part of how the tree-alphabet idea arose in the first place. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-uath", "url": "https://www.teclamairme.com/#g-og-uath", "name": "the letter Uath", "symbol": "ᚆ", "chapter": 8, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘horror’ or ‘fear’; the medieval glossators read it as the whitethorn.", "more": "Úath means ‘horror, fear’. It is unattested in the orthodox Irish inscriptions, and both its etymology and its original sound value are unclear; McManus has suggested [y]. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give condál cúan, “assembly of packs of hounds”; bánad gnúise, “blanching of faces”; and ansam aidche, “most difficult at night.” The Auraicept glosses sce, whitethorn, and gives both readings at once: “a meet of hounds is huath, that is, white-thorn; or because it is formidable for its thorns”—the Irish for formidable being uathmar, from the letter name itself. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-dair", "url": "https://www.teclamairme.com/#g-og-dair", "name": "the letter Dair", "symbol": "ᚇ", "chapter": 9, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘oak’, the highest of the trees in its first kenning.", "more": "Dair means ‘oak’, from the Indo-European root *doru-, and is one of the five names the medieval tradition read without glossing. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give ardam dosae, “highest tree”; grés soír, “handicraft of a craftsman”; and slechtam soíre, “most carved of craftsmanship.” The Auraicept keeps the first: “higher than bushes is an oak.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-tinne", "url": "https://www.teclamairme.com/#g-og-tinne", "name": "the letter Tinne", "symbol": "ᚈ", "chapter": 10, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘a bar of metal’ or ‘an ingot’; the medieval glossators read it as the holly.", "more": "Tinne is not a tree name. On the evidence of the kennings it means ‘bar of metal, ingot’, and it is probably cognate with Old Irish tend, ‘strong’, or tind, ‘brilliant’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give trian roith, “one of three parts of a wheel”; smiur gúaile, “marrow of charcoal”; and trian n-airm, “one of three parts of a weapon.” The arboreal tradition glosses cuileand, holly, and reaches it through the wheel: “a third of a wheel is tinne, that is, because holly is one of the three timbers of the chariot-wheel.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-coll", "url": "https://www.teclamairme.com/#g-og-coll", "name": "the letter Coll", "symbol": "ᚉ", "chapter": 11, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘hazel’, the fairest and the sweetest of the trees in its kennings.", "more": "Coll means ‘hazel-tree’, cognate with Welsh collen and with Latin corylus, and is the fifth of the transparent tree names. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give caíniu fedaib, “fairest tree”; carae blóesc, “friend of nutshells”; and milsem fedo, “sweetest tree.” The Auraicept’s epithet for it, cainfidh, fair-wood, is apt, and it adds the plainest of its remarks: “coll, that is, every one is eating from its nuts.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-cert", "url": "https://www.teclamairme.com/#g-og-cert", "name": "the letter Ceirt", "symbol": "ᚊ", "chapter": 12, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘a bush’, confused early with ceirt, ‘rag’; the medieval glossators read it as the apple.", "more": "Cert is cognate with Welsh perth, ‘bush’, and with Latin quercus, ‘oak’, from Indo-European *perkwos. It was confused with a different Old Irish word, ceirt, ‘rag’, and the kennings carry the confusion. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give clithar baiscill, “shelter of a lunatic” (the reading is uncertain); bríg anduini, “substance of an insignificant person”; and dígu fethail, “dregs of clothing.” The Auraicept glosses aball, apple, and reads the first kenning as shelter for a wild hind. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-muin", "url": "https://www.teclamairme.com/#g-og-muin", "name": "the letter Muin", "symbol": "ᚋ", "chapter": 13, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name pulling three ways at once: ‘neck’, ‘ruse’, and ‘love’; the medieval glossators read it as the vine.", "more": "Muin is not a tree name, and the three kennings do not agree on which word it is: Old Irish has muin, ‘neck, upper part of the back’, muin, ‘wile, ruse’, and muin, ‘love, esteem’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tressam fedmae, “strongest in exertion”; árusc n-airlig, “proverb of slaughter”; and conar gotha, “path of the voice”—one for each sense. The arboreal tradition glosses finemhain, vine, with a kenning that reads the third sense: “highest of beauty is muin, that is, because it grows aloft.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-gort", "url": "https://www.teclamairme.com/#g-og-gort", "name": "the letter Gort", "symbol": "ᚌ", "chapter": 14, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘a field’; the medieval glossators forced it into the ivy.", "more": "Gort means ‘field’, and is cognate with the English word garden. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give milsiu féraib, “sweetest grass”; ined erc, “suitable place for cows”; and sásad ile, “sating of multitudes”—all three about pasture, not about a tree. The Auraicept nonetheless glosses edind, ivy: “greener than pastures is gort.” McManus names gort, with ruis and úath, as one of the letter names the medieval glossators reinterpreted as a tree epithet more or less by force. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-getal", "url": "https://www.teclamairme.com/#g-og-getal", "name": "the letter nGéadal", "symbol": "ᚍ", "chapter": 15, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘killing’; the medieval glossators read it as broom or fern, and modern lists as the reed.", "more": "Gétal is not a tree name. The kennings give it the sense ‘killing’, and it is perhaps cognate with gonid, ‘slays’; its Primitive Irish value was accordingly a voiced labiovelar. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give lúth lego, “sustenance of a leech”; étiud midach, “raiment of physicians”; and tosach n-échto, “beginning of slaying.” The Auraicept glosses cilcach, read as broom or fern: “a leech’s sustenance is getal, that is, broom.” In modern Irish that gloss word is giolcach, which also means reed, the plant the letters table carries. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-straif", "url": "https://www.teclamairme.com/#g-og-straif", "name": "the letter Straif", "symbol": "ᚎ", "chapter": 16, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘sulphur’; the medieval glossators read it as the blackthorn.", "more": "Straif, Old Irish Straiph, means ‘sulphur’. Its Primitive Irish sound value is uncertain: it may have been a sibilant distinct from the s that sail already held, perhaps a reflex of /st/ or /sw/. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give tressam rúamnai, “strongest reddening, that is, dye”; mórad rún, “increase of secrets”; and saigid nél, “seeking of clouds.” The Auraicept glosses draighin, blackthorn: “the hedge of a stream is sraibh.” Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-ailm", "url": "https://www.teclamairme.com/#g-og-ailm", "name": "the letter Ailm", "symbol": "ᚐ", "chapter": 17, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name of uncertain meaning, possibly ‘pine’; its kennings are all about sound.", "more": "Ailm is of uncertain meaning. McManus concedes it may mean ‘pine-tree’, since an eighth-century poem appears to use it so—the hermit Marbán saying caine ailmi ardom-peitet, “beautiful are the pines that make music for me,” of the sound of wind in the needles. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give ardam íachta, “loudest groan”; tosach frecrai, “beginning of an answer”; and tosach garmae, “beginning of calling.” The Auraicept glosses crand giuis or ochtach, fir or pine, so the plant column and the philology point the same way here, unusually for a name this uncertain. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-onn", "url": "https://www.teclamairme.com/#g-og-onn", "name": "the letter Onn", "symbol": "ᚑ", "chapter": 18, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘ash tree’; the medieval gloss, furze, is the one McManus calls wrong.", "more": "Onn means ‘ash-tree’, and McManus counts it a sixth secure tree name beyond the Auraicept’s own five. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give congnaid ech, “wounder of horses”; féthem soíre, “smoothest of craftsmanship”; and lúth fían, “[equipment] of warrior bands.” Here the medieval tradition goes astray: the Auraicept glosses aiten, furze or gorse, which is the plant the letters table carries, and which McManus records flatly as wrong. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-ur", "url": "https://www.teclamairme.com/#g-og-ur", "name": "the letter Úr", "symbol": "ᚒ", "chapter": 19, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name meaning ‘earth, clay, soil’; the medieval glossators read it as the heath.", "more": "Úr is not a tree name. On the evidence of the kennings it means ‘earth, clay, soil’. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give úaraib adbaib, “in cold dwellings”; sílad cland, “propagation of plants”; and forbbaid ambí, “shroud of a lifeless one”—a grave, a seedbed, and a shroud, all of them soil. The Auraicept glosses fraech, heath, which is where the heather of the letters table comes from. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-edad", "url": "https://www.teclamairme.com/#g-og-edad", "name": "the letter Eadhadh", "symbol": "ᚓ", "chapter": 20, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name of unknown meaning, paired with Iodhadh; the medieval glossators read it as the aspen.", "more": "Edad is of unknown meaning, and it is one of a pair: edad and idad look made to match each other. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give érgnaid fid, “discerning tree”; commaín carat, “exchange of friends”; and bráthair bethi, “brother of birch,” the last uncertain. The Auraicept glosses crand fir no crithach, test-tree or aspen, and reads the name itself as ed uath, “horrible grief”—which is glossing by sound, not by evidence. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-idad", "url": "https://www.teclamairme.com/#g-og-idad", "name": "the letter Iodhadh", "symbol": "ᚔ", "chapter": 21, "type": "og", "badge": "Ogham letter", "quick": "Ogham letter name probably made from ibhar, ‘yew’, to pair with Eadhadh; its kennings are all about age.", "more": "Idad is of uncertain meaning. McManus believes it is probably an artificial form of ibhar, ‘yew’, coined to make a pair with edad, and notes that the kennings support that sense. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give sinem fedo, “oldest tree”; caínem sen, “fairest of the ancients”; and lúth lobair, “energy of an infirm person,” the last uncertain. The Auraicept glosses ibhar, yew, which is the meaning the arboreal tradition gives it and the one the kennings will bear. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.19" ], "links_to_cards": [] }, { "id": "og-ebad", "url": "https://www.teclamairme.com/#g-og-ebad", "name": "the letter Ébad", "symbol": "ᚕ", "chapter": 22, "type": "og", "badge": "Ogham letter", "quick": "A supplementary letter of unknown meaning; its kennings point at the salmon, and the medieval glossators read it as the aspen.", "more": "Ébad is the first of the five forfeda, the supplementary letters invented in the Old Irish period, centuries after the peak of ogham use, and the sounds they stood for are uncertain. The meaning of this one is unknown, and the name looks modeled on edad and idad. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give snámchaín feda, “fair-swimming letter”; cosc lobair, “admonishing of an infirm person,” uncertain; and caínem éco, “fairest fish”—two of the three pointing to éo, the word for salmon. It is the one forfid that is common in the orthodox inscriptions, where it carries a consonant value inside the formula word KOI, ‘here’, the ogham equivalent of hic iacet. The Auraicept glosses crithach, aspen; the elecampane of the letters table is the other meaning alleged for it, though the Irish for elecampane is elenn.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.18" ], "links_to_cards": [] }, { "id": "og-or", "url": "https://www.teclamairme.com/#g-og-or", "name": "the letter Ór", "symbol": "ᚖ", "chapter": 23, "type": "og", "badge": "Ogham letter", "quick": "A supplementary letter whose name means ‘gold’, from Latin aurum.", "more": "Ór means ‘gold’, borrowed from Latin aurum, and is the second of the five forfeda. Only two of the three word-ogham lists have an entry for it: sruithem aicde, “most venerable substance” and lí crotha, “splendour of form”; the Cú Chulainn list is silent. The Auraicept glosses feorus no edind, spindle tree or ivy. It appears twice in the inscribed corpus, at Killogrone in County Kerry and on a cross slab at Formaston in Aberdeenshire—which is to say, once in Ireland and once in Pictland. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.18", "r-2.19" ], "links_to_cards": [] }, { "id": "og-uilleann", "url": "https://www.teclamairme.com/#g-og-uilleann", "name": "the letter Uilleann", "symbol": "ᚗ", "chapter": 24, "type": "og", "badge": "Ogham letter", "quick": "A supplementary letter whose name means ‘elbow’, and whose kenning is a play on the name.", "more": "Uilleand means ‘elbow’, and is the third of the five forfeda. Two lists have it: túthmar fid, “fragrant tree” and cubat oll, “great elbow, or cubit,” the second of which is a play on the letter name rather than a clue to anything beyond it. The Auraicept glosses edleand, honeysuckle. It appears in only one orthodox inscription, at Teeromoyle in County Kerry, though some later scholastic inscriptions use all five forfeda. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.18", "r-2.19" ], "links_to_cards": [] }, { "id": "og-ifin", "url": "https://www.teclamairme.com/#g-og-ifin", "name": "the letter Ifín", "symbol": "ᚘ", "chapter": 25, "type": "og", "badge": "Ogham letter", "quick": "A supplementary letter, first called Pín: the sign added for p, its name from Latin pinus, ‘pine’, or spina, ‘thorn’.", "more": "This letter was first called Pín and was added because ogham, devised for Primitive Irish, had no sign for [p] at all. Sources differ on the name. The Forfeda page, reporting McManus, allows the influence of Latin pinus, pine, but calls Latin spina, thorn, the likelier source, since the kennings describe a shrub with sweet fruit rather than a conifer; Hayden and Stifter (2022) take it as probably pinus, after the model of the other tree names. The kennings are milsem fedo, “sweetest tree” and amram mlais, “most wonderful taste.” The Auraicept glosses spinan no ispin, gooseberry or thorn, and Kelly finds spín in the Old Irish tree lists with just that range, so the medieval gloss may be right on this occasion. When the later oghamists reassigned all five forfeda to diphthongs, this letter lost its consonant value and its first name, and became Ifín (also spelled Iphín). Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.18", "r-2.19" ], "links_to_cards": [] }, { "id": "og-emancholl", "url": "https://www.teclamairme.com/#g-og-emancholl", "name": "the letter Eamhancholl", "symbol": "ᚙ", "chapter": 26, "type": "og", "badge": "Ogham letter", "quick": "A supplementary letter named for its shape: ‘twin of Coll’, two Colls side by side.", "more": "Emancholl means ‘twin of coll’. The name is taken from the shape of the sign, which is two of the letter Coll set together, and perhaps also from its sound, the fricative of the sound Coll carries. Two lists give it a kenning, and both say the same thing: lúad sáethaig, “groan of a sick person” and mol galraig, “groan of a sick person,” which fits the consonant value it held before the decision that all five forfeda should be vowels. It is the one forfid the Auraicept does not gloss with a plant, and it does not appear in the orthodox inscriptions at all, though some later scholastic inscriptions use all five forfeda; the witch hazel of the letters table belongs to a modern list.", "cites": [ "r-2.15", "r-2.16", "r-2.17", "r-2.18", "r-2.19" ], "links_to_cards": [ "og-coll" ] }, { "id": "og-peith", "url": "https://www.teclamairme.com/#g-og-peith", "name": "the letter Peith", "symbol": "ᚚ", "chapter": 27, "type": "og", "badge": "Ogham letter", "quick": "A late addition: beithe bog, ‘soft beithe’, invented in the manuscripts to carry p once more.", "more": "Peith is not one of the five forfeda but a later addition, and it is a letter of the manuscripts only. When the later oghamists insisted on treating all five supplementary letters as vowels, the alphabet was left with no sign for [p] for the second time, so [p] had to be written again as a modified [b]: peithe, after beithe, also called beithe bog, ‘soft beithe’, or tautologically peithbog. Being an invention of the scribes rather than a name of anything, the kenning and tree-gloss lists give it no word-ogham and no medieval plant; the dwarf elder of the letters table is a dictionary sense of the word, not a medieval gloss.", "cites": [ "r-2.15", "r-2.17", "r-2.18", "r-2.19" ], "links_to_cards": [] }, { "id": "og-feather-rev", "url": "https://www.teclamairme.com/#g-og-feather-rev", "name": "the reversed feather mark", "symbol": "᚜", "chapter": 28, "type": "og", "badge": "Ogham mark", "quick": "Not a letter but punctuation: the reversed ogham feather mark closes the text.", "more": "U+169C OGHAM REVERSED FEATHER MARK is classed as close punctuation and is the partner of the feather mark; the Unicode Standard gives the two as marking the beginning and the end of ogham text, particularly in manuscript ogham. Like its partner it is not one of the twenty-six letters, so it has no name-meaning, no word-ogham, and no plant. And like its partner it spends no letter, which is how the piece comes to have twenty-five lettered chapters standing between the two marks, with Ruis outside them on the left.", "cites": [ "r-2.13", "r-2.15", "r-2.17" ], "links_to_cards": [ "og-feather" ] }, { "id": "rose", "url": "https://www.teclamairme.com/#g-rose", "name": "Rose’s number", "symbol": "ᚏ4.16", "symbol_markup": "ᚏ_{4.16}", "chapter": 1, "type": "undef", "badge": "Undefinable", "quick": "ᚏ4.16 := x/4.16: one chosen Ruis number, labeled with Rose’s April 16 birthday and fixed throughout this book.", "quick_markup": "ᚏ_{4.16} := x/4.16: one chosen Ruis number, labeled with Rose’s April 16 birthday and fixed throughout this book.", "more": "Rose’s number is the name attached to the label 4.16 in the Ruis notation. Its value here is x/4.16 for this book’s chosen x. A formula outside this book may choose another x and use the same birthday label for a different value. Here, the original choice stays fixed wherever Rose’s number returns, including Chapter ᚋ(M). The date inspired the label; it does not tell us the number’s value. The label cannot smuggle in a definition: if x/4.16 had one in the book’s original language, then “that number times 4.16” would define x, and x has none. Many of the first things we can say about her number are things it is not, and the list is longer than it first looks. Zero has a definition, so it was never in the pool; but so does every fraction, every root of a polynomial with integer coefficients, every number a program could print digit by digit, and even Chaitin’s Ω, which no algorithm computes but one sentence specifies. So her number is irrational, transcendental, and uncomputable, without a single proof about it in particular: the fence around D does all the work, and hers is the only symbol in the equation with the Undefinable badge. The one ordinary question the book leaves open is its sign: x was chosen from ℝ ∖ D, not from the positive reals, and the arithmetic does not care, since a negative finite number over −∞ is 0 just the same. In fact the left side needs none of this; any finite number over −∞ comes home as 0. The undefinability is not there for the logic. It is there so that the one number in the piece that carries her name is also a number whose value our original language cannot specify without parameters. A side note, from a parent. Read the left side’s −∞ as the surreal −ω instead, in the field of Chapter ᚙ(X), and her number does not vanish: ᚏ_{4.16}/(−ω) is a nonzero infinitesimal, smaller than every positive real and never zero, and the equation would not balance until a standard part brought it home to 0. The book’s left side lives in the extended reals, where −∞ is not a field element and a/(−∞) = 0 does that job on its own. So through ordinary eyes, dividing her by infinity reduces her to 0. Through a parent’s eyes she lives in a richer field, and there even that division does not reduce her to nothing. Type: undefinable relative to the book’s fixed setting.", "cites": [ "r-1.1", "r-1.10", "r-1.2", "r-1.4", "r-1.9", "r-26.9" ], "links_to_cards": [ "defreals", "extreals", "infinitesimal", "neginf", "omega", "omegaS", "reals", "ruis", "stpart", "uncomputable" ] }, { "id": "neginf", "url": "https://www.teclamairme.com/#g-neginf", "name": "negative infinity", "symbol": "−∞", "chapter": 1, "type": "inf", "badge": "Infinity or limit device", "quick": "The value the tropical bracket hands to the extended reals; any finite number divided by −∞ is 0 by convention.", "more": "The value the tropical bracket hands to the extended reals, ℝ ∪ {−∞, +∞}, the number system the left side is read in. The extended reals are the standard setting of measure theory and analysis, and their arithmetic conventions, including finite/±∞ = 0, are what make the left side equal 0. It is chosen over +∞ so that the piece contains a negative number, which is why the max-plus semiring is used rather than min-plus, whose zero is +∞. It is not a surreal number: in Conway’s surreals, dividing by ω gives an infinitesimal rather than 0, and ∞ itself is a gap rather than a number. What that alternative would do to Rose’s number is a side note on her card. It is the one value the equation is sure is negative: nothing on the right side comes home negative. Some selected negative real values are read through absolute-value bars, including the double torus’s χ, Feigenbaum’s α, ζ(0), ζ(−1), and ψ(1). Every denominator is nonzero in the framework where its fraction is evaluated. Completed chapter contributions on the right are ordinary finite real numbers before joining the book’s product; their internal calculations can contain nonreal or infinite values. And the zero it produces is adopted rather than proved, unlike the settled zeros on the right; the book’s own model for such a convention is 0! = 1 in the garden, which everything else leans on. Type: infinity or limit device, a convention rather than a value.", "cites": [ "r-1.10", "r-1.4", "r-1.9", "r-26.14" ], "links_to_cards": [ "chiS2g", "digamma1", "extreals", "feigA", "inf", "infinitesimal", "minplus", "omegaS", "reals", "rose", "surgap", "tropical", "union", "zerofact", "zeta0", "zetam1" ] }, { "id": "tropical", "url": "https://www.teclamairme.com/#g-tropical", "name": "the tropical zero", "symbol": "𝟘", "chapter": 1, "type": "obj", "badge": "Non-real object", "quick": "In the max-plus tropical semiring, where addition is max and multiplication is +, the additive zero is −∞: an arithmetic whose “nothing” is infinite.", "more": "The additive identity of the max-plus tropical semiring, where a ⊕ b = max(a, b) and a ⊗ b = a + b. The identity for max is −∞, because max(x, −∞) = x for every x, so the tropical zero is literally negative infinity, and the tropical one is the ordinary 0. The min-plus version was introduced by the Brazilian computer scientist Imre Simon in 1978 for problems about finite automata; the adjective tropical was coined afterwards by French colleagues in his honor, Jean-Éric Pin crediting Dominique Perrin with it and Simon himself crediting Christian Choffrut, and means nothing deeper than the French view of Brazil. Max-plus algebra grew up separately from the early 1960s, with applications that include job scheduling. Tropical geometry proper, in which polynomials become piecewise-linear shapes, took off around 2000 with Mikhalkin, Sturmfels, and others. Type: non-real object, because 𝟘 lives in a semiring, and only its resolved value, −∞, comes out of the bracket into the extended reals.", "cites": [ "r-1.5", "r-1.6" ], "links_to_cards": [ "extreals", "minplus", "neginf", "tropone", "tropplus", "troptimes" ] }, { "id": "reals", "url": "https://www.teclamairme.com/#g-reals", "name": "the real numbers", "symbol": "ℝ", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "The complete ordered field: every decimal expansion, rational or not, and the line the whole ladder of numbers is drawn on.", "more": "ℝ, the real numbers, is the set of all points on the number line: the rationals together with every limit of rationals. Dedekind built it from cuts and Cantor from Cauchy sequences, both in 1872, and it is the unique complete ordered field. It is uncountable (Cantor, 1874), which is the fact the governing sentence rests on. The blackboard-bold letter is the usual notation, with ℕ, ℤ, ℚ, and ℂ for the naturals, integers, rationals, and complex numbers, whose examples fill Chapters ᚂ(L) and ᚔ(I). Type: a tool of the framework rather than a value.", "cites": [ "r-1.3", "r-4.21" ], "links_to_cards": [] }, { "id": "defreals", "url": "https://www.teclamairme.com/#g-defreals", "name": "the definable real numbers", "symbol": "D", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "The reals that a finite formula picks out uniquely, without parameters, in the fixed setting; counted from outside, they form a countable set within the uncountable real line.", "more": "D collects the reals that a finite formula picks out uniquely, without parameters, in the fixed formal language and mathematical setting used here. A parameter is an extra object supplied as input; allowing arbitrary reals as parameters would make every real definable using itself. Familiar integer and rational labels already have definitions, so they require no such extra input. With the language and setting fixed, there are only countably many finite formulas; viewed externally against the ordinary uncountable real line, D is countable and has measure zero. Familiar algebraic and computable numbers, and definable uncomputable numbers such as Chaitin’s Ω, are on this side. Familiar examples include π and e. The author proposes Ruis numbers as a name for the reals outside D. Hamkins, Linetsky, and Reitz explain why the external viewpoint matters: if ZFC is consistent, there are models in which every real is definable without parameters. A new constant naming x expands the language; it does not give a definition in the original one. The adopted setting. For this book, fix the set-theoretic structure ⟨V_{ω+ω}, ∈⟩ and use its first-order language with no parameters, counting its definable reals from outside against the ordinary uncountable real line. V_{ω+ω} is a fixed level of the cumulative hierarchy of sets, rich enough for the familiar constructions used here, including algebraic and computable reals and the halting probability of our fixed optimal universal prefix-free machine. Hamkins, Linetsky, and Reitz discuss this structure as a background for classical definitions; our use of it is the book’s declared choice. Type: a tool of the framework.", "cites": [ "r-1.1", "r-1.17", "r-1.2" ], "links_to_cards": [ "e", "omega", "pi", "reals", "ruis", "zfc" ] }, { "id": "setminus", "url": "https://www.teclamairme.com/#g-setminus", "name": "set difference", "symbol": "∖", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "A ∖ B is everything in A that is not in B; here ℝ ∖ D is the reals with the definable ones removed.", "more": "The set-difference symbol, A ∖ B, denotes the members of A that are not members of B. In the governing sentence ℝ ∖ D is the collection this book calls Ruis numbers; it is from that collection that this book’s fixed x is chosen. The related symbols of set theory, ∈ for membership, ⊆ for subset, ∪ and ∩ for union and intersection, and ∅ for the empty set, belong to the same language; ∅ is in Chapter ᚚ(P). Type: a tool of the framework.", "cites": [ "r-1.8" ], "links_to_cards": [ "defreals", "elem", "reals", "ruis", "union" ] }, { "id": "elem", "url": "https://www.teclamairme.com/#g-elem", "name": "is an element of", "symbol": "∈", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "x ∈ ℝ ∖ D reads: x is an element of the reals minus the definable reals.", "more": "The membership symbol, introduced by Peano in 1889 as an abbreviation for the Latin est, in a lunate epsilon rather than the stylized one now used, is the basic relation of set theory: x ∈ A means x is a member of A. The governing sentence uses it once, to say that the number is chosen from the undefinable reals. Type: a tool of the framework.", "cites": [ "r-1.12" ], "links_to_cards": [ "defreals", "reals", "setminus" ] }, { "id": "extreals", "url": "https://www.teclamairme.com/#g-extreals", "name": "the extended real numbers", "symbol": "ℝ ∪ {−∞, +∞}", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "The reals with two infinities added, the setting chosen for the left side of the equation.", "more": "The extended real line adds −∞ and +∞ to ℝ, ordered below and above every real, with conventions such as x + ∞ = ∞ and x/±∞ = 0 for finite x, while ∞ − ∞ and 0·∞ are left undefined here. It is the standard setting of limits in analysis and of measure theory—with the caveat that measure theory does give 0·∞ a value, setting it to 0, the natural choice for integration. This book does not, because nothing in the equation needs it. It is the framework the left side of the equation is evaluated in: a finite number over −∞ is 0. Type: a tool of the framework.", "cites": [ "r-1.4" ], "links_to_cards": [ "neginf", "reals" ] }, { "id": "defsym", "url": "https://www.teclamairme.com/#g-defsym", "name": "is defined as", "symbol": ":=", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "The colon-equals defines the relationship ᚏ4.16 := x/4.16 using the chosen x; it does not specify x’s value.", "quick_markup": "The colon-equals defines the relationship ᚏ_{4.16} := x/4.16 using the chosen x; it does not specify x’s value.", "more": "The symbol := marks a definition rather than an equation: ᚏ_{4.16} := x / 4.16 introduces the name on the left for the thing on the right. In programming it was the assignment operator of Algol 58 and then of Algol 60; other mathematical forms are ≝ and the phrase “let.” In this piece it defines an individual Ruis symbol in terms of the chosen x. Labels used together in one formula share that x, and this book keeps its choice fixed throughout. A formula outside this book may choose another x. The relationship does not supply a parameter-free definition of the individual value in the original setting. Type: a tool of the framework.", "cites": [ "r-1.12", "r-1.8" ], "links_to_cards": [ "ruis" ] }, { "id": "tropplus", "url": "https://www.teclamairme.com/#g-tropplus", "name": "tropical addition", "symbol": "⊕", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "In the max-plus semiring a ⊕ b = max(a, b); its identity is the tropical zero, −∞.", "more": "Tropical addition, written ⊕, is the maximum: a ⊕ b = max(a, b). It is associative, commutative, and idempotent (a ⊕ a = a), and its identity element is −∞, which is why the tropical zero is infinite. In the min-plus version, the kind Simon put to work in 1978, ⊕ is the minimum and the zero is +∞. Type: an operator of the framework.", "cites": [ "r-1.5", "r-1.6" ], "links_to_cards": [ "minplus", "neginf", "tropical" ] }, { "id": "troptimes", "url": "https://www.teclamairme.com/#g-troptimes", "name": "tropical multiplication", "symbol": "⊗", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "In the tropical semiring a ⊗ b = a + b, ordinary addition; its identity, the tropical one, is 0.", "more": "Tropical multiplication, written ⊗, is ordinary addition: a ⊗ b = a + b. It distributes over ⊕ because a + max(b, c) = max(a + b, a + c). Its identity is 0, the tropical one, and every element except the zero has an inverse, −a, which makes the tropical semiring a semifield. Tropical polynomials such as x ⊗ x ⊕ 3 ⊗ x ⊕ 5 are piecewise-linear functions, which is the door into tropical geometry. Type: an operator of the framework.", "cites": [ "r-1.5" ], "links_to_cards": [ "tropone", "tropplus" ] }, { "id": "tropone", "url": "https://www.teclamairme.com/#g-tropone", "name": "the tropical one", "symbol": "𝟙", "chapter": 1, "type": "int", "badge": "Integer or rational", "quick": "The multiplicative identity of the tropical semiring, which is the ordinary number 0.", "more": "In the max-plus semiring the multiplicative identity, written 𝟙, is the real number 0, since a ⊗ 0 = a + 0 = a. So the framework that puts its zero at −∞ puts its one at 0: both identities of ordinary arithmetic have moved down the number line. It is not used in the formula, but it is the reason the tropical zero had to be written 𝟘 rather than 0. Type: integer, exactly 0.", "cites": [ "r-1.5" ], "links_to_cards": [ "neginf", "tropical", "troptimes" ] }, { "id": "union", "url": "https://www.teclamairme.com/#g-union", "name": "union", "symbol": "∪", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "A ∪ B is everything in A or in B; the extended reals are ℝ ∪ {−∞, +∞}.", "more": "The union A ∪ B collects the members of either set; the extended real line of the left side is the union of ℝ with the two-element set {−∞, +∞}. The symbols ∪ and ∩ were introduced by Peano in 1888. Type: a tool of the framework.", "cites": [ "r-1.12", "r-1.4", "r-1.8" ], "links_to_cards": [ "extreals", "neginf", "reals" ] }, { "id": "richard", "url": "https://www.teclamairme.com/#g-richard", "name": "Richard’s paradox", "symbol": "Richard’s paradox", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "List every English phrase that defines a real number, then diagonalize: the number you build seems to be defined by that very sentence, yet it is not on the list. Jules Richard, 1905.", "more": "Jules Richard observed in 1905 that the phrases of a language that define real numbers can be listed, and that Cantor’s diagonal recipe then produces a real number differing from every listed one, although the recipe itself is a phrase that defines a real number. The resolution is that “defines a real number” is not itself a precise predicate of the language; it is the same fact the Ruis numbers card records, that definability has to be judged from outside the language. Type: a tool of the framework.", "cites": [ "r-1.7" ], "links_to_cards": [ "diagonal", "reals" ] }, { "id": "berry", "url": "https://www.teclamairme.com/#g-berry", "name": "Berry’s paradox", "symbol": "Berry’s paradox", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "”The smallest positive integer not definable in under sixty letters” has only fifty-seven letters. G. G. Berry’s paradox, first published by Russell in 1906.", "more": "Bertrand Russell published the paradox in 1906 and credited it to G. G. Berry, a junior librarian at the Bodleian. The sixty-letters form in the headline is the usual modern wording; Russell’s of 1908 is “the least integer not nameable in fewer than nineteen syllables,” itself a name of eighteen syllables. Like Richard’s, it shows that “definable” cannot be used inside the language it talks about; Chaitin turned it into a proof of Gödel’s incompleteness theorem in 1971. Type: a tool of the framework.", "cites": [ "r-1.11" ], "links_to_cards": [ "godel", "richard" ] }, { "id": "minplus", "url": "https://www.teclamairme.com/#g-minplus", "name": "the min-plus semiring", "symbol": "(ℝ ∪ {+∞}, min, +)", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "The min-plus form, the kind Imre Simon put to work in 1978, whose zero is +∞; the alternate reading of the left side, with a positive infinity instead of a negative one.", "more": "The min-plus semiring takes minimum as addition and ordinary addition as multiplication, so its zero is +∞ and its one is 0; it is the kind Imre Simon put to work in 1978 and the one shortest-path algorithms compute in. Use min-plus to interpret the tropical symbol and the denominator becomes +∞; evaluate the quotient in the extended reals and its value is still 0. The piece loses its one negative number that way, which is why max-plus was chosen. Type: a tool of the framework.", "cites": [ "r-1.5", "r-1.6" ], "links_to_cards": [ "extreals", "inf", "tropical" ] }, { "id": "ruis", "url": "https://www.teclamairme.com/#g-ruis", "name": "Ruis numbers", "symbol": "ᚏb", "symbol_markup": "ᚏ_{b}", "chapter": 1, "type": "undef", "badge": "Undefinable", "quick": "The author’s proposed name for real numbers our fixed original language cannot individually define in the book’s chosen mathematical setting.", "more": "Ruis numbers are the reals outside D: those with no definition without parameters in the adopted original language and setting. A parameter is an extra object supplied to a formula; allowing an arbitrary real to be supplied would let “the number equal to r” identify any r. Our definitions must work without that extra input. There are only countably many finite formulas in the fixed language, so the definable collection has measure zero in the ordinary uncountable real line. Why the different words? Asaf Karagila uses undefinable and unnameable, while explaining that intuitive nameability and mathematical definability are not automatically the same. Anne Newstead and James Franklin use indefinables and compare them to the dark matter of mathematics. These descriptions emphasize definitions, names, and an analogy; they are collected here because they inform this chapter’s vocabulary. “Ruis numbers” has the specific fixed-language definition above. “The dark pool” is descriptive language in this book; “normal” here means typical in everyday speech, rather than the technical digit-frequency property of a normal number. The notation. The name covers the whole collection ℝ ∖ D; the symbol ᚏ_{b} := x/b presents an individual chosen member, using a nonzero rational label b. Labels used together in one formula share the same x. This book holds its choice fixed throughout the master equation and its later appearances, including Chapter ᚋ(M). A formula outside this book may choose another x, so equal labels in independently chosen formulas need not represent equal values. Dividing x by a nonzero rational preserves undefinability: a definition of x/b in the original setting would define x there by multiplication. Since x is nonzero, distinct labels sharing it give distinct values. The birthday label 4.16 gives Rose’s number. The setting matters. Hamkins, Linetsky, and Reitz explain that, if ZFC is consistent, some models have every real definable without parameters. This book adopts an external view of a fixed setting containing the ordinary uncountable real line. Adding a new constant interpreted as our chosen x changes the language; it does not supply a definition in the original one. Ruis is the Pictish Ogham letter associated with the elder tree. Type: undefinable relative to the book’s fixed setting.", "cites": [ "r-1.1", "r-1.17", "r-1.18", "r-1.19", "r-1.2" ], "links_to_cards": [ "defreals", "defsym", "og-ruis", "rose", "zfc" ] }, { "id": "ints", "url": "https://www.teclamairme.com/#g-ints", "name": "the integers", "symbol": "ℤ", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "ℤ is the whole numbers together with their negatives and zero, the smallest ring built from the naturals by adding subtraction.", "more": "ℤ = {…, −2, −1, 0, 1, 2, …} extends the natural numbers ℕ by giving every element an additive inverse, forming a ring rather than a mere semiring: closed under addition, subtraction, and multiplication, though not division. It sits between ℕ and the rational numbers ℚ, which sit inside the real numbers ℝ, the field the book’s governing sentence is written in. The blackboard-bold letter abbreviates the German Zahlen, numbers.", "cites": [ "r-1.14" ], "links_to_cards": [ "nats", "rats", "reals" ] }, { "id": "nats", "url": "https://www.teclamairme.com/#g-nats", "name": "the natural numbers", "symbol": "ℕ", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "ℕ is the counting numbers, 0, 1, 2, 3, and so on, the starting point that ℤ, ℚ, and ℝ are built from.", "more": "Whether ℕ begins at 0 or at 1 is a convention that differs by author and by field; either way it is closed under addition and multiplication but not subtraction, which is exactly why the integers ℤ have to exist. Giuseppe Peano gave the natural numbers their modern axiomatic footing in 1889, defining them by a starting element and a successor operation rather than by counting objects. Every other number system in the chapter’s tower, ℤ, ℚ, ℝ, is built from ℕ by adjoining something it lacks: negatives, then ratios, then limits.", "cites": [ "r-1.15" ], "links_to_cards": [ "ints" ] }, { "id": "rats", "url": "https://www.teclamairme.com/#g-rats", "name": "the rational numbers", "symbol": "ℚ", "chapter": 1, "type": "fn", "badge": "Function or operator", "quick": "ℚ is every ratio of two integers, the numbers that can be written as a terminating or eventually repeating decimal.", "more": "Built as fractions a/b with integers a and b ≠ 0, ℚ is a field: every nonzero element has a multiplicative inverse, unlike the integers it is built from. It is countably infinite, unlike the real numbers ℝ it sits inside, even though between any two rationals there is always another; the definable real numbers of Chapter ᚏ(R) are also a countable set, because there are only countably many formulas. The letter Q abbreviates quotient, and the gaps the rationals leave on the number line, at numbers like √2, are exactly what forces the construction of ℝ.", "cites": [ "r-1.16" ], "links_to_cards": [ "defreals", "ints", "reals" ] }, { "id": "probability-one", "url": "https://www.teclamairme.com/#g-probability-one", "name": "100%—probability 1", "symbol": "P(X ∉ D) = 1", "chapter": 1, "type": "int", "badge": "Integer or rational", "quick": "Under an ideal uniform draw from [0, 1], the probability of hitting a real outside the book’s definable collection is exactly 1, or 100%.", "more": "Why exactly 100%? In the book’s adopted setting, finite formulas in the fixed original language can be listed, and each formula picks out at most one real uniquely. So D is countable. Under the uniform continuous distribution on [0, 1], a single point has probability zero; countable additivity gives the entire countable collection probability zero too. Therefore P(X ∈ D) = 0 and P(X ∉ D) = 1 − 0 = 1. This is exact, rather than a percentage rounded up from something smaller. Probability 1 means “almost surely.” A definable outcome is still logically possible: 1/2 remains a point in the interval even though its individual probability is zero. The claim concerns the ideal continuous model. A physical dart or a computer choosing among finite-precision outputs uses a different model. What about a million languages? Their rules and interpretations must be fixed independently of the target, with each using a finite or countable stock of symbols and finite descriptions. A million countable collections together are still countable, so their combined describable targets still have probability zero. If we choose the target first and then invent a special name interpreted as that target, we have changed the question. More time and computers do not change the countability argument. The everyday word “normal.” Here it means typical. Mathematical normality is a separate property of digit frequencies; a normal number can be computable and definable. Sources and deduction. Hamkins, Linetsky, and Reitz, Pointwise Definable Models of Set Theory, introduction, supplies the language/setting distinction. Curtis McMullen’s Harvard Math 114 Real Analysis notes, §§1.1–1.2, gives an analogous countable-set probability argument for algebraic numbers and develops measure and countable additivity. The application to this book’s D is our deduction from those facts; neither source introduces Ruis notation.", "cites": [ "r-1.17" ], "links_to_cards": [ "defreals" ] }, { "id": "mayazero", "url": "https://www.teclamairme.com/#g-mayazero", "name": "the Mayan zero", "symbol": "𝋠", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "A shell glyph, base twenty, an invention of zero independent of the Old World, in use by the fourth century.", "more": "The Maya wrote numbers in base twenty with dots and bars, and used a stylized shell for zero. The Long Count calendar, which needs a positional zero to work, is attested before the Maya: the oldest dates are Epi-Olmec, Stela 2 at Chiapa de Corzo (36 BC) and Stela C at Tres Zapotes (32 BC), and the oldest surviving shell-glyph zeros are fourth-century. It is one of the independent inventions of zero. Type: integer, exactly 0.", "cites": [ "r-2.4", "r-2.9" ], "links_to_cards": [ "longcount", "twenty" ] }, { "id": "babyzero", "url": "https://www.teclamairme.com/#g-babyzero", "name": "the Babylonian placeholder", "symbol": "𒑊", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "Two slanted wedges marking an empty sexagesimal place, in use by ~300 BC; a zero that could hold a column but never stood alone.", "more": "Babylonian scribes counted in base sixty and for centuries left empty places blank, so 61 and 3601 looked alike. A tablet from Kish of perhaps 700 BC marks the gap with three hooks; by ~300 BC, in the Seleucid period, two slanted wedges had become the standard placeholder. When the wedges came in is not settled: three sources give three dates, about 700 BC, about 400 BC, and 300 BC, and none of them argues its date, so the book gives the latest, by which all three have the sign in use. It is a zero, but only a placeholder: it never appears alone and was not used at the end of a number, so 1 and 60 still had to be told apart by context. Type: integer, exactly 0 in the place it holds.", "cites": [ "r-2.22", "r-2.23", "r-2.3", "r-2.9" ], "links_to_cards": [ "sixty" ] }, { "id": "devzero", "url": "https://www.teclamairme.com/#g-devzero", "name": "Brahmagupta’s zero", "symbol": "०", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "The first zero with rules: 628 AD, a − a = 0, a·0 = 0, and the Sanskrit name śūnya, emptiness.", "more": "In 628 AD, in the Brāhmasphuṭasiddhānta, Brahmagupta gave zero the first written rules: a − a = 0, a + 0 = a, a × 0 = 0, and rules for negative numbers as debts. He also claimed 0/0 = 0, which is wrong, and wrote a/0 only as a fraction with zero below, never assigning it a value, which is as close to “undefined” as the seventh century got. The Sanskrit name śūnya, emptiness, is the ancestor of the names cipher and zero. The oldest Indian zero in stone that everyone agrees is genuine is the small circle in the Chaturbhuj temple inscription at Gwalior, 876 AD, which has its own entry; the Bakhshali manuscript’s dot is on birch bark that Oxford’s re-dating places in the eighth to eleventh centuries. Type: integer; this is the moment zero became a number rather than a gap.", "cites": [ "r-2.1", "r-2.5", "r-2.6" ], "links_to_cards": [ "bakhzero", "gwalior" ] }, { "id": "khmerzero", "url": "https://www.teclamairme.com/#g-khmerzero", "name": "the Khmer zero", "symbol": "០", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "One of the oldest securely dated zeros in the Hindu-Arabic positional tradition, on a stone at Sambor, Cambodia, 683 AD.", "more": "Inscription K-127, on a stone from Sambor on the Mekong in Cambodia, is dated to the Śaka year that corresponds to 683 AD, and the year’s three digits carry a dot for zero in the middle position. George Cœdès published it in 1931, and an Old Malay inscription at Kedukan Bukit in Sumatra carries a zero in the same Śaka year, so the two cannot be ranked against each other. The stone vanished during the Khmer Rouge years and was tracked down in 2013 by Amir Aczel, who told the story in Finding Zero. Type: integer, exactly 0.", "cites": [ "r-2.14", "r-2.6" ], "links_to_cards": [] }, { "id": "arabzero", "url": "https://www.teclamairme.com/#g-arabzero", "name": "ṣifr", "symbol": "٠", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "The Arabic zero, whose name became both cipher and zero.", "more": "The Indian digit reached the Arabic world by the ninth century; al-Khwarizmi’s treatise on Indian reckoning, around 825, spread it westward. The Arabic name ṣifr translates śūnya, emptiness, and became the Latin zephirum, the Italian zefiro, and the English cipher and zero, so both of those English names are fossils of the Sanskrit one. Type: integer, exactly 0.", "cites": [ "r-2.7" ], "links_to_cards": [ "devzero", "latinzero" ] }, { "id": "chinazero", "url": "https://www.teclamairme.com/#g-chinazero", "name": "the Chinese circle zero", "symbol": "〇", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "A circle for the empty place, in use by the thirteenth century, still used in Chinese and Japanese today.", "more": "Chinese mathematicians used counting rods and a blank space for an empty place for centuries. Qin Jiushao’s Mathematical Treatise in Nine Sections of 1247 is the oldest surviving Chinese mathematical text to write zero as the circle 〇, which also survives in the Suzhou numerals, the last living form of the rod system; the book itself circulated in manuscript and was not printed until 1842. The origin of the circle is unknown; one hypothesis makes it a modified □, the text-space filler. 〇 is still used in Chinese and Japanese for zero today. Type: integer, exactly 0.", "cites": [ "r-2.7" ], "links_to_cards": [] }, { "id": "jpzero", "url": "https://www.teclamairme.com/#g-jpzero", "name": "rei", "symbol": "零", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "A Japanese and Chinese word for zero, written as a character rather than a digit.", "more": "The character 零, read rei in Japanese and líng in Chinese, originally meant a drizzle of rain or a small leftover, and was pressed into service for zero as a word rather than a digit. It is the ordinary word for zero in Chinese; in Japanese the preferred spoken reading of 0 is the English loanword zero, with rei beside it. Type: integer, exactly 0, written as a word rather than a glyph.", "cites": [ "r-2.12" ], "links_to_cards": [] }, { "id": "bedezero", "url": "https://www.teclamairme.com/#g-bedezero", "name": "Bede’s nulla", "symbol": "N", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "The letter N for nulla or nihil, both meaning nothing, used by Bede around 725 AD in tables that Roman numerals could not otherwise fill.", "more": "Bede, writing in Northumbria in 725 in De temporum ratione, built tables for calculating the date of Easter, and needed to record that a remainder, the epact, was nothing. Roman numerals have no zero, so he or his colleagues wrote N, the initial of nulla or of nihil. He was following Dionysius Exiguus, who had written the word nulla in his Easter table two centuries earlier, in 525; Bede’s N may be that word abbreviated to a letter. A zero invented on the spot by people whose number system did not have one. Type: integer, exactly 0.", "cites": [ "r-2.8" ], "links_to_cards": [ "dionzero" ] }, { "id": "latinzero", "url": "https://www.teclamairme.com/#g-latinzero", "name": "zephirum", "symbol": "0", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "Fibonacci’s Latin for the Arabic ṣifr, 1202, the zero the West finally kept.", "more": "Leonardo of Pisa, Fibonacci, popularized the Indian digits in Latin Europe in the Liber Abaci of 1202, and Latinized ṣifr as zephirum. It was not widely used for a long time, but the digit won, and the word shortened through the Italian zefiro to zero. This is the 0 the rest of the book uses. Type: integer, exactly 0.", "cites": [ "r-2.7" ], "links_to_cards": [ "arabzero" ] }, { "id": "greekzero", "url": "https://www.teclamairme.com/#g-greekzero", "name": "Ptolemy’s zero", "symbol": "ο", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "A small circle with a bar, used by Ptolemy in the Almagest around 150 AD as the empty place in sexagesimal fractions; the Greek zero.", "more": "Ptolemy, writing the Almagest in Alexandria around 150 AD, tabulated astronomy in Babylonian base-sixty fractions and marked an empty place with a small circle carrying a long overbar, the Greek zero sign (Unicode 𐆊). It was a placeholder in tables rather than a number to calculate with, and its origin is argued over: the popular story derives it from omicron, the first letter of οὐδέν, nothing, but Neugebauer rejected that, since omicron already meant 70 in Greek alphabetic numerals; it was free to serve as zero only because a sexagesimal place never runs past 59. In ninth-century Byzantine manuscripts it is still a full-size uncial omicron with a macron, and the form alters markedly only in some late medieval copies. Type: integer, exactly 0.", "cites": [ "r-2.3" ], "links_to_cards": [ "sixty" ] }, { "id": "bakhzero", "url": "https://www.teclamairme.com/#g-bakhzero", "name": "the Bakhshali dot", "symbol": "•", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "A dot for zero in the Bakhshali manuscript, an Indian birch-bark text; Oxford’s 2024 re-dating places the sampled leaves in about 773–1032 AD and models the manuscript’s production at roughly 799–1102 AD.", "more": "The Bakhshali manuscript, seventy birch-bark leaves of arithmetic found near Peshawar in 1881 and kept in the Bodleian Library, writes zero as a dot, hundreds of times. In 2017 the Bodleian radiocarbon-dated three leaves to 224–383, 680–779, and 885–993 AD and announced the oldest known written zero; Plofker, Wujastyk, Hayashi, Keller, and Montelle replied that a manuscript is dated by its latest leaf, so the text as we have it may be ninth-century or later. Oxford’s Radiocarbon Accelerator Unit then re-dated the manuscript and published the revised results on 14 October 2024: the earlier Folio 16 determination was inaccurate, Folio 17 was treated as an outlier, and Folios 15, 16, 23, 33, and 55 date to 773–986, 901–1032, 890–1014, 885–995, and 774–991 AD at 95.4% confidence, with a phase model putting production at roughly 799–1102 AD. Whatever its date, the dot is a written zero used in calculation and not only a placeholder, and so it belongs in the product. Type: integer, exactly 0.", "cites": [ "r-2.1", "r-2.10", "r-2.2" ], "links_to_cards": [] }, { "id": "dionzero", "url": "https://www.teclamairme.com/#g-dionzero", "name": "Dionysius Exiguus’s nulla", "symbol": "nulla", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "The Latin word for nothing, written in an Easter table in 525 AD where a Roman numeral could not go; Bede’s N may abbreviate it.", "more": "Dionysius Exiguus, the monk who invented the Anno Domini count, drew up an Easter table in 525 AD in which some entries had to be nothing. Roman numerals cannot say so, so he wrote the word nulla. It is a zero as a word rather than a glyph, like 零, and it may be the ancestor of Bede’s N two centuries later. Type: integer, exactly 0.", "cites": [ "r-2.8" ], "links_to_cards": [ "bedezero", "jpzero" ] }, { "id": "gwalior", "url": "https://www.teclamairme.com/#g-gwalior", "name": "the Gwalior zero", "symbol": "○", "chapter": 2, "type": "int", "badge": "Integer or rational", "quick": "The oldest Indian zero in stone that everyone agrees is genuine: a small circle in an inscription of 876 AD at the Chaturbhuj temple in Gwalior.", "more": "An inscription of 876 AD at the Chaturbhuj temple in the fort of Gwalior records a land grant of 270 by 187 hastas and a gift of 50 garlands, and writes the zeros of 270 and 50 as small circles. It is the oldest dated Indian zero in stone that nobody disputes: older Indian inscriptions have been proposed, among them inscriptions of AD 672, 683, and 684, but their dates or their authenticity are questioned. It is later than the Khmer stone of 683. Type: integer, exactly 0.", "cites": [ "r-2.6" ], "links_to_cards": [ "khmerzero" ] }, { "id": "placevalue", "url": "https://www.teclamairme.com/#g-placevalue", "name": "positional notation", "symbol": "dₙ·bⁿ", "chapter": 2, "type": "fn", "badge": "Function or operator", "quick": "The column a digit stands in tells you what it is worth, so a small fixed set of signs writes every number—and an empty column has to be written down.", "more": "In positional notation the value of a digit is its face value times a power of the base, fixed by the column it occupies, so a small fixed alphabet of signs writes every number and arithmetic becomes a matter of columns. The Babylonian system, base 60, is credited as the first positional one, and it is exactly the system that made a zero necessary: with no sign for an empty column, 61 and 3601 look alike, which is why three hooks by about 700 BC, and later two slanted wedges, were pressed into service; they were never written at the end of a number, so 1 and 60 still looked alike. The Indians were the first to build a base-10 positional system, and the one on this page is its descendant. This is the invention that the chapter’s three separate zeros—placeholder, digit, number—are three stages of: a sign-value system like Roman numerals has no empty column to name, and so never needs the first of them. A notation of the framework, uncolored.", "cites": [ "r-2.22", "r-2.3" ], "links_to_cards": [ "babyzero" ] }, { "id": "longcount", "url": "https://www.teclamairme.com/#g-longcount", "name": "the Long Count calendar", "symbol": "13.0.0.0.0", "chapter": 2, "type": "fn", "badge": "Function or operator", "quick": "A positional, mostly base-twenty count of days used for Mesoamerican dates, needing its own zero place to work at all.", "more": "The Long Count numbers days from a fixed starting point using five or more place values, all base twenty except the second, which is base eighteen, so that the third place counts 360 days, roughly a year, rather than 400. Its two oldest surviving dates, Stela 2 at Chiapa de Corzo and Stela C at Tres Zapotes, are Epi-Olmec rather than Maya and predate the Common Era; the Maya inherited the system and came to write its zero place with a stylized shell glyph. A positional count needs a symbol for an empty place to be positional at all, which is why the calendar and the Mayan zero belong to the same story.", "cites": [ "r-2.24" ], "links_to_cards": [ "mayazero", "twenty" ] }, { "id": "Lambda", "url": "https://www.teclamairme.com/#g-Lambda", "name": "cosmological constant", "symbol": "Λ", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The cosmological constant in Einstein’s field equations, with units m⁻²; the associated vacuum-energy density is derived from Λ, c, and G.", "more": "Einstein added Λ to general relativity in 1917 to hold the universe static. It returned to the center of cosmology after the 1998 discovery of accelerated expansion, recognized by the 2011 Nobel Prize. Cosmological observations imply Λ ≈ 1.1 × 10^{−52} m^{−2}; the dimensionless combination ΛGℏ/c³ is about 2.9 × 10^{−122}. Λ itself is not an energy density: the corresponding vacuum-energy density is obtained from Λ, c, and G, with the exact factor depending on whether mass or energy density is meant. The enormous mismatch with naive quantum-field estimates is the cosmological constant problem. Physics: measured through cosmological fits.", "cites": [ "r-3.2", "r-3.21", "r-3.9" ], "links_to_cards": [ "G", "c", "ccproblem", "dimensionless", "efe" ] }, { "id": "G", "url": "https://www.teclamairme.com/#g-G", "name": "Newton’s gravitational constant", "symbol": "G", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "One of the least precisely known fundamental constants, measured to about one part in fifty thousand.", "more": "Newton’s Principia of 1687 stated the inverse-square law but never wrote G as a number; Cavendish’s torsion-balance experiment of 1798 weighed the Earth and fixed it in effect. CODATA 2018 and 2022 both give G = 6.67430(15) × 10^{−11} m³ kg^{−1} s^{−2}, a relative uncertainty of 22 parts per million, poor by the standards of the CODATA table, because gravity is so weak that laboratory masses barely attract each other. Physics: measured.", "cites": [ "r-3.20", "r-3.7" ], "links_to_cards": [] }, { "id": "h", "url": "https://www.teclamairme.com/#g-h", "name": "Planck’s constant", "symbol": "h", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "Fixed exactly at 6.62607015×10⁻³⁴ joule-seconds by the 2019 redefinition of the SI.", "more": "Max Planck introduced h in 1900 to explain black-body radiation, calling it a mathematical trick; Einstein took it literally in 1905. Since 20 May 2019 it is fixed by definition at exactly 6.62607015 × 10^{−34} joule-seconds, and the kilogram is defined as whatever mass makes that true, measured by a Kibble balance. Physics: fixed by definition.", "cites": [ "r-3.1", "r-3.20" ], "links_to_cards": [ "si2019" ] }, { "id": "c", "url": "https://www.teclamairme.com/#g-c", "name": "speed of light", "symbol": "c", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "Fixed exactly at 299,792,458 meters per second since 1983.", "more": "Ole Rømer made the first estimate in 1676 from the moons of Jupiter. Since 1983 the meter has been defined as the distance light travels in 1/299,792,458 of a second, so c = 299,792,458 m/s exactly and can never be measured again, only the meter can. Physics: fixed by definition, a generation before the rest of them.", "cites": [ "r-3.1", "r-3.18" ], "links_to_cards": [] }, { "id": "kB", "url": "https://www.teclamairme.com/#g-kB", "name": "Boltzmann constant", "symbol": "kB", "symbol_markup": "k_{B}", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "Converts temperature into energy, and was fixed exactly by decree in 2019 to define the kelvin; that changed where we put the ruler, not what the constant does.", "more": "Planck introduced the constant in the same 1900 paper as h; it came to be called Boltzmann’s, after the 1877 relation S = k log W it appears in. It converts temperature to energy, one kelvin being about 1.38 × 10^{−23} joules, and since 2019 it has been fixed at exactly 1.380649 × 10^{−23} J/K, so that the kelvin is now defined by it. Physics: fixed by definition, because temperature is energy in disguise.", "cites": [ "r-3.1", "r-3.21" ], "links_to_cards": [ "h" ] }, { "id": "Tcmb", "url": "https://www.teclamairme.com/#g-Tcmb", "name": "cosmic microwave background temperature", "symbol": "TCMB", "symbol_markup": "T_{CMB}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The measured temperature of the oldest light in the universe, 2.7255 kelvin.", "more": "Penzias and Wilson found the cosmic microwave background by accident in ~1964, as a hiss their antenna could not shake; the sources give two years for two events: the measurement that clearly showed it was made on 20 May 1964, and the Astrophysical Journal letter that announced it appeared in 1965. It is light released when the universe became transparent, 380,000 years after the beginning, cooled by expansion to 2.7255 ± 0.0006 K (Fixsen, 2009, combining the published measurements). Physics: measured, and the most perfect black-body spectrum ever observed.", "cites": [ "r-3.3" ], "links_to_cards": [] }, { "id": "me", "url": "https://www.teclamairme.com/#g-me", "name": "electron mass", "symbol": "me", "symbol_markup": "m_{e}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The measured rest mass of the electron, whose rest energy mec² is 511 keV.", "quick_markup": "The measured rest mass of the electron, whose rest energy m_{e}c² is 511 keV.", "more": "J. J. Thomson identified the electron in 1897 and measured its charge-to-mass ratio. CODATA 2022 gives m_{e} = 9.1093837139(28) × 10^{−31} kg, and its rest energy m_{e}c² is 0.51099895 MeV, half the 1.022 MeV released when an electron and a positron annihilate at rest. In the heat bracket it is the yardstick against which the cold of the universe is measured. Physics: measured.", "cites": [ "r-3.22", "r-3.7" ], "links_to_cards": [ "emc2" ] }, { "id": "NA", "url": "https://www.teclamairme.com/#g-NA", "name": "Avogadro constant", "symbol": "NA", "symbol_markup": "N_{A}", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "Exactly 6.02214076×10²³ per mole since 2019.", "more": "Avogadro’s 1811 hypothesis that equal volumes of gas hold equal numbers of molecules did not give the number; Jean Perrin, who measured it by several independent methods, named it after Avogadro in 1909. Since 2019 it is exactly 6.02214076 × 10^{23} per mole, and the mole is defined as that many entities. Physics: fixed by definition; times one mole it is a pure count.", "cites": [ "r-3.1", "r-3.22" ], "links_to_cards": [ "mol" ] }, { "id": "mol", "url": "https://www.teclamairme.com/#g-mol", "name": "one mole", "symbol": "1 mol", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "An exact amount of substance, present only to cancel Avogadro’s per-mole.", "more": "The SI base unit for amount of substance, adopted in 1971 and redefined in 2019 as exactly 6.02214076 × 10^{23} elementary entities, no longer tied to 12 grams of carbon-12. It appears here only to cancel the per-mole in N_{A}, leaving a pure number. Physics: fixed by definition.", "cites": [ "r-3.1", "r-3.22" ], "links_to_cards": [] }, { "id": "sigma", "url": "https://www.teclamairme.com/#g-sigma", "name": "Stefan–Boltzmann constant", "symbol": "σ", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "Derived from Planck’s law as σ = π²kB⁴/(60ℏ³c²), so its bracket is secretly π²/60.", "quick_markup": "Derived from Planck’s law as σ = π²k_{B}⁴/(60ℏ³c²), so its bracket is secretly π²/60.", "more": "Jožef Stefan found in 1879 that a hot body radiates in proportion to the fourth power of its temperature; his student Boltzmann derived it from thermodynamics in 1884, and Planck’s 1900 law gave the constant its value, σ = π²k_{B}⁴/(60ℏ³c²) ≈ 5.67 × 10^{−8} W m^{−2} K^{−4}. So the bracket σℏ³c²/k_{B}⁴ is exactly π²/60 ≈ 0.1645. Physics: derived; it is not a separate fact about nature.", "cites": [ "r-3.23", "r-3.3" ], "links_to_cards": [] }, { "id": "hbar", "url": "https://www.teclamairme.com/#g-hbar", "name": "reduced Planck constant", "symbol": "ℏ", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "Planck’s constant divided by 2π; derived, not independent.", "more": "h divided by 2π, written ℏ, a symbol Dirac introduced in his 1930 Principles of Quantum Mechanics, having written plain h for the same quantity since 1926, because the combination appears in almost every quantum formula. Since h is exact, ℏ = 1.054571817… × 10^{−34} J s is exact too, though its decimal expansion never ends. The gravity bracket ΛGℏ/c³ is Λ times the square of the Planck length, ℓ_{P}² = ℏG/c³. Physics: derived.", "cites": [ "r-3.1", "r-3.21" ], "links_to_cards": [ "Lambda", "h", "plancklen" ] }, { "id": "dimensionless", "url": "https://www.teclamairme.com/#g-dimensionless", "name": "dimensionless numbers", "symbol": "dimensionless", "chapter": 3, "type": "pmeas", "badge": "Measured or derived ratio", "quick": "A quantity whose units cancel, so its numerical value does not depend on the chosen units.", "more": "Dimensionless numbers can still describe real physical relationships. Ratios such as the fine-structure constant and the proton-to-electron mass ratio compare quantities after their units have canceled.", "cites": [ "r-3.17" ], "links_to_cards": [ "alpha", "mpme" ] }, { "id": "alpha", "url": "https://www.teclamairme.com/#g-alpha", "name": "fine-structure constant", "symbol": "α", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The dimensionless strength of electromagnetism, about 1/137, measured and still unexplained.", "more": "Arnold Sommerfeld introduced α in 1916 to explain the fine splitting of spectral lines, as e²/(4πε₀ℏc), the ratio of the electron’s speed in the first Bohr orbit to the speed of light. It is dimensionless, the same in every unit system, and no theory predicts its value; Feynman called it one of the greatest damn mysteries of physics. CODATA 2022 gives 1/α = 137.035999177(21), a relative uncertainty of 1.6 parts in 10¹⁰, from the electron’s magnetic moment and atom-interferometry measurements of h/m; the Rydberg constant and the electron g-factor are both known better. Physics: measured.", "cites": [ "r-3.18", "r-3.4", "r-3.5" ], "links_to_cards": [ "a0", "ae", "c", "dimensionless", "h", "rydberg" ] }, { "id": "echarge", "url": "https://www.teclamairme.com/#g-echarge", "name": "elementary charge", "symbol": "e", "chapter": 3, "type": "pdef", "badge": "Physics: fixed by definition", "quick": "The charge of one proton, fixed exactly at 1.602176634×10⁻¹⁹ coulombs in 2019; not Euler’s e, which lives in Chapter ᚂ(L).", "more": "The charge of one proton, or minus that of one electron. Millikan and Fletcher’s oil-drop experiment, begun in 1909 and published in final form in 1913, measured it and showed that charge comes in whole multiples of it; since 2019 it is fixed at exactly 1.602176634 × 10^{−19} coulombs, and the ampere is defined from it. Not Euler’s e, which lives in Chapter ᚂ(L). Physics: fixed by definition.", "cites": [ "r-3.1", "r-3.19" ], "links_to_cards": [ "e" ] }, { "id": "eps0", "url": "https://www.teclamairme.com/#g-eps0", "name": "vacuum permittivity", "symbol": "ε0", "symbol_markup": "ε_{0}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "How strongly empty space resists an electric field; since 2019 it is derived from α, e, ℏ, and c rather than defined.", "more": "The permittivity of free space, the constant in Coulomb’s law that says how strongly charges attract across empty space. It was exact before 2019, tied to the old definition of the ampere; since then it is derived, ε₀ = e²/(2αhc) ≈ 8.854 × 10^{−12} F/m, and inherits the tiny uncertainty of α. Physics: derived.", "cites": [ "r-3.1" ], "links_to_cards": [ "alpha" ] }, { "id": "mu0", "url": "https://www.teclamairme.com/#g-mu0", "name": "vacuum permeability", "symbol": "μ0", "symbol_markup": "μ_{0}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The magnetic twin of ε₀; μ₀ε₀c² = 1 exactly, so the magnetism bracket is secretly one.", "more": "The permeability of free space, the magnetic twin of ε₀. From 1948 to 2019 it was exactly 4π × 10^{−7} H/m by definition of the ampere; since 2019 it is μ₀ = 2αh/(e²c) ≈ 1.25663706 × 10^{−6} H/m, with the uncertainty of α. Maxwell’s 1865 result c² = 1/(μ₀ε₀) makes the bracket μ₀ε₀c² exactly 1. Physics: derived.", "cites": [ "r-3.22", "r-3.4" ], "links_to_cards": [ "alpha", "eps0" ] }, { "id": "RK", "url": "https://www.teclamairme.com/#g-RK", "name": "von Klitzing constant", "symbol": "RK", "symbol_markup": "R_{K}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The quantum of electrical resistance, h/e², so e²R_K/h = 1 exactly; another famous R, and the collision with ᚏ is deliberate.", "more": "Klaus von Klitzing discovered in 1980 that the Hall resistance of a two-dimensional electron gas in a strong magnetic field comes in exact steps of h/e², the quantum Hall effect, which won the 1985 Nobel Prize. R_{K} = h/e² = 25,812.80745… ohms, exact since 2019 because h and e are, so e²R_{K}/h is 1 identically. Another famous R, and the collision with ᚏ is deliberate. Physics: derived.", "cites": [ "r-3.23", "r-3.4" ], "links_to_cards": [ "echarge", "h", "og-ruis" ] }, { "id": "Z0", "url": "https://www.teclamairme.com/#g-Z0", "name": "impedance of free space", "symbol": "Z0", "symbol_markup": "Z_{0}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "About 376.73 ohms, equal to μ₀c, so Z₀/(μ₀c) = 1 exactly.", "more": "The impedance of free space, the ratio of electric to magnetic field in a light wave traveling through vacuum, Z₀ = μ₀c ≈ 376.730313 ohms. It was exact before 2019 and is now derived from α; either way Z₀/(μ₀c) is 1 by definition. Physics: derived.", "cites": [ "r-3.24", "r-3.4" ], "links_to_cards": [ "alpha" ] }, { "id": "wien", "url": "https://www.teclamairme.com/#g-wien", "name": "Wien’s displacement constant", "symbol": "b", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The wavelength at which a hot body glows brightest, times its temperature: b = 2.897771955… mm·K, exact since 2019 because it is hc/(x·k_B).", "more": "Wilhelm Wien found in 1893 that the peak wavelength of black-body radiation times the temperature is a constant, b = λ_{max}T. Planck’s law fixes it as b = hc/(x_{W}k_{B}), where x_{W} = 4.965114… is the root of x = 5(1 − e^{−x}), so since 2019 b = 2.897771955… × 10^{−3} m K is exact; the “…” marks only that its digits, like the root’s, run on past those printed. For the 2.7255 K background it gives a peak near 1.06 mm, which is why the oldest light is microwave. The bracket b·x_{W}·k_{B}/(hc) is exactly 1. Physics: derived.", "cites": [ "r-3.21", "r-3.24", "r-3.3" ], "links_to_cards": [] }, { "id": "wienx", "url": "https://www.teclamairme.com/#g-wienx", "name": "Wien’s root", "symbol": "xW", "symbol_markup": "x_{W}", "chapter": 3, "type": "tr", "badge": "Proven transcendental", "quick": "The number 4.965114231…, the root of x = 5(1 − e^(−x)), which sits inside Wien’s constant; it is 5 + W(−5e^(−5)) and transcendental.", "more": "The peak of Planck’s black-body curve in wavelength is where x = hc/(λk_{B}T) satisfies x = 5(1 − e^{−x}); the nonzero root is 4.965114231…, and in closed form x_{W} = 5 + W_{0}(−5e^{−5}) with W the Lambert function of Chapter ᚐ(A). It is transcendental by Lindemann–Weierstrass: if it were algebraic, e^{−x} = 1 − x/5 would be algebraic too, which is impossible for a nonzero algebraic exponent. Type: proven transcendental.", "cites": [ "r-3.3" ], "links_to_cards": [ "lindweier", "wien" ] }, { "id": "gasR", "url": "https://www.teclamairme.com/#g-gasR", "name": "the molar gas constant", "symbol": "R", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The R of PV = nRT, exactly N_A·k_B = 8.314462618… J/(mol·K) since 2019; one more famous R.", "more": "The gas constant of Clapeyron’s ideal-gas law PV = nRT (1834), which gathers Boyle, Charles, and Avogadro into one line. It is Boltzmann’s constant per molecule scaled up to a mole, R = N_{A}k_{B}, so since 2019 it is exact, 8.31446261815324 J mol^{−1} K^{−1}, and the bracket R/(N_{A}k_{B}) is exactly 1. The letter is one more famous R in the piece, after ᚏ, R_{K}, and R_{∞}. Physics: derived.", "cites": [ "r-3.1", "r-3.20" ], "links_to_cards": [ "kB", "og-ruis" ] }, { "id": "faraday", "url": "https://www.teclamairme.com/#g-faraday", "name": "the Faraday constant", "symbol": "F", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The charge of one mole of electrons, exactly N_A·e = 96485.33212… C/mol since 2019.", "more": "Faraday’s laws of electrolysis of 1834 say that the mass deposited at an electrode is proportional to the charge passed; the constant of proportionality per mole of electrons is F = N_{A}e, later named in his honor. We could not confirm when: no source we found dates the naming. Since 2019 it is exact, 96485.33212331… C/mol, and the bracket F/(N_{A}e) is exactly 1. Physics: derived.", "cites": [ "r-3.1", "r-3.19" ], "links_to_cards": [ "echarge", "mol" ] }, { "id": "coulombk", "url": "https://www.teclamairme.com/#g-coulombk", "name": "Coulomb’s constant", "symbol": "ke", "symbol_markup": "k_{e}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The constant in Coulomb’s law of 1785, k_e = 1/(4πε₀) ≈ 8.98755 × 10⁹ N·m²/C², so k_e·4πε₀ = 1 exactly.", "more": "Charles-Augustin de Coulomb measured the inverse-square law of electric force with a torsion balance in 1785. The constant in F = k_{e}q₁q₂/r² is k_{e} = 1/(4πε₀) = 8.9875517862(14) × 10⁹ N m² C^{−2} (computed from CODATA 2022’s ε₀), and the bracket k_{e}·4πε₀ is 1 by definition. Physics: derived.", "cites": [ "r-3.19", "r-3.4" ], "links_to_cards": [] }, { "id": "rydberg", "url": "https://www.teclamairme.com/#g-rydberg", "name": "the Rydberg constant", "symbol": "R∞", "symbol_markup": "R_{∞}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The constant of atomic spectra, 10973731.568157(12) per meter, among the most precisely measured constants in physics, to about one part in 10¹²; equal to α²m_ec/(2h).", "more": "Johannes Rydberg’s 1888 formula for the lines of hydrogen-like spectra has one constant, which Bohr’s 1913 atom explained as R_{∞} = α²m_{e}c/(2h) = m_{e}e⁴/(8ε₀²h³c). CODATA 2022 gives 10 973 731.568 157(12) m^{−1}, a relative uncertainty of 1.1 parts in 10¹², from hydrogen and deuterium spectroscopy; among the most precisely measured constants in physics, though CODATA 2022 knows the electron g-factor better still, to 1.8 parts in 10^{13}. The bracket 2hR_{∞}/(α²m_{e}c) is exactly 1 and is the identity that ties the spectra to the fine-structure constant. Physics: measured.", "cites": [ "r-3.23", "r-3.4" ], "links_to_cards": [ "alpha", "c" ] }, { "id": "fluxq", "url": "https://www.teclamairme.com/#g-fluxq", "name": "the magnetic flux quantum", "symbol": "Φ0", "symbol_markup": "Φ_{0}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The smallest amount of magnetic flux a superconducting ring can hold, h/(2e) = 2.067833848… × 10⁻¹⁵ Wb, exact since 2019.", "more": "In 1961 Deaver and Fairbank at Stanford and Doll and Näbauer in Herrsching independently found that the magnetic flux through a superconducting ring comes in steps of h/2e, the 2 because electrons pair up. Φ_{0} = h/(2e) = 2.067833848… × 10^{−15} Wb is exact since 2019, and its product with the Josephson constant 2e/h is exactly 1. Physics: derived.", "cites": [ "r-3.19", "r-3.20", "r-3.4" ], "links_to_cards": [ "h", "josephson" ] }, { "id": "josephson", "url": "https://www.teclamairme.com/#g-josephson", "name": "the Josephson constant", "symbol": "KJ", "symbol_markup": "K_{J}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The frequency-to-voltage ratio of a Josephson junction, 2e/h = 483597.8484… GHz per volt, exact since 2019; the reciprocal of the flux quantum.", "more": "Brian Josephson predicted in 1962 that a supercurrent tunnels between two superconductors and that a voltage V across the junction produces an oscillation at frequency 2eV/h. The constant K_{J} = 2e/h = 483 597.8484… × 10⁹ Hz V^{−1} is exact since 2019 and underlies the Josephson voltage standard, which gives the standard representation of the volt. Its product with Φ_{0} = h/2e is 1 identically. Physics: derived.", "cites": [ "r-3.21", "r-3.4" ], "links_to_cards": [] }, { "id": "alphaG", "url": "https://www.teclamairme.com/#g-alphaG", "name": "the gravitational coupling constant", "symbol": "αG", "symbol_markup": "α_{G}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "Gravity’s answer to α: Gm_e²/(ℏc) ≈ 1.75 × 10⁻⁴⁵, the dimensionless strength of gravity between two electrons, 10⁴² times weaker than electromagnetism.", "more": "The gravitational coupling constant α_{G} = Gm_{e}²/(ℏc) = (m_{e}/m_{P})² ≈ 1.7518 × 10^{−45} is the dimensionless strength of gravity between two electrons, the exact analogue of the fine-structure constant α ≈ 1/137 for electromagnetism. Their ratio, about 4 × 10⁴², is one statement of the hierarchy problem, and one of Dirac’s large numbers. It closes the formula’s particle-physics brace, and, with the gravity bracket, it is one of the two numbers that drag the whole factor down to 10^{−154}. Physics: derived.", "cites": [ "r-3.12", "r-3.7" ], "links_to_cards": [ "alpha", "dimensionless", "hierarchy" ] }, { "id": "mpme", "url": "https://www.teclamairme.com/#g-mpme", "name": "the proton-to-electron mass ratio", "symbol": "mp/me", "symbol_markup": "m_{p}/m_{e}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "1836.152673426(32), the dimensionless ratio of the proton’s mass to the electron’s.", "more": "The proton is 1836.152673426(32) times heavier than the electron (CODATA 2022), a fundamental dimensionless constant, like α: no theory predicts it, and it must be measured. It opens the formula’s particle-physics brace. Physics: measured.", "cites": [ "r-3.7" ], "links_to_cards": [ "alpha", "dimensionless", "mp" ] }, { "id": "ae", "url": "https://www.teclamairme.com/#g-ae", "name": "the electron’s anomalous magnetic moment", "symbol": "ae", "symbol_markup": "a_{e}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The electron’s magnetic moment exceeds Dirac’s prediction by 0.00115965218…, which quantum electrodynamics computes and experiment confirms to more than ten significant figures.", "more": "Dirac’s 1928 equation gives the electron a g-factor of exactly 2; quantum electrodynamics corrects it by a_{e} = (g − 2)/2 = 0.00115965218…, first computed by Schwinger in 1948 as α/2π. The 2022 Northwestern measurement gives 1.15965218059(13) × 10^{−3}, and the quantum-electrodynamic prediction agrees with it to more than ten significant figures. One of the five ratios in the formula’s particle-physics brace. Physics: measured.", "cites": [ "r-3.18", "r-3.5" ], "links_to_cards": [ "alpha", "bohrmag", "qed" ] }, { "id": "alphas", "url": "https://www.teclamairme.com/#g-alphas", "name": "the strong coupling constant", "symbol": "αs", "symbol_markup": "α_{s}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The strength of the strong nuclear force at the Z boson’s energy, 0.1180(9): the coupling of the Yang–Mills theory of the epilogue once quarks are added.", "more": "The coupling of quantum chromodynamics, the SU(3) Yang–Mills theory of the epilogue with quarks added; the epilogue’s mass-gap problem is about the pure theory, without quarks. It runs with energy, and the standard reference value is at the Z boson mass: α_{s}(M_{Z}) = 0.1180(9) (Particle Data Group, Review of Particle Physics, 2026). It grows at low energies, the regime in which quarks are confined, though confinement has not been proved from the theory. One of the five ratios in the formula’s particle-physics brace. Physics: measured.", "cites": [ "r-3.5" ], "links_to_cards": [] }, { "id": "OmegaL", "url": "https://www.teclamairme.com/#g-OmegaL", "name": "the dark-energy density parameter", "symbol": "ΩΛ", "symbol_markup": "Ω_{Λ}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The dark-energy density divided by critical density; about 0.685 in Planck’s base flat ΛCDM fit.", "more": "Ω_{Λ} is the dark-energy density divided by the critical density. Planck’s base flat ΛCDM fit gives a value near 0.685, with spatial curvature fixed to zero and the radiation share, tiny today, normally left out when the rounded dark-energy and matter values are quoted as adding to 1. The exact symbolic Friedmann identity Ω_{Λ} + Ω_{m} + Ω_{r} + Ω_{k} = 1 is retained, but numerical values from different model fits must not be mixed. Physics: measured within a specified model.", "cites": [ "r-3.2" ], "links_to_cards": [ "Omegar", "lcdm" ] }, { "id": "Omegam", "url": "https://www.teclamairme.com/#g-Omegam", "name": "the matter density parameter", "symbol": "Ωm", "symbol_markup": "Ω_{m}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "Total matter divided by critical density; about 0.315 in Planck’s base flat ΛCDM fit.", "more": "Ω_{m} is total matter density—dark matter plus baryonic matter and the model’s neutrino contribution—over critical density. Planck’s base flat ΛCDM fit gives a value near 0.315. That result belongs to the same model assumptions as its quoted dark-energy complement; it must not be combined numerically with a curvature value from an extended model. Physics: measured within a specified model.", "cites": [ "r-3.2" ], "links_to_cards": [ "lcdm" ] }, { "id": "Omegar", "url": "https://www.teclamairme.com/#g-Omegar", "name": "the radiation density parameter", "symbol": "Ωr", "symbol_markup": "Ω_{r}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The share of the universe that is light and neutrinos today: about 9 × 10⁻⁵, negligible now, dominant in the first fifty thousand years.", "more": "Ω_{r} is the density of photons and relativistic neutrinos over critical density. Its present value is of order 9 × 10^{−5}, derived from the background temperature, neutrino assumptions, and the chosen Hubble parameter. It belongs in the exact Friedmann balance, but a rounded value should be quoted only with the same cosmological model as the other Ω terms. Physics: derived from measured inputs.", "cites": [ "r-3.2" ], "links_to_cards": [ "H0" ] }, { "id": "Omegak", "url": "https://www.teclamairme.com/#g-Omegak", "name": "the curvature density parameter", "symbol": "Ωk", "symbol_markup": "Ω_{k}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The curvature term in the Friedmann balance; fixed to zero in flat ΛCDM and fitted only in curved extensions.", "more": "Ω_{k} is the spatial-curvature term in the Friedmann equation. It is fixed to zero in the base flat ΛCDM model and allowed to vary in curved extensions, which produce different correlated estimates for the other parameters. Algebraically Ω_{k} := 1 − Ω_{Λ} − Ω_{m} − Ω_{r}, so the four-symbol bracket is exactly one. The book keeps that identity and does not attach a curved-fit number to flat-fit values. Physics: derived within the displayed identity.", "cites": [ "r-3.2" ], "links_to_cards": [ "lcdm" ] }, { "id": "weakmix", "url": "https://www.teclamairme.com/#g-weakmix", "name": "the weak mixing angle", "symbol": "sin²θW", "symbol_markup": "sin²θ_{W}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "How the two neutral electroweak fields, W³ and B, are mixed to make the photon and the Z boson: 0.22305(23) on-shell (CODATA 2022), a number the Standard Model does not predict.", "more": "The Weinberg angle θ_{W} of the 1967 electroweak theory sets the ratio of the W and Z boson masses, cos θ_{W} = m_{W}/m_{Z}, and is the angle through which the theory’s two neutral fields, W^{3} and B, are rotated into the photon and the Z. CODATA 2022 gives sin²θ_{W} = 0.22305(23) in the on-shell scheme; the value depends on the scheme and the energy, so the formula quotes the on-shell value. Physics: measured.", "cites": [ "r-3.23", "r-3.5" ], "links_to_cards": [ "smodel" ] }, { "id": "ccproblem", "url": "https://www.teclamairme.com/#g-ccproblem", "name": "the cosmological constant problem", "symbol": "", "chapter": 3, "type": "open", "badge": "Status open", "quick": "The gap between the measured Λ and the value quantum field theory naively predicts: about 120 orders of magnitude, and unexplained.", "more": "In Planck units the measured cosmological constant is about 3 × 10⁻¹²², while a naive quantum-field estimate of the vacuum energy gives something of order 1. The mismatch of roughly 120 orders of magnitude has been called the worst theoretical prediction in the history of physics, a phrase attributed to Hobson, Efstathiou, and Lasenby’s General Relativity: An Introduction for Physicists (2006), p. 187. No accepted resolution exists: proposals range from modified gravity to anthropic selection in a landscape of vacua. Status: open.", "cites": [ "r-3.2" ], "links_to_cards": [ "Lambda" ] }, { "id": "hierarchy", "url": "https://www.teclamairme.com/#g-hierarchy", "name": "the hierarchy problem", "symbol": "", "chapter": 3, "type": "open", "badge": "Status open", "quick": "Why gravity is so much weaker than the other forces: between two electrons, electromagnetism beats gravity by about 10⁴², and no accepted theory says why.", "more": "Dividing the fine-structure constant by the gravitational coupling constant gives α/α_G ≈ 4 × 10⁴², the factor by which electromagnetism exceeds gravity between two electrons. The same disparity appears in the Standard Model as the gap between the electroweak scale and the Planck scale, where it takes the form of the Higgs-mass fine-tuning problem: quantum corrections should drive the Higgs mass up to the Planck scale unless something cancels them to many decimal places. Proposed answers include supersymmetry, large extra dimensions, and compositeness; none is established. Not to be confused with the fast-growing hierarchy of Chapter ᚂ(L). Status: open.", "cites": [ "r-3.12" ], "links_to_cards": [ "alpha", "alphaG", "fgh", "plancklen", "smodel" ] }, { "id": "si2019", "url": "https://www.teclamairme.com/#g-si2019", "name": "the 2019 redefinition of the SI", "symbol": "h, e, kʙ, Nₐ", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "The day the kilogram stopped being a lump of metal: four base units redefined by fixing exact values for four constants, effective 20 May 2019.", "more": "The 26th General Conference on Weights and Measures voted unanimously on 16 November 2018 to redefine four of the seven SI base units, and the change took effect on 20 May 2019, the 144th anniversary of the Metre Convention. The kilogram, ampere, kelvin, and mole are now defined by fixing exact numerical values for the Planck constant, the elementary charge, the Boltzmann constant, and the Avogadro constant; the second, meter, and candela were already defined by constants. This is what the blue entries in this chapter’s key mean: a constant marked as fixed by definition has no uncertainty because a committee gave it a number, and every quantity built only from such constants inherits that exactness. It is also why several entries here say “since 2019”—the same number can have been measured before that date and derived after it. An act of definition, uncolored.", "cites": [ "r-3.16" ], "links_to_cards": [ "NA", "echarge", "h", "kB" ] }, { "id": "dslight", "url": "https://www.teclamairme.com/#g-dslight", "name": "the interval along a light ray", "symbol": "ds²light", "symbol_markup": "ds²_{light}", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "In relativity the spacetime interval between two events on a light ray is exactly 0. In the formula it is added in as the zero it is.", "more": "Minkowski’s 1908 reformulation of special relativity gives spacetime a distance, ds² = c²dt² − dx² − dy² − dz², that all observers agree on. For two events joined by a light ray it is exactly 0, a null interval: in spacetime terms light goes nowhere and takes no time. Every observer agrees, which is another way of saying the speed of light is the same for everyone. Type: exactly 0, written as a law rather than a measurement.", "cites": [ "r-3.11", "r-3.6" ], "links_to_cards": [ "c" ] }, { "id": "divB", "url": "https://www.teclamairme.com/#g-divB", "name": "no magnetic monopoles", "symbol": "∇·𝐁", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "Maxwell’s second equation: in classical electromagnetism, with no magnetic monopoles, the divergence of the magnetic field is 0 everywhere. In the formula it is added in as the zero it is.", "more": "Gauss’s law for magnetism, ∇·𝐁 = 0, one of the four Maxwell equations assembled in 1861–65 and put into vector form by Heaviside in the 1880s. It says the magnetic field has no sources or sinks: field lines never begin or end, and there are no magnetic monopoles. Searches for monopoles continue, and every one so far has found nothing, which is the point. Type: exactly 0.", "cites": [ "r-3.19", "r-3.6" ], "links_to_cards": [ "maxwell" ] }, { "id": "noether", "url": "https://www.teclamairme.com/#g-noether", "name": "Noether’s conservation", "symbol": "dQ/dt", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "Under the theorem’s hypotheses, every continuous symmetry of an action gives a quantity whose rate of change is 0; energy, momentum, and charge are conserved for this reason. In the formula it is added in as the zero it is.", "more": "Emmy Noether’s work on the theorem began in 1915, when she was helping Felix Klein and David Hilbert with general relativity; the invariant-variation theorem was presented for her by Klein, presumably because she was not a member of the Göttingen society, and published in 1918. Under the theorem’s hypotheses, every differentiable symmetry of an action yields a conservation law: time translation gives energy, space translation momentum, and rotational symmetry angular momentum. The conserved quantity Q satisfies dQ/dt = 0. Type: exactly 0 within its stated scope.", "cites": [ "r-3.22", "r-3.6" ], "links_to_cards": [] }, { "id": "emc2", "url": "https://www.teclamairme.com/#g-emc2", "name": "rest energy", "symbol": "E − mc²", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "Einstein’s 1905 equation written as a zero: a body at rest has energy E = mc², so E − mc² = 0. In the formula it is added in as the zero it is.", "more": "In a paper received in September 1905 and published that November, Einstein showed that a body’s inertia depends on its energy content, the result now written E = mc²; for a body at rest the total energy is exactly mc², so E − mc² is exactly 0, and in motion E = γmc² with γ the Lorentz factor. It has been called the world’s most famous equation, and it explains the source of the Sun’s energy and the energy released by fission in nuclear weapons and nuclear power. Type: exactly 0, written as a law.", "cites": [ "r-3.19", "r-3.6" ], "links_to_cards": [ "lorentz" ] }, { "id": "bianchi", "url": "https://www.teclamairme.com/#g-bianchi", "name": "the contracted Bianchi identity", "symbol": "∇μGμν", "symbol_markup": "∇_{μ}G^{μν}", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "The Einstein tensor has zero divergence, identically, which underlies local energy-momentum conservation in general relativity. In the formula it is added in as the zero it is.", "more": "The Bianchi identity, which Luigi Bianchi published in 1902 and which Levi-Civita says Ricci had already found in about 1889, says the covariant derivatives of the Riemann curvature tensor cancel in a cyclic sum; contracted twice it gives ∇_{μ}G^{μν} = 0 for the Einstein tensor G^{μν} = R^{μν} − ½Rg^{μν} — and that contracted form, the one this entry is named for, had been published by Aurel Voss in 1880, twenty-two years before the identity got its name. Einstein’s field equations set G^{μν} + Λg^{μν} proportional to the stress–energy tensor, so the identity forces ∇_{μ}T^{μν} = 0: energy and momentum are conserved because geometry has no choice. Type: exactly 0, a law.", "cites": [ "r-3.6" ], "links_to_cards": [ "efe" ] }, { "id": "continuity", "url": "https://www.teclamairme.com/#g-continuity", "name": "conservation of charge", "symbol": "∂μjμ", "symbol_markup": "∂_{μ}j^{μ}", "chapter": 3, "type": "int", "badge": "Integer or rational", "quick": "The continuity equation: the four-divergence of the electric current is 0, so charge is never created or lost. In the formula it is added in as the zero it is.", "more": "∂_{μ}j^{μ} = ∂ρ/∂t + ∇·𝐉 = 0 says that charge leaving a region must flow out through its boundary. It follows from Maxwell’s equations of 1865 (take the divergence of the Ampère–Maxwell law) and, by Noether’s theorem, from the phase symmetry of electromagnetism. Type: exactly 0, a law.", "cites": [ "r-3.19", "r-3.6" ], "links_to_cards": [ "maxwell", "noether" ] }, { "id": "plancklen", "url": "https://www.teclamairme.com/#g-plancklen", "name": "the Planck length", "symbol": "ℓP", "symbol_markup": "ℓ_{P}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "√(ℏG/c³) = 1.616255(18) × 10⁻³⁵ m, a scale at which gravity and quantum mechanics are expected to meet; the gravity bracket is Λ times its square.", "more": "Max Planck noticed in 1899 that G, c, and his own new constant combine into one natural length; in the modern form, which uses ℏ where he used h, it is ℓ_{P} = √(ℏG/c³) = 1.616255(18) × 10^{−35} m (CODATA 2022), a scale at which quantum effects of gravity are expected to dominate. The gravity bracket ΛGℏ/c³ is exactly Λℓ_{P}², which is what “the cosmological constant in Planck units” means. Not in the formula; a key entry. Physics: derived, with the uncertainty of G.", "cites": [ "r-3.23", "r-3.7" ], "links_to_cards": [ "G", "Lambda", "c", "h", "hbar" ] }, { "id": "plancktime", "url": "https://www.teclamairme.com/#g-plancktime", "name": "the Planck time", "symbol": "tP", "symbol_markup": "t_{P}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "ℓ_P/c = 5.391247(60) × 10⁻⁴⁴ s, a scale at which quantum-gravity effects are expected to become essential.", "more": "The time light takes to cross a Planck length, t_{P} = √(ℏG/c⁵) = 5.391247(60) × 10^{−44} s. It is a natural scale built from G, ℏ, and c, not a proved smallest possible tick. Present theories indicate that quantum-gravity effects should become essential near this scale, but do not establish a literal boundary at which known physics stops. Physics: derived.", "cites": [ "r-3.7" ], "links_to_cards": [ "G", "c", "hbar", "plancklen" ] }, { "id": "planckmass", "url": "https://www.teclamairme.com/#g-planckmass", "name": "the Planck mass", "symbol": "mP", "symbol_markup": "m_{P}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "√(ℏc/G) = 2.176434(24) × 10⁻⁸ kg, about a flea’s egg; roughly the mass at which a particle’s Compton wavelength equals its Schwarzschild radius.", "more": "The Planck mass m_{P} = √(ℏc/G) = 2.176434(24) × 10^{−8} kg, about 22 micrograms, is the one Planck base unit on a human scale. It is the mass at which a particle’s quantum size and its black-hole size roughly coincide, and the ratio (m_{e}/m_{P})² is the gravitational coupling constant. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.21", "r-3.7" ], "links_to_cards": [ "G", "alphaG", "compton", "schwarzschild" ] }, { "id": "H0", "url": "https://www.teclamairme.com/#g-H0", "name": "the Hubble constant", "symbol": "H0", "symbol_markup": "H_{0}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The expansion rate of the universe today, 67.4 ± 0.5 km/s per megaparsec from the microwave background and 73.0 ± 1.0 from nearby supernovae; the gap is the Hubble tension.", "more": "Hubble’s 1929 law says galaxies recede at a speed proportional to their distance, v = H_{0}d. The Planck satellite’s 2018 fit to the microwave background gives H_{0} = 67.4 ± 0.5 km s^{−1} Mpc^{−1}; the SH0ES team’s 2022 distance ladder of Cepheids and supernovae gives 73.04 ± 1.04. The disagreement, about five standard deviations, is the Hubble tension, the most statistically significant tension in the standard cosmological model. Its reciprocal is about 14 billion years, the age scale of the universe. Not in the formula; a key entry. Physics: measured.", "cites": [ "r-3.20", "r-3.7" ], "links_to_cards": [ "age" ] }, { "id": "a0", "url": "https://www.teclamairme.com/#g-a0", "name": "the Bohr radius", "symbol": "a0", "symbol_markup": "a_{0}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "The radius of the hydrogen atom in Bohr’s 1913 model, 5.29177210544(82) × 10⁻¹¹ m, equal to ℏ/(α m_e c).", "more": "Bohr’s 1913 model of hydrogen puts the electron’s innermost orbit at a_{0} = 4πε₀ℏ²/(m_{e}e²) = ℏ/(αm_{e}c) = 5.29177210544(82) × 10^{−11} m (CODATA 2022), about half an ångström, and quantum mechanics keeps it as the most probable distance of the electron from the proton. It is the unit of length of atomic physics. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.18", "r-3.7" ], "links_to_cards": [ "c", "hbar" ] }, { "id": "compton", "url": "https://www.teclamairme.com/#g-compton", "name": "the Compton wavelength", "symbol": "λC", "symbol_markup": "λ_{C}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "h/(mec) = 2.42631023538(76) × 10⁻¹² m, the length scale at which an electron stops being a particle you can locate.", "quick_markup": "h/(m_{e}c) = 2.42631023538(76) × 10⁻¹² m, the length scale at which an electron stops being a particle you can locate.", "more": "Arthur Compton explained the scattering of X-rays by electrons in 1923 with a shift proportional to h/(m_{e}c); the constant, λ_{C} = 2.42631023538(76) × 10^{−12} m (CODATA 2022), is the wavelength of a photon with the electron’s rest energy, and the scale below which quantum field theory becomes essential. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.18", "r-3.7" ], "links_to_cards": [ "c", "emc2", "h" ] }, { "id": "re", "url": "https://www.teclamairme.com/#g-re", "name": "the classical electron radius", "symbol": "re", "symbol_markup": "r_{e}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "e²/(4πε₀mec²) = 2.8179403205(13) × 10⁻¹⁵ m, the size an electron would have if its mass were all electrostatic energy; it is α² times the Bohr radius.", "quick_markup": "e²/(4πε₀m_{e}c²) = 2.8179403205(13) × 10⁻¹⁵ m, the size an electron would have if its mass were all electrostatic energy; it is α² times the Bohr radius.", "more": "The classical electron radius, 2.8179403205(13) × 10^{−15} m (CODATA 2022), comes from setting the electrostatic energy of a charged sphere equal to m_{e}c²; the real electron is pointlike as far as experiment can tell, but r_{e} still sets the Thomson scattering cross-section. The three lengths of the electron nest by powers of α: r_{e} = αλ_{C}/2π = α²a₀. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.7" ], "links_to_cards": [ "a0", "alpha" ] }, { "id": "bohrmag", "url": "https://www.teclamairme.com/#g-bohrmag", "name": "the Bohr magneton", "symbol": "μB", "symbol_markup": "μ_{B}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "eℏ/(2me) = 9.2740100657(29) × 10⁻²⁴ J/T, the natural unit of magnetic moment; the electron’s is very slightly more than one of them.", "quick_markup": "eℏ/(2m_{e}) = 9.2740100657(29) × 10⁻²⁴ J/T, the natural unit of magnetic moment; the electron’s is very slightly more than one of them.", "more": "The Bohr magneton μ_{B} = eℏ/(2m_{e}), obtained by Procopiu and by Bohr in 1913 and named by Pauli in 1920, is the magnetic moment of an electron in the first Bohr orbit; CODATA 2022 gives 9.2740100657(29) × 10^{−24} J T^{−1}. The electron’s actual moment is (1 + a_{e}) of it, which is where the anomalous moment in this key comes from. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.18", "r-3.7" ], "links_to_cards": [ "a0" ] }, { "id": "fermiG", "url": "https://www.teclamairme.com/#g-fermiG", "name": "the Fermi coupling constant", "symbol": "GF", "symbol_markup": "G_{F}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "The strength of the weak nuclear force, GF/(ℏc)³ = 1.1663787(6) × 10⁻⁵ GeV⁻²; the weak force’s own coupling, which the formula leaves out in favor of the mixing angle.", "quick_markup": "The strength of the weak nuclear force, G_{F}/(ℏc)³ = 1.1663787(6) × 10⁻⁵ GeV⁻²; the weak force’s own coupling, which the formula leaves out in favor of the mixing angle.", "more": "Enrico Fermi’s 1934 theory of beta decay had one constant, the strength of the four-fermion contact interaction; in modern units G_{F}/(ℏc)³ = 1.1663787(6) × 10^{−5} GeV^{−2} (CODATA 2022), measured from the muon lifetime. It is the weak force’s answer to α and α_{s}, and it has units because the weak interaction is carried by heavy bosons. Not in the formula; a key entry. Physics: measured.", "cites": [ "r-3.19", "r-3.7" ], "links_to_cards": [ "alpha" ] }, { "id": "planckT", "url": "https://www.teclamairme.com/#g-planckT", "name": "the Planck temperature", "symbol": "TP", "symbol_markup": "T_{P}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "√(ℏc⁵/GkB²) = 1.416784(16) × 10³² K, the temperature at which the wavelength of thermal radiation reaches the Planck length.", "quick_markup": "√(ℏc⁵/Gk_{B}²) = 1.416784(16) × 10³² K, the temperature at which the wavelength of thermal radiation reaches the Planck length.", "more": "The Planck temperature, m_{P}c²/k_{B} = 1.416784(16) × 10^{32} K (CODATA 2022), is the temperature at which the wavelength of thermal radiation reaches the Planck length. Present theories are not expected to hold above it, though nothing proves a hard boundary there, and it is the temperature usually attached to the universe one Planck time after the beginning. It completes the set of Planck units in this key with the length, time, and mass. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.7" ], "links_to_cards": [ "plancktime" ] }, { "id": "age", "url": "https://www.teclamairme.com/#g-age", "name": "the age of the universe", "symbol": "t0", "symbol_markup": "t_{0}", "chapter": 3, "type": "pmeas", "badge": "Physics: measured", "quick": "13.787 ± 0.020 billion years since the beginning, from the Planck 2018 fit with baryon acoustic oscillations.", "more": "The age of the universe follows from H₀ and the density parameters through the Friedmann equation; the Planck 2018 fit gives t₀ = 13.787 ± 0.020 billion years once the microwave background is combined with baryon acoustic oscillations, and the oldest stars agree. It is the one number in this key with a date attached to everything. Not in the formula; a key entry. Physics: measured.", "cites": [ "r-3.18", "r-3.7" ], "links_to_cards": [ "H0" ] }, { "id": "lorentz", "url": "https://www.teclamairme.com/#g-lorentz", "name": "the Lorentz factor", "symbol": "γ", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "1/√(1 − v²/c²): how much time slows and mass-energy grows at speed v; equal to 1 at rest, which is why E − mc² is 0 there.", "more": "The Lorentz factor γ = 1/√(1 − v²/c²) appeared in Lorentz’s transformations of 1904 and Einstein’s special relativity of 1905; it multiplies time dilation, length contraction, and energy, E = γmc². At the LHC it is about 7000. Type: a function of the framework.", "cites": [ "r-3.21", "r-3.6" ], "links_to_cards": [ "emc2" ] }, { "id": "schwarzschild", "url": "https://www.teclamairme.com/#g-schwarzschild", "name": "the Schwarzschild radius", "symbol": "rs", "symbol_markup": "r_{s}", "chapter": 3, "type": "pder", "badge": "Physics: derived", "quick": "2GM/c², the radius a mass must be squeezed inside to become a black hole: 3 km for the Sun, 9 mm for the Earth.", "more": "Karl Schwarzschild solved Einstein’s field equations for a point mass in 1916, months after they were published and months before his death; the solution has a horizon at r_{s} = 2GM/c², about 3 km for the Sun and 9 mm for the Earth. The Planck mass is the mass whose Schwarzschild radius equals its Compton wavelength, up to a factor. Not in the formula; a key entry. Physics: derived.", "cites": [ "r-3.21" ], "links_to_cards": [ "compton", "efe", "planckmass" ] }, { "id": "efe", "url": "https://www.teclamairme.com/#g-efe", "name": "the Einstein field equations", "symbol": "Gμν + Λgμν = 8πG Tμν/c⁴", "symbol_markup": "G_{μν} + Λg_{μν} = 8πG T_{μν}/c⁴", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "General relativity in one line, 1915: curvature on the left, matter and energy on the right, and Λ, the cosmological constant of the gravity bracket, in between.", "more": "Einstein’s field equations of November 1915 equate the Einstein tensor, which measures curvature, to the stress–energy tensor, which measures matter and energy, with 8πG/c⁴ as the conversion; Λ was added in 1917. They are ten coupled nonlinear equations, they contain Newton’s law as a limit, and their contracted Bianchi identity is one of the zeros in the laws brace. Type: a law of the framework.", "cites": [ "r-3.19", "r-3.2", "r-3.6" ], "links_to_cards": [ "Lambda", "bianchi" ] }, { "id": "smodel", "url": "https://www.teclamairme.com/#g-smodel", "name": "the Standard Model", "symbol": "SU(3)×SU(2)×U(1)", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "The theory that owns three of the four forces, defined by its gauge group, and carrying 19 numbers it cannot predict—among them the gauge couplings behind this chapter’s strong coupling and weak mixing angle.", "more": "The Standard Model is the quantum field theory of the strong, weak, and electromagnetic interactions, and what defines it is a symmetry rather than a list of particles: the local gauge group SU(3) × SU(2) × U(1), whose three factors give the three forces. Writing down the most general Lagrangian with that symmetry leaves 19 numbers the theory does not fix—quark and lepton masses, mixing angles, the three couplings, the Higgs mass—and every one of them has to be measured. That is the sense in which the strong coupling and the weak mixing angle in the chapter’s particle-physics brace are measurements and not derivations. Gravity is not in it; neutrino mass is believed to need another seven or eight constants. A theory, uncolored.", "cites": [ "r-3.14" ], "links_to_cards": [] }, { "id": "lcdm", "url": "https://www.teclamairme.com/#g-lcdm", "name": "the ΛCDM model", "symbol": "ΛCDM", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "Six fitted numbers, a cosmological constant, and cold dark matter: the model that the chapter’s four density parameters are read out of, and the reason they may be added.", "more": "ΛCDM is the standard cosmological model—a cosmological constant Λ for dark energy, cold dark matter, and a flat, expanding universe. The Planck collaboration’s version is fitted with six parameters: the baryon density, the dark-matter density, the angular size of the sound horizon, the amplitude and the spectral index of the primordial fluctuations, and the reionization optical depth. Six is the smallest number that fits the data; everything else is either fixed at a natural value (total density 1, dark-energy equation of state −1) or calculated, including the Hubble constant and the dark-energy density themselves. This matters for the Ω bracket: the fit fixes the total density at 1, so the four density parameters add to 1 within that one fit. A model, uncolored.", "cites": [ "r-3.15" ], "links_to_cards": [ "H0", "Lambda" ] }, { "id": "qed", "url": "https://www.teclamairme.com/#g-qed", "name": "quantum electrodynamics", "symbol": "QED", "chapter": 3, "type": "fn", "badge": "Function or operator", "quick": "The quantum field theory of light and charged matter, whose prediction for the electron’s anomalous magnetic moment agrees with experiment to more than ten significant figures.", "more": "Quantum electrodynamics unites quantum mechanics with special relativity to describe how electrons, positrons, and photons interact; Feynman, Schwinger, and Tomonaga each developed it independently in the late 1940s and shared the 1965 Nobel Prize for it. Its central success is the electron’s anomalous magnetic moment, first computed by Schwinger in 1948 and now matched by measurement to more than ten significant figures, one of the most accurately verified predictions in the history of physics. A theory, uncolored.", "cites": [ "r-3.26" ], "links_to_cards": [ "ae" ] }, { "id": "taxicab", "url": "https://www.teclamairme.com/#g-taxicab", "name": "Hardy–Ramanujan taxicab number", "symbol": "1729", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The smallest number that is a sum of two positive cubes in two ways, 1³ + 12³ = 9³ + 10³.", "more": "Hardy took a taxi to visit Ramanujan, who was ill at Putney, and remarked that its number, 1729, seemed dull; Ramanujan replied that it is the smallest number expressible as a sum of two positive cubes in two different ways, 1³ + 12³ = 9³ + 10³. The story is Hardy’s own, from his 1940 book on Ramanujan. Written both ways and subtracted it is exactly 0, which is why it is the integers’ numerator. An integer, so untinted.", "cites": [ "r-4.2", "r-4.34" ], "links_to_cards": [ "ints" ] }, { "id": "squares", "url": "https://www.teclamairme.com/#g-squares", "name": "the first two square numbers after 1", "symbol": "4, 9", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The squares of 2 and 3, the smallest curiosities a reader can check by hand.", "more": "4 and 9 are 2² and 3², the first two square numbers after 1. They stand at the head of the small-numbers brace, behind only 0!, because they are the smallest curiosities a reader can check by hand.", "cites": [ "r-4.1" ], "links_to_cards": [] }, { "id": "fortytwo", "url": "https://www.teclamairme.com/#g-fortytwo", "name": "the Answer", "symbol": "42", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Deep Thought’s Answer to the Ultimate Question of Life, the Universe and Everything, useless because nobody ever knew the Question (Douglas Adams, The Hitchhiker’s Guide to the Galaxy, 1979).", "more": "“The Answer to the Great Question… Of Life, the Universe and Everything… Is… Forty-two,” said Deep Thought, with infinite majesty and calm. Douglas Adams, The Hitchhiker’s Guide to the Galaxy, 1979; the joke had already aired in the 1978 radio series. The Answer is useless because nobody knows the Question, and Adams said in 1993 that he simply wanted an ordinary, smallish number. It happens to be the Catalan number C₅ (counting from C₀: 1, 1, 2, 5, 14, 42), which counts the ways to bracket six factors, but it stands here for the reason that has nothing to do with mathematics. Type: integer.", "cites": [ "r-4.1", "r-4.3" ], "links_to_cards": [] }, { "id": "perfect", "url": "https://www.teclamairme.com/#g-perfect", "name": "the first four perfect numbers", "symbol": "496, 8128", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Each equals the sum of its proper divisors: 6, 28, 496, and 8128, all known to the ancient Greeks.", "more": "A perfect number equals the sum of its proper divisors: 6 = 1 + 2 + 3, then 28, 496, and 8128 are the first four, all known to the ancient Greeks; Euclid proved in Elements IX.36 that 2^(p−1)(2^p − 1) is perfect whenever 2^p − 1 is prime. Euler proved that every even perfect number has that form, in a paper published in 1849, long after his death. Whether an odd perfect number exists is still open; Sylvester called one “little short of a miracle” in 1888. Plain integers, untinted.", "cites": [ "r-4.3", "r-4.33" ], "links_to_cards": [] }, { "id": "cyclic", "url": "https://www.teclamairme.com/#g-cyclic", "name": "the cyclic number", "symbol": "142857", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The repeating period of 1/7; multiplying it by 1 through 6 permutes its digits.", "more": "142857 is the repeating block of 1/7 = 0.142857142857…; multiplying it by 2, 3, 4, 5, or 6 gives 285714, 428571, 571428, 714285, and 857142, the same digits rotated, and multiplying by 7 gives 999999. It is the smallest cyclic number, and it comes from 7 being a full-reptend prime in base 10. An integer.", "cites": [ "r-4.3" ], "links_to_cards": [] }, { "id": "kaprekar", "url": "https://www.teclamairme.com/#g-kaprekar", "name": "Kaprekar’s constant", "symbol": "6174", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Any four-digit number with at least two distinct digits reaches 6174 within seven steps of Kaprekar’s routine, if leading zeros are kept.", "more": "D. R. Kaprekar found in 1949 that if you take any four-digit number with at least two different digits, sort its digits descending and ascending, keeping any leading zeros, and subtract, then repeat, you reach 6174 within seven steps and stay there: 7641 − 1467 = 6174. An integer, and a fixed point.", "cites": [ "r-4.3", "r-4.31" ], "links_to_cards": [] }, { "id": "m31", "url": "https://www.teclamairme.com/#g-m31", "name": "the Mersenne prime 2³¹ − 1", "symbol": "M31", "symbol_markup": "M_{31}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "2147483647, Euler’s prime of 1772 and the largest signed 32-bit integer; the smallest giant by an absurd margin.", "more": "M₃₁ = 2³¹ − 1 = 2,147,483,647, proven prime by Euler in 1772 by trial division, and the largest known prime for 95 years, until Landry’s 3,203,431,780,337 in 1867. It is also the largest value of a signed 32-bit integer, 7FFFFFFF in hexadecimal, which is why old software fails at it and why 32-bit Unix time runs out in 2038. The smallest member of the GARGANTUAN brace by an absurd margin.", "cites": [ "r-4.4" ], "links_to_cards": [ "lkp" ] }, { "id": "sand", "url": "https://www.teclamairme.com/#g-sand", "name": "the Sand Reckoner", "symbol": "1063", "symbol_markup": "10^{63}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Archimedes’ count of grains of sand that would fill the universe, third century BC.", "more": "In The Sand Reckoner, third century BC, Archimedes set out to show that the number of grains of sand needed to fill the universe, as Aristarchus had sized it, was not infinite but nameable, and built a system of orders and periods to name it: about 10^{63}. It uses the same trick as the later giants, powers of powers. Type: integer.", "cites": [ "r-4.4" ], "links_to_cards": [] }, { "id": "muryo", "url": "https://www.teclamairme.com/#g-muryo", "name": "muryōtaisū", "symbol": "無量大数", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "10⁶⁸, the immeasurable large number, the top of the traditional Japanese number names.", "more": "無量大数, muryōtaisū, the immeasurable large number, is the largest of the traditional Japanese number names, fixed in the Jinkōki, Japan’s oldest arithmetic book, first printed in 1627 with many errors, corrected in 1631, and changed again in 1634. The editions disagree on whether the names climb by 10⁴ or 10⁸ above 10⁴⁸, which is why the value is 10⁶⁸ in the modern scheme and 10⁸⁸ in some editions; this key uses 10⁶⁸. Type: integer.", "cites": [ "r-4.32", "r-4.33", "r-4.4" ], "links_to_cards": [] }, { "id": "googol", "url": "https://www.teclamairme.com/#g-googol", "name": "googol", "symbol": "10100", "symbol_markup": "10^{100}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "10¹⁰⁰, a one followed by a hundred zeros; named by Kasner’s nine-year-old nephew in 1920 and published in 1940.", "more": "10¹⁰⁰, a one followed by a hundred zeros. Edward Kasner asked his nine-year-old nephew Milton Sirotta to name it in 1920 and published the name in Mathematics and the Imagination in 1940. Larger than the number of particles in the observable universe.", "cites": [ "r-4.31", "r-4.4" ], "links_to_cards": [] }, { "id": "googolplex", "url": "https://www.teclamairme.com/#g-googolplex", "name": "googolplex", "symbol": "1010100", "symbol_markup": "10^{10^{100}}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Ten to the googol, named at the same time in 1920; it has more digits than the universe has particles.", "more": "10 to the googol, named at the same time in 1920. Writing it out in full would need more digits than there are particles in the universe, so no one ever has; it is an integer in exactly the sense that 7 is one, and it exists just as firmly.", "cites": [ "r-4.31", "r-4.4" ], "links_to_cards": [ "googol" ] }, { "id": "skewes", "url": "https://www.teclamairme.com/#g-skewes", "name": "Skewes’ number", "symbol": "Skewes’", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The rounded form 10^(10^(10^34)) of Skewes’ 1933 bound e^(e^(e^79)) for the first place the prime-counting function π(x) exceeds the logarithmic integral li(x); the true crossing is now known to lie below about 1.4×10³¹⁶.", "more": "Stanley Skewes proved in 1933, assuming the Riemann Hypothesis, that the prime-counting function π(x) first exceeds the logarithmic integral li(x) somewhere below e^(e^(e^79)), usually rounded to 10^(10^(10^34)), which is the integer this brace uses; in 1955 he gave an unconditional bound of about 10^(10^(10^964)). G. H. Hardy called it the largest number that had ever served any definite purpose in mathematics. The true first crossing is now known to lie below about 1.397 × 10³¹⁶ (Bays and Hudson 2000, sharpened by Saouter and Demichel 2010) and above 10¹⁹ (Büthe 2015), but the original bound keeps its place in the folklore.", "cites": [ "r-4.34", "r-4.4" ], "links_to_cards": [ "li", "primecount", "rh" ] }, { "id": "moser", "url": "https://www.teclamairme.com/#g-moser", "name": "Moser’s number", "symbol": "Moser’s", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "A 2 inside a mega-gon in Steinhaus–Moser polygon notation.", "more": "Moser’s number is a 2 inside a mega-gon in the Steinhaus–Moser polygon notation, devised by Hugo Steinhaus and extended by Leo Moser, at a date we could not confirm: a number in a triangle means n^n, in a square means n inside n triangles, and so on up. Vastly larger than Skewes’ number, and still smaller than Graham’s.", "cites": [ "r-4.4" ], "links_to_cards": [ "skewes" ] }, { "id": "g64", "url": "https://www.teclamairme.com/#g-g64", "name": "Graham’s number", "symbol": "G64", "symbol_markup": "G_{64}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Ronald Graham’s bound for a problem in Ramsey theory, which Martin Gardner published in 1977 as the largest number ever used in a serious proof.", "more": "Graham’s number bounds a problem in Ramsey theory about coloring the edges of a hypercube. The bound Ronald Graham and Bruce Rothschild published in 1971 is smaller; the number that carries Graham’s name is the larger and simpler one he gave Martin Gardner, who put it in Scientific American in November 1977 as the largest number ever used in a serious proof. It is g₆₄, where g₁ = 3↑↑↑↑3 and each gₙ is 3 with gₙ₋₁ arrows 3, so it sits below f_{ω+1}(64) in the fast-growing hierarchy and between 3→3→64→2 and 3→3→65→2 in Conway’s chains. The answer to the original problem is now known to lie between 13 (Barkley, 2008) and a small tower of twos, 2↑↑(2↑↑(3 + 2↑↑8)) (Lavrov, Lee, and Mackey, 2014, refined in 2019). Its last ten digits, …2464195387, are known; its first are not.", "cites": [ "r-4.5" ], "links_to_cards": [ "chain", "fgh" ] }, { "id": "tree3", "url": "https://www.teclamairme.com/#g-tree3", "name": "TREE(3)", "symbol": "TREE(3)", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "From Kruskal’s tree theorem of 1960 and Friedman’s TREE function of the 2000s; incomparably larger than Graham’s number.", "more": "TREE(3) comes from Kruskal’s tree theorem (1960) and Harvey Friedman’s finite version: every sequence of trees on three labels, the i-th having at most i vertices, in which no earlier tree embeds in a later one must end, and the length of the longest such sequence is TREE(3), a number so large that Graham’s number is negligible beside it; in the fast-growing hierarchy it lies beyond f_{Γ₀}. It is a definite finite integer. It is used twice on purpose: here on the ladder of giants, and in Chapter ᚃ(F) beside Rayo’s number.", "cites": [ "r-4.34", "r-4.35", "r-4.5" ], "links_to_cards": [ "fgh", "g64", "kruskal", "rayo" ] }, { "id": "belphegor", "url": "https://www.teclamairme.com/#g-belphegor", "name": "Belphegor’s prime", "symbol": "Belphegor’s", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "1, thirteen zeros, 666, thirteen zeros, 1: a 31-digit palindromic prime, written as a name because the digits wreck the line.", "more": "Belphegor’s prime is 1000000000000066600000000000001: a 1, thirteen zeros, 666, thirteen zeros, and a 1, a 31-digit palindromic prime named by Clifford Pickover after a demon, for the 666 and the two thirteens. It is written in the formula as a name because its digits would wreck the line, and it stands among the small famous integers as the piece’s one deliberate superstition.", "cites": [ "r-4.3" ], "links_to_cards": [] }, { "id": "twenty", "url": "https://www.teclamairme.com/#g-twenty", "name": "the Mayan base", "symbol": "20", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Vigesimal, fingers and toes.", "more": "The Maya, like several Mesoamerican cultures, counted in base twenty, fingers and toes together; their numerals used dots for ones, bars for fives, and a shell for zero, stacked in powers of twenty (with one calendar-driven exception at the third place). It lives with the small integers, where a base belongs. Type: integer.", "cites": [ "r-2.4" ], "links_to_cards": [] }, { "id": "sixty", "url": "https://www.teclamairme.com/#g-sixty", "name": "the Babylonian base", "symbol": "60", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Sexagesimal, the reason there are 60 minutes and 360 degrees.", "more": "The Sumerians were counting in base sixty in the third millennium BC, and the Babylonians inherited it; sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, which makes fractions easy. It survives in the sixty minutes of an hour, the sixty seconds of a minute, and the 360 degrees of a circle. It lives with the small integers, where a base belongs. Type: integer.", "cites": [ "r-2.3", "r-4.1" ], "links_to_cards": [] }, { "id": "amicable", "url": "https://www.teclamairme.com/#g-amicable", "name": "the first amicable pair", "symbol": "220 + 284", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Each is the sum of the other’s proper divisors: 220 → 284 and 284 → 220, known to the Pythagoreans.", "more": "220 and 284 are the smallest amicable pair: the proper divisors of 220 add to 284 and those of 284 add to 220. The Pythagoreans knew them and credited them with mystical properties; Fermat and Descartes found further pairs, rediscoveries of pairs known to Arab mathematicians, and Euler raised the count to about 60; the second smallest pair, 1184 and 1210, was missed by all of them and found by the sixteen-year-old Nicolò Paganini and published in the Turin Academy’s proceedings for 1866–67. Plain integers.", "cites": [ "r-4.29", "r-4.3" ], "links_to_cards": [] }, { "id": "carmichael", "url": "https://www.teclamairme.com/#g-carmichael", "name": "the first Carmichael number", "symbol": "561", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "3 · 11 · 17, a composite that passes Fermat’s primality test for every base coprime to it, the smallest such liar.", "more": "561 = 3 · 11 · 17 is the smallest Carmichael number: a composite n with b^n ≡ b (mod n) for every integer b, so Fermat’s little theorem from Chapter ᚄ(S) cannot expose it. Václav Šimerka found the first seven in 1885 in a Czech journal, where his work went unnoticed, Alwin Korselt gave the criterion in 1899 without an example, and Robert Carmichael published 561 in 1910 and got the name. Alford, Granville, and Pomerance proved in 1994 that there are infinitely many. An integer.", "cites": [ "r-4.16" ], "links_to_cards": [ "fermatlittle" ] }, { "id": "fib144", "url": "https://www.teclamairme.com/#g-fib144", "name": "the only square Fibonacci number", "symbol": "F12 = 144", "symbol_markup": "F_{12} = 144", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The twelfth Fibonacci number, 144 = 12², and apart from 0 and 1 the only Fibonacci number that is a perfect square (Cohn, 1964).", "more": "The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, … are Leonardo of Pisa’s rabbits of 1202, each the sum of the two before, and their ratios tend to the golden ratio φ of the algebraic column. F₁₂ = 144 is, apart from 0 and 1, the only perfect square among them, proved by J. H. E. Cohn in 1964; 144 is also the square of its own index, F₁₂ = 12². The notation Fₙ is the one the reader will meet everywhere. An integer.", "cites": [ "r-4.17" ], "links_to_cards": [ "phi" ] }, { "id": "fact7", "url": "https://www.teclamairme.com/#g-fact7", "name": "seven factorial", "symbol": "7! = 5040", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "1 · 2 · 3 · 4 · 5 · 6 · 7 = 5040, the factorial notation’s small famous value, Plato’s number of citizens in the Laws.", "more": "n! is the product of the integers from 1 to n, the notation Christian Kramp introduced in 1808, and 7! = 5040 is its most famous small value: Plato chose 5040 in the Laws as the ideal number of citizens because it divides so many ways—by every number from 1 to 12 except 11, and by 60 numbers in all — and it is the largest factorial known to be one less than a square (5040 + 1 = 71²); whether any larger factorial is one less than a square is Brocard’s problem, and that is open. Wilson’s theorem in Chapter ᚄ(S) is a factorial too. An integer.", "cites": [ "r-4.1", "r-4.24" ], "links_to_cards": [ "permfact", "wilson" ] }, { "id": "primorial5", "url": "https://www.teclamairme.com/#g-primorial5", "name": "the primorial of five", "symbol": "5# = 30", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "2 · 3 · 5 = 30, the product of the primes up to 5, in the primorial notation p#.", "more": "The primorial n#, a name coined by Harvey Dubner in 1987, is the product of all primes up to n, so 5# = 2 · 3 · 5 = 30; primorials are what Euclid’s proof of the infinitude of primes multiplies before adding one. 30 is also the largest number all of whose smaller coprimes other than 1 are prime. An integer.", "cites": [ "r-4.1", "r-4.24" ], "links_to_cards": [] }, { "id": "fermat4", "url": "https://www.teclamairme.com/#g-fermat4", "name": "the largest known Fermat prime", "symbol": "65537", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "2¹⁶ + 1, the fifth Fermat prime and the largest known; by Gauss’s criterion the 65537-gon can be drawn with compass and straightedge.", "more": "65537 = 2¹⁶ + 1 = F₄ is the largest known prime of Fermat’s form 2^(2ⁿ) + 1; Fermat conjectured in 1640 that all such numbers are prime, and Euler broke F₅ = 4294967297 = 641 · 6700417 in 1732. Gauss proved in the Disquisitiones Arithmeticae of 1801 that a regular polygon with a Fermat-prime number of sides can be drawn, and stated without proof that no other prime-sided one can, which Wantzel proved in 1837; so the 65537-gon can be drawn, and Johann Gustav Hermes spent ten years writing out the construction, finished in 1894. It is also the standard public exponent in RSA. An integer.", "cites": [ "r-4.30", "r-4.6" ], "links_to_cards": [] }, { "id": "knuth3", "url": "https://www.teclamairme.com/#g-knuth3", "name": "three tetrated to three, in Knuth’s arrows", "symbol": "3↑↑3", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "3^(3^3) = 3²⁷ = 7,625,597,484,987: the smallest giant written in Knuth’s up-arrow notation of 1976.", "more": "Donald Knuth introduced the up-arrow notation in 1976, in Science, to write the hyperoperations: a↑b is a^b, a↑↑b is a tower of b copies of a, a↑↑↑b iterates towers, and so on. 3↑↑3 = 3^(3^3) = 3²⁷ = 7,625,597,484,987 is the first value that leaves the calculator, and 3↑↑4 = 3^7625597484987 has about 3.6 trillion digits. The notation is what Graham’s number is written in. An integer.", "cites": [ "r-4.31", "r-4.5" ], "links_to_cards": [ "g64", "hyperop", "knuth34", "uparrow" ] }, { "id": "eddington", "url": "https://www.teclamairme.com/#g-eddington", "name": "the Eddington number", "symbol": "136·2256", "symbol_markup": "136·2^{256}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Arthur Eddington’s 1938 count of the protons in the observable universe, 136 × 2²⁵⁶ ≈ 1.57 × 10⁷⁹.", "more": "Arthur Eddington claimed in his Tarner lectures of 1938, published in 1939, to have derived the number of protons in the universe from pure theory: 136 × 2²⁵⁶ ≈ 1.575 × 10⁷⁹, with, he said, the same number of electrons. The derivation is not accepted, but the number is the classic estimate of how many particles there are, about 10⁸⁰, and it marks the point on this ladder where numbers stop counting anything physical. An integer.", "cites": [ "r-4.30", "r-4.4" ], "links_to_cards": [] }, { "id": "shannon", "url": "https://www.teclamairme.com/#g-shannon", "name": "the Shannon number", "symbol": "10120", "symbol_markup": "10^{120}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Claude Shannon’s 1950 lower bound for the number of possible chess games, 10¹²⁰.", "more": "In his 1950 paper Programming a Computer for Playing Chess, Claude Shannon estimated the game tree of chess at about 10¹²⁰ games, from roughly a thousand choices per pair of moves over forty pairs, to show that brute force could never play chess. It is larger than the googol and vastly larger than the number of particles, and it is the reason chess programs search rather than enumerate. An integer.", "cites": [ "r-4.34", "r-4.4" ], "links_to_cards": [ "googol" ] }, { "id": "lkp", "url": "https://www.teclamairme.com/#g-lkp", "name": "the largest known prime", "symbol": "M136279841", "symbol_markup": "M_{136279841}", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "2¹³⁶²⁷⁹⁸⁴¹ − 1, found on 12 October 2024 by Luke Durant’s GPU search for GIMPS; 41,024,320 digits.", "more": "M₁₃₆₂₇₉₈₄₁ = 2¹³⁶²⁷⁹⁸⁴¹ − 1 is the largest prime anyone has found, discovered on 12 October 2024 by Luke Durant using a cloud of graphics processors for the Great Internet Mersenne Prime Search; it has 41,024,320 decimal digits and is the 52nd known Mersenne prime. Euclid proved there is no largest prime, so this item is a record, not a limit; it will be replaced. An integer.", "cites": [ "r-4.18" ], "links_to_cards": [] }, { "id": "sscg3", "url": "https://www.teclamairme.com/#g-sscg3", "name": "Friedman’s SSCG(3)", "symbol": "SSCG(3)", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The longest sequence of simple subcubic graphs with no earlier one embedded in a later one; finite by Robertson–Seymour, and larger than TREE(3).", "more": "Harvey Friedman’s SSCG(k) is the length of the longest sequence of simple subcubic graphs G₁, G₂, … with Gᵢ on at most i + k vertices and no earlier graph homeomorphically embeddable in a later one; the Robertson–Seymour theorem says every such sequence ends, so the number exists. SSCG(0) = 2, SSCG(1) = 5, SSCG(2) = 3·2^(3·2⁹⁵) − 8, and SSCG(3) is larger than TREE(3). It is the last of the giants in the brace. An integer.", "cites": [ "r-4.5" ], "links_to_cards": [ "tree3" ] }, { "id": "zerofact", "url": "https://www.teclamairme.com/#g-zerofact", "name": "zero factorial", "symbol": "0!", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "1: the empty product. There is exactly one way to arrange nothing, and the convention makes the binomial theorem and every Taylor series come out right.", "more": "The factorial n! counts the orderings of n things, and the recursion n! = n·(n−1)! forces 0! = 1, the empty product, in the same way that an empty sum is 0. It is the value that makes C(n, 0) = 1, e^{x} = Σx^{n}/n! start correctly, and Γ(1) = 1 agree with the factorial. An integer.", "cites": [ "r-4.1" ], "links_to_cards": [ "binomthm", "permfact" ] }, { "id": "lookandsay", "url": "https://www.teclamairme.com/#g-lookandsay", "name": "the look-and-say sequence", "symbol": "1, 11, 21, 1211, 111221, ...", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "The integer sequence built by reading the previous term aloud, run by run: 1, 11, 21, 1211, 111221, and onward.", "more": "Starting from 1, each term of the look-and-say sequence describes the digit runs of the term before it: one 1 gives 11, then 11, read as two 1s, gives 21, and so on. John Conway analyzed the sequence in a 1986 paper, The Weird and Wonderful Chemistry of Audioactive Decay, and showed that the length of the n-th term grows like Conway’s constant, about 1.3036, raised to the n. The digit sequence itself, distinct from that growth-rate constant. An integer sequence, uncolored.", "cites": [ "r-4.49" ], "links_to_cards": [ "conway" ] }, { "id": "third", "url": "https://www.teclamairme.com/#g-third", "name": "one third", "symbol": "⅓", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Three of them make exactly one in the rationals, with no decimal expansion needed.", "more": "One third is the rational number 1/3. Its decimal expansion never ends, but it does not need one: three of them make exactly 1 in the rationals by the definition of a fraction, so 3 · ⅓ − 1 is exactly 0. Integer or rational, so untinted.", "cites": [ "r-4.1" ], "links_to_cards": [] }, { "id": "nines", "url": "https://www.teclamairme.com/#g-nines", "name": "zero point nine repeating", "symbol": "0.999…", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Exactly equal to 1, not approximately: let x = 0.999…, then 10x − x = 9.", "more": "0.999… is the rational number 1 written with an infinite decimal, in the same way that 0.5 and ½ are one number written twice. Let x = 0.999…; then 10x − x = 9.999… − 0.999… = 9, so x = 1, and the rigorous version says the two decimals have the same limit. The confusion is old and the answer is not up for debate, which is why 0.999… − 1 belongs beside 3 · ⅓ − 1 as a zero of the rationals.", "cites": [ "r-4.1" ], "links_to_cards": [] }, { "id": "piapprox", "url": "https://www.teclamairme.com/#g-piapprox", "name": "rational approximations of π", "symbol": "22/7, 355/113", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Archimedes’ upper bound and Zu Chongzhi’s fraction, which is correct to six decimal places.", "more": "22/7 is Archimedes’ upper bound for π from about 250 BC, obtained by trapping the circle between polygons of 96 sides; 355/113 is Zu Chongzhi’s fraction from the fifth century AD, correct to six decimal places and not bettered by any fraction with a denominator under 16,604. Both are rationals, which is why they lead the denominator of the rationals column, and their sum is positive.", "cites": [ "r-4.33", "r-4.7" ], "links_to_cards": [] }, { "id": "bern2", "url": "https://www.teclamairme.com/#g-bern2", "name": "the Bernoulli number B₂", "symbol": "B2 = ⅙", "symbol_markup": "B_{2} = ⅙", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "B₂ is ⅙, and the Bernoulli numbers 1, −½, ⅙, 0, −1/30, … are the rationals inside every sum of powers and every even zeta value.", "more": "The Bernoulli numbers Bₙ are the rational coefficients that make Σ kᵐ a polynomial; Jacob Bernoulli published them in the Ars Conjectandi of 1713 and Seki Takakazu independently a year earlier. B₂ = ⅙, B₄ = −1/30, B₆ = 1/42, and every odd one after B₁ is 0. They give ζ(2n) in Chapter ᚑ(O), and their numerators decide which primes are regular in Kummer’s attack on Fermat. This B₂ is the Bernoulli number, not Brun’s constant B₂ of Chapter ᚃ(F). A rational.", "cites": [ "r-4.29", "r-4.7" ], "links_to_cards": [ "bernoulli", "brun" ] }, { "id": "h4", "url": "https://www.teclamairme.com/#g-h4", "name": "the fourth harmonic number", "symbol": "H4 = 25/12", "symbol_markup": "H_{4} = 25/12", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "1 + ½ + ⅓ + ¼ = 25/12; no harmonic number after the first is a whole number (Theisinger, 1915).", "more": "The harmonic numbers Hₙ = 1 + ½ + … + 1/n grow like ln n plus Euler’s γ from Chapter ᚃ(F), and never come out whole again: Leopold Theisinger proved in 1915 that Hₙ is an integer only for n = 1. H₄ = 25/12 is the first one whose denominator is not a factorial, a hint that the harmonic numbers keep their own arithmetic. A rational.", "cites": [ "r-4.31", "r-4.7" ], "links_to_cards": [ "gamma", "harmonicnum", "permfact" ] }, { "id": "contfrac", "url": "https://www.teclamairme.com/#g-contfrac", "name": "continued fractions and convergents", "symbol": "[a₀; a₁, a₂, …]", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Peel off the integer part, invert the remainder, repeat. The truncations are the convergents, giving a sequence of rational approximations.", "more": "A simple continued fraction writes a number as a₀ + 1/(a₁ + 1/(a₂ + ⋯)) with integer partial quotients, obtained by taking the integer part, inverting what is left, and repeating. Truncating gives the convergents, a systematic sequence of rational approximations. Two famous convergents of π are 22/7 and 355/113. Counting the initial 3 as the first, the first four convergents of π = [3;7, 15, 1,…] are 3, 22/7, 333/106, and 355/113. The process stops if and only if the number is rational, so an expansion that never stops is a proof of irrationality. A tool of the framework, uncolored.", "cites": [ "r-4.26" ], "links_to_cards": [ "pi" ] }, { "id": "bernoulli", "url": "https://www.teclamairme.com/#g-bernoulli", "name": "Bernoulli numbers", "symbol": "Bₙ", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "A sequence of rational numbers, 1, −½, ⅙, 0, −1/30, ..., that appear in every closed formula for a sum of integer powers and every even value of the Riemann zeta function.", "more": "The Bernoulli numbers Bn are the rational coefficients in the expansion x/(e^x − 1) = Σ Bn x^n/n!; Jacob Bernoulli published them in the Ars Conjectandi of 1713, and Seki Takakazu had reached the same numbers in Japan a year earlier. They give closed formulas for 1^k + 2^k + ... + n^k for any power k, and ζ(2n) is always a rational multiple of π^(2n) built from them. B₂ = 1/6 has a card of its own in this chapter. General sequence, uncolored.", "cites": [ "r-4.41" ], "links_to_cards": [ "rats", "zeta" ] }, { "id": "harmonicnum", "url": "https://www.teclamairme.com/#g-harmonicnum", "name": "harmonic numbers", "symbol": "Hₙ = Σ 1/k", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "The partial sums of the harmonic series, 1, 3/2, 11/6, 25/12, ..., always rational and, after the first, never a whole number.", "more": "Hn = 1 + 1/2 + 1/3 + ... + 1/n is always rational, and Leopold Theisinger proved in 1915 that it is an integer only when n = 1: from the second term on, the denominators never fully cancel. H₄ = 25/12 has a card of its own in this chapter, and the harmonic numbers grow like ln n, which is why the harmonic series itself diverges however slowly. A general sequence, uncolored.", "cites": [ "r-4.44" ], "links_to_cards": [] }, { "id": "phi", "url": "https://www.teclamairme.com/#g-phi", "name": "golden ratio", "symbol": "φ", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "The positive root of x² − x − 1, about 1.618.", "more": "The golden ratio φ = (1 + √5)/2 ≈ 1.618, the positive root of x² − x − 1, defined by Euclid as the division of a line so that the whole is to the larger part as the larger is to the smaller—a line cut in extreme and mean ratio, which is the first known definition of it. It is algebraic of degree 2 and constructible with compass and straightedge, since square roots are.", "cites": [ "r-4.1", "r-4.24" ], "links_to_cards": [ "constructible" ] }, { "id": "sqrt2", "url": "https://www.teclamairme.com/#g-sqrt2", "name": "square root of two", "symbol": "√2", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "The Pythagoreans’ scandal, by tradition the first number shown not to be a fraction.", "more": "√2 ≈ 1.4142 is the diagonal of a unit square. Its irrationality is credited by tradition to the Pythagoreans in the fifth century BC, probably the first number shown not to be a fraction; no original proof survives, and the reference works that report the tradition also record that neither the discoverer nor the date can be fixed. It is algebraic of degree 2 and constructible, and it appears in the formula twice: once among the constructible numbers of its own column, and once in the transcendental column’s numerator, as the number geometry can build.", "cites": [ "r-4.8", "r-8.9" ], "links_to_cards": [ "constructible" ] }, { "id": "plastic", "url": "https://www.teclamairme.com/#g-plastic", "name": "plastic number", "symbol": "ρ", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "The real root of x³ = x + 1, about 1.3247, van der Laan’s ratio.", "more": "The plastic number ρ ≈ 1.3247 is the real root of x³ = x + 1, found by Gérard Cordonnier in 1924 and by the architect Hans van der Laan, who came to it in 1928; van der Laan, who named it the plastic number only from the 1950s, saw it as a three-dimensional cousin of the golden ratio. It is algebraic of degree 3, and because 3 is not a power of 2 it cannot be built with compass and straightedge.", "cites": [ "r-4.33", "r-4.8" ], "links_to_cards": [ "phi" ] }, { "id": "silver", "url": "https://www.teclamairme.com/#g-silver", "name": "silver ratio", "symbol": "δS", "symbol_markup": "δ_{S}", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "1 + √2, the positive root of x² − 2x − 1, about 2.414.", "more": "The silver ratio 1 + √2 ≈ 2.414 is the positive root of x² − 2x − 1 and the limit of ratios of consecutive Pell numbers, as φ is for Fibonacci. Algebraic of degree 2 and constructible.", "cites": [ "r-4.1" ], "links_to_cards": [ "constructible", "sqrt2" ] }, { "id": "conway", "url": "https://www.teclamairme.com/#g-conway", "name": "Conway’s constant", "symbol": "Conway’s", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "The growth rate of the look-and-say sequence, about 1.3036, algebraic of degree 71.", "more": "Conway’s constant λ ≈ 1.3036 is the rate at which the look-and-say sequence 1, 11, 21, 1211, 111221, … grows in length. John Conway showed in 1986, in a paper reprinted in 1987, that it is the largest root of a specific polynomial of degree 71, so it is algebraic, and since 71 is not a power of 2 it is not constructible. The surprise is that a game about reading digits aloud has an algebraic answer at all.", "cites": [ "r-4.29", "r-4.8" ], "links_to_cards": [ "constructible", "lookandsay" ] }, { "id": "cbrt2", "url": "https://www.teclamairme.com/#g-cbrt2", "name": "cube root of two", "symbol": "∛2", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "The classical non-constructible number: doubling the cube with compass and straightedge is impossible (Wantzel 1837).", "more": "∛2 ≈ 1.2599 is the edge of a cube with twice the volume of the unit cube. The Delian problem, doubling the cube with compass and straightedge, was open from the Greeks until Pierre Wantzel proved in 1837 that it is impossible, because ∛2 is algebraic of degree 3 and a constructible number must have degree a power of 2. It is the exponent on the transcendental numerator’s little factory precisely because geometry cannot build it.", "cites": [ "r-4.29", "r-4.8" ], "links_to_cards": [ "constructible", "nonconstructible" ] }, { "id": "phipoly", "url": "https://www.teclamairme.com/#g-phipoly", "name": "the golden ratio’s own equation", "symbol": "φ² − φ − 1", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "φ is defined as the positive root of x² − x − 1, so the polynomial vanishes at φ exactly.", "more": "φ² − φ − 1 is the golden ratio’s defining polynomial evaluated at the golden ratio, and so it is exactly 0: φ is by definition the positive root of x² − x − 1. Every algebraic number has such an equation, and this is the algebraic column’s way of saying nothing. Its value is the integer 0.", "cites": [ "r-4.1" ], "links_to_cards": [ "phi" ] }, { "id": "cos17", "url": "https://www.teclamairme.com/#g-cos17", "name": "the heptadecagon’s cosine", "symbol": "cos(2π/17)", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 0.9325, algebraic of degree 8 and constructible: Gauss found in 1796 that the regular 17-gon can be drawn.", "more": "cos(2π/17) ≈ 0.93247 is a root of a degree-8 polynomial and is reached by three successive square roots, which is why Gauss could show on 30 March 1796 that the regular 17-gon is constructible with compass and straightedge, the first such polygon found since the Greeks; a story, probably apocryphal, has Gauss asking for the 17-gon on his gravestone. Its expression in nested square roots fills a line. Algebraic, constructible.", "cites": [ "r-4.8" ], "links_to_cards": [ "constructible" ] }, { "id": "cos7", "url": "https://www.teclamairme.com/#g-cos7", "name": "the heptagon’s cosine", "symbol": "2cos(2π/7)", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.2470, a root of x³ + x² − 2x − 1: degree 3, so the regular heptagon cannot be drawn with compass and straightedge.", "more": "2cos(2π/7) ≈ 1.24698 is a root of the irreducible cubic x³ + x² − 2x − 1, and a constructible number must have degree a power of 2, so the regular heptagon is the smallest polygon that compass and straightedge cannot draw; Gauss knew which polygons could be drawn in 1801, and Wantzel proved in 1837 that the others cannot. Algebraic, not constructible.", "cites": [ "r-4.29", "r-4.8" ], "links_to_cards": [ "constructible" ] }, { "id": "tribonacci", "url": "https://www.teclamairme.com/#g-tribonacci", "name": "the tribonacci constant", "symbol": "τ3", "symbol_markup": "τ_{3}", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.8393, the real root of x³ = x² + x + 1, the growth rate of the sequence in which each term is the sum of the three before.", "more": "The tribonacci constant ≈ 1.83929 is the real root of x³ − x² − x − 1, the ratio that consecutive tribonacci numbers 1, 1, 2, 4, 7, 13, 24, … approach, as the golden ratio is for Fibonacci. The word was coined in 1963 by Mark Feinberg, a fourteen-year-old, in the Fibonacci Quarterly; the sequence itself turns up in Darwin’s elephant calculation in On the Origin of Species and in an 1892 puzzle about sheep. Algebraic of degree 3, so not constructible, and the tetranacci constant 1.9276 follows it.", "cites": [ "r-4.35", "r-4.8" ], "links_to_cards": [ "constructible", "phi" ] }, { "id": "lehmer", "url": "https://www.teclamairme.com/#g-lehmer", "name": "Lehmer’s number", "symbol": "λL", "symbol_markup": "λ_{L}", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.17628, the largest root of Lehmer’s degree-10 polynomial and the smallest known Salem number; whether anything smaller exists is Lehmer’s open question.", "more": "Lehmer’s number ≈ 1.176280818 is the largest real root of x¹⁰ + x⁹ − x⁷ − x⁶ − x⁵ − x⁴ − x³ + x + 1, found by Derrick Lehmer in 1933. It is the smallest known Salem number and the smallest known Mahler measure greater than 1, and Lehmer’s conjecture, still open, says nothing smaller exists; the same number appears in the Coxeter group E₁₀ and in knot theory. Algebraic of degree 10, not constructible.", "cites": [ "r-4.32", "r-4.8" ], "links_to_cards": [ "constructible", "salem" ] }, { "id": "salem", "url": "https://www.teclamairme.com/#g-salem", "name": "a Salem number", "symbol": "θ > 1, |θⁱ| ≤ 1", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "An algebraic integer above 1 whose other conjugates crowd into the closed unit disc with at least one exactly on the circle; Lehmer’s 1.1762808 is the smallest known.", "more": "A Salem number is a real algebraic integer greater than 1 all of whose other conjugates lie in the closed unit disc, with at least one of them on the unit circle itself. That one root on the circle forces the minimal polynomial to be reciprocal, so 1/θ is a root too, the degree is even and at least 4, and θ is a unit. They sit just beside the Pisot numbers, where every other conjugate is strictly inside. The smallest known one is the largest real root of Lehmer’s degree-10 polynomial, about 1.1762808, which is also the smallest known Mahler measure above 1; whether a smaller Salem number exists is Lehmer’s problem and is still open. A class of numbers, uncolored.", "cites": [ "r-4.27" ], "links_to_cards": [] }, { "id": "classnum", "url": "https://www.teclamairme.com/#g-classnum", "name": "the class number", "symbol": "h(ℚ(√−d))", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "How far a ring of integers is from unique factorization. It is 1 for exactly nine imaginary quadratic fields, the last of them ℚ(√−163).", "more": "The class number counts the ideal classes of a number field, and it is 1 exactly when the ring of integers has unique factorization. For imaginary quadratic fields ℚ(√−d) the answer is startlingly short: by the Stark–Heegner theorem there are precisely nine values of d with class number 1—1, 2, 3, 7, 11, 19, 43, 67, and 163. Gauss conjectured the list, Kurt Heegner proved it in 1952 with a gap that was later judged minor, and Alan Baker and Harold Stark proved it independently in 1966. The last entry is why this chapter’s most famous almost-integer is almost an integer: class number 1 makes the j-invariant at that point an exact integer, which is why e^(π√163) misses an integer so narrowly. An integer invariant, uncolored.", "cites": [ "r-4.28" ], "links_to_cards": [ "jinv", "ram", "ufd" ] }, { "id": "lehmermahler", "url": "https://www.teclamairme.com/#g-lehmermahler", "name": "Lehmer’s Mahler measure problem", "symbol": "M(θ) < 1.176…?", "chapter": 4, "type": "open", "badge": "Status open", "quick": "The open question of whether any algebraic integer has Mahler measure strictly between 1 and Lehmer’s number, about 1.1762808, the smallest value ever found above 1.", "more": "Derrick Lehmer asked in 1933 whether the Mahler measure of an algebraic integer, once it exceeds 1, can come arbitrarily close to 1 or must jump by some fixed minimum amount. Lehmer’s number, the largest root of his degree-10 polynomial, has Mahler measure about 1.1762808 and is still the smallest example above 1 ever found, despite computer searches over polynomials of far higher degree. The problem is distinct from Lehmer’s number itself, which is the record holder rather than the question. Status open.", "cites": [ "r-4.45" ], "links_to_cards": [ "lehmer" ] }, { "id": "champ", "url": "https://www.teclamairme.com/#g-champ", "name": "Champernowne’s constant", "symbol": "Champernowne", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "0.123456789101112…, normal in base 10 and transcendental (Mahler 1937).", "more": "Champernowne’s constant 0.123456789101112… is the positive integers written one after another as decimals. David Champernowne, an undergraduate, showed in 1933 that it is normal in base 10, meaning every digit string occurs with the expected frequency, and Kurt Mahler proved it transcendental in 1937. Built by hand rather than found in nature, which is its brace.", "cites": [ "r-4.10", "r-4.29" ], "links_to_cards": [ "normalnum" ] }, { "id": "gelfond", "url": "https://www.teclamairme.com/#g-gelfond", "name": "Gelfond–Schneider constant", "symbol": "2√2", "symbol_markup": "2^{√2}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Transcendental by the 1934 theorem that settled Hilbert’s seventh problem.", "more": "The Gelfond–Schneider constant 2^√2 ≈ 2.665 was one of the two explicit examples in Hilbert’s seventh problem of 1900, which asked whether a^b is transcendental when a is algebraic (not 0 or 1) and b is algebraic and irrational. Aleksandr Gelfond and Theodor Schneider proved it independently in 1934, so 2^√2 is transcendental. Here it plays the number built by theorem.", "cites": [ "r-4.9" ], "links_to_cards": [ "gelfschn", "hilbert7", "sqrt2" ] }, { "id": "sinpi", "url": "https://www.teclamairme.com/#g-sinpi", "name": "sine of pi", "symbol": "sin π", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: π is where sine first vanishes after zero, so the transcendental column has a zero of its own.", "more": "sin π = 0 exactly, because π is the first positive zero of the sine function: the half-turn of the unit circle. It is the transcendental column’s own zero, and a reminder that the most famous transcendental number is characterized by where a function vanishes. The value is the integer 0, so it is untinted.", "cites": [ "r-4.13", "r-4.9" ], "links_to_cards": [] }, { "id": "e", "url": "https://www.teclamairme.com/#g-e", "name": "Euler’s number", "symbol": "e", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "The base of natural logarithms, proven transcendental by Hermite in 1873.", "more": "Euler’s number e ≈ 2.71828, the base of natural logarithms, first appearing in Bernoulli’s 1683 study of compound interest and given its letter by Euler; he used it in a letter to Goldbach in 1731. Charles Hermite proved it transcendental in 1873, the first naturally occurring number to be proven so, by a method Lindemann then extended to π.", "cites": [ "r-4.29", "r-4.30", "r-4.9" ], "links_to_cards": [] }, { "id": "pi", "url": "https://www.teclamairme.com/#g-pi", "name": "pi", "symbol": "π", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Proven transcendental by Lindemann in 1882, which ended the quest to square the circle.", "more": "π ≈ 3.14159, the ratio of a circle’s circumference to its diameter. Lambert proved it irrational in 1761; Ferdinand von Lindemann proved it transcendental in 1882, which settled the ancient question of squaring the circle in the negative, since a constructible number is algebraic.", "cites": [ "r-4.33", "r-4.9" ], "links_to_cards": [ "constructible" ] }, { "id": "epi", "url": "https://www.teclamairme.com/#g-epi", "name": "Gelfond’s constant", "symbol": "eπ", "symbol_markup": "e^{π}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 23.14, transcendental by Gelfond–Schneider.", "more": "Gelfond’s constant e^π ≈ 23.1407 is transcendental: Gelfond proved this case directly in 1929, five years before the general Gelfond–Schneider theorem, using e^π = (−1)^(−i) with −1 algebraic and −i algebraic and irrational. It is one of the few expressions mixing e and π whose status is settled; π^e, by contrast, is in Chapter ᚃ(F).", "cites": [ "r-4.9" ], "links_to_cards": [ "gelfschn" ] }, { "id": "ii", "url": "https://www.teclamairme.com/#g-ii", "name": "i to the i", "symbol": "ii", "symbol_markup": "i^{i}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "An imaginary raised to an imaginary is real: e^(−π/2), about 0.2079.", "more": "i^i = e^(−π/2) ≈ 0.2079 by Euler’s formula, since i = e^(iπ/2) and (e^(iπ/2))^i = e^(−π/2). An imaginary number raised to an imaginary power is a real number, and a transcendental one, by Gelfond–Schneider. The principal value; other branches give e^(−π/2 + 2πk).", "cites": [ "r-4.9" ], "links_to_cards": [ "eulerformula", "gelfschn" ] }, { "id": "twoverpi", "url": "https://www.teclamairme.com/#g-twoverpi", "name": "two over pi", "symbol": "2/π", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Transcendental because π is; Buffon’s needle probability and the average of |cos| over a period.", "more": "2/π ≈ 0.6366 is the probability in Buffon’s needle problem (1777) that a needle dropped on ruled lines whose spacing equals its length crosses a line, and also the average value of |cos x| over a period. Transcendental because π is: a non-constant rational function with algebraic coefficients, applied to a transcendental number, gives a transcendental number.", "cites": [ "r-13.6", "r-4.9" ], "links_to_cards": [] }, { "id": "ram", "url": "https://www.teclamairme.com/#g-ram", "name": "Ramanujan’s constant", "symbol": "eπ√163", "symbol_markup": "e^{π√163}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "262537412640768743.99999999999925, an almost-integer explained by the class number of ℚ(√−163).", "more": "e^(π√163) = 262537412640768743.99999999999925…, an almost-integer first noticed by Charles Hermite in 1859. The name comes from Martin Gardner’s April Fool column of 1975, which claimed that Ramanujan had predicted the number to be an integer; Ramanujan had in fact studied cousins such as e^(π√58) in a paper of 1913–14. The near miss is explained by the class number 1 of ℚ(√−163) and the j-invariant, whose q-expansion gives e^(π√163) ≈ 640320³ + 744. Transcendental by Gelfond–Schneider, since it equals (−1)^(−i√163).", "cites": [ "r-4.33", "r-4.34", "r-4.9" ], "links_to_cards": [ "classnum", "gelfschn", "jinv" ] }, { "id": "lemn", "url": "https://www.teclamairme.com/#g-lemn", "name": "lemniscate constant", "symbol": "ϖ", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "π’s twin for the lemniscate curve, about 2.622, with its own rare symbol.", "more": "The lemniscate constant ϖ ≈ 2.6221 plays for the lemniscate of Bernoulli the role π plays for the circle: the arc length of the lemniscate is 2ϖ. Gauss studied it from 1798 and found in 1799 that ϖ/π = 1/agm(1, √2). Carl Ludwig Siegel proved it transcendental in 1932 and Theodor Schneider again later in the 1930s, through their work on periods of elliptic functions, and Chudnovsky showed in the 1970s that ϖ and π are algebraically independent.", "cites": [ "r-4.11", "r-4.12", "r-4.32" ], "links_to_cards": [ "sqrt2" ] }, { "id": "ln2", "url": "https://www.teclamairme.com/#g-ln2", "name": "natural logarithm of 2", "symbol": "ln 2", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Transcendental by Lindemann–Weierstrass, because e^(ln 2) = 2 is algebraic.", "more": "ln 2 ≈ 0.6931, the natural logarithm of 2, is transcendental by the Lindemann–Weierstrass theorem: e raised to a nonzero algebraic power is transcendental, so if ln 2 were algebraic, e^(ln 2) = 2 would be transcendental, which it is not. It is also the sum of the alternating harmonic series 1 − ½ + ⅓ − ….", "cites": [ "r-4.9" ], "links_to_cards": [ "lindweier" ] }, { "id": "sincos1", "url": "https://www.teclamairme.com/#g-sincos1", "name": "sine and cosine of one radian", "symbol": "sin 1, cos 1", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Two numbers on every calculator, both transcendental by Lindemann–Weierstrass.", "more": "sin 1 ≈ 0.8415 and cos 1 ≈ 0.5403, the sine and cosine of one radian, are on every calculator and are both transcendental by Lindemann–Weierstrass, which makes the sine and cosine of every nonzero algebraic number transcendental. Their sum, about 1.3818, is a positive number in the denominator.", "cites": [ "r-4.9" ], "links_to_cards": [ "lindweier" ] }, { "id": "laplacelim", "url": "https://www.teclamairme.com/#g-laplacelim", "name": "the Laplace limit", "symbol": "Laplace limit", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 0.6627, the largest eccentricity for which Kepler’s equation can be solved by a power series; transcendental by Hermite–Lindemann, from its own defining equation.", "more": "The Laplace limit ≈ 0.6627434 is the eccentricity beyond which the series solution of Kepler’s equation M = E − e sin E stops converging. Francesco Carlini had the limit about five years before Laplace computed 0.66195 in 1827, and Cauchy sharpened it to 0.66274 in 1829; the name is Laplace’s all the same. It is the real number x with x e^√(1 + x²)/(1 + √(1 + x²)) = 1, and that equation settles its arithmetic in three lines. Write λ for the limit and s = √(1 + λ²), so that λe^s = 1 + s and e^s = (1 + s)/λ. If λ were algebraic, s would be algebraic and not zero, and (1 + s)/λ would be algebraic too—but Hermite–Lindemann makes e^s transcendental for every nonzero algebraic s. So λ is transcendental, and with that it is not a fraction either.", "cites": [ "r-4.15" ], "links_to_cards": [] }, { "id": "liou", "url": "https://www.teclamairme.com/#g-liou", "name": "Liouville’s constant", "symbol": "Liouville’s", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Σ 10^(−n!) = 0.110001000…, the first decimal constant ever proven transcendental; Liouville announced the method in 1844 and gave this number in 1851.", "more": "Liouville’s constant Σ 10^(−n!) = 0.110001000000000000000001…, with a 1 in each factorial position, is the first decimal constant ever proven transcendental. Joseph Liouville announced in 1844 that numbers approximated by rationals far better than any algebraic number can be must be transcendental, and gave this constant in his full paper of 1851; the approximation property is the whole proof. It is the founding member of the built-by-hand brace.", "cites": [ "r-4.32" ], "links_to_cards": [ "permfact" ] }, { "id": "tm", "url": "https://www.teclamairme.com/#g-tm", "name": "Prouhet–Thue–Morse constant", "symbol": "Thue–Morse", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "Its binary digits follow the Thue–Morse sequence; transcendental (Mahler 1929).", "more": "The Prouhet–Thue–Morse constant 0.4124… has for its binary digits the Thue–Morse sequence 0110100110010110…, which is obtained by repeatedly appending the complement of what has been written so far. Kurt Mahler proved it transcendental in 1929. Built by hand, like Liouville’s and Champernowne’s.", "cites": [ "r-4.10", "r-4.34" ], "links_to_cards": [ "champ", "liou" ] }, { "id": "dottie", "url": "https://www.teclamairme.com/#g-dottie", "name": "Dottie number", "symbol": "Dottie", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "The fixed point of cosine, 0.7390…, found by anyone who presses cos repeatedly; transcendental.", "more": "The Dottie number 0.7390851… is the unique real fixed point of cosine, the number a calculator settles on if you press the cos key over and over in radian mode; it is named for a professor of French who noticed it. It is transcendental by Lindemann–Weierstrass, since cos of a nonzero algebraic number is transcendental and so could not equal the number itself.", "cites": [ "r-4.9" ], "links_to_cards": [ "lindweier" ] }, { "id": "gaussc", "url": "https://www.teclamairme.com/#g-gaussc", "name": "Gauss’s constant", "symbol": "Gauss’s", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "The reciprocal of the arithmetic–geometric mean of 1 and √2, about 0.8346, proven transcendental.", "more": "Gauss’s constant G = 1/agm(1, √2) ≈ 0.8346, where agm is the arithmetic–geometric mean, found by Gauss in 1799 when he noticed that it matches 2/π times the lemniscate integral. It equals the lemniscate constant ϖ divided by π, and Theodor Schneider proved it transcendental in 1941; Chudnovsky later showed more, that ϖ and π are algebraically independent. Not to be confused with Catalan’s G in Chapter ᚃ(F).", "cites": [ "r-4.11", "r-4.12", "r-4.30" ], "links_to_cards": [ "lemn", "sqrt2" ] }, { "id": "parab", "url": "https://www.teclamairme.com/#g-parab", "name": "universal parabolic constant", "symbol": "parabolic P", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "ln(1 + √2) + √2, about 2.2955, the parabola’s answer to π; transcendental by Lindemann–Weierstrass.", "more": "The universal parabolic constant P = ln(1 + √2) + √2 ≈ 2.2955 is, for every parabola, the ratio of the arc length of the segment cut off by the latus rectum to the focal parameter, the parabola’s version of π. It is transcendental by Lindemann–Weierstrass: if P were algebraic, ln(1 + √2) = P − √2 would be algebraic too, and then e raised to it, 1 + √2, would be transcendental, which it is not.", "cites": [ "r-4.9" ], "links_to_cards": [ "lindweier", "sqrt2" ] }, { "id": "cahen", "url": "https://www.teclamairme.com/#g-cahen", "name": "Cahen’s constant", "symbol": "Cahen’s", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 0.6434, an alternating sum over Sylvester’s sequence, transcendental.", "more": "Cahen’s constant ≈ 0.6434 is the alternating sum Σ (−1)ᵏ/(sₖ − 1) over Sylvester’s sequence sₖ = 2, 3, 7, 43, 1807, …, in which each term is one more than the product of all the terms before it. Eugène Cahen introduced it in 1891, and Davison and Shallit proved it transcendental in 1991 by way of its continued fraction.", "cites": [ "r-4.11" ], "links_to_cards": [ "contfrac" ] }, { "id": "log23", "url": "https://www.teclamairme.com/#g-log23", "name": "the logarithm of three in base two", "symbol": "log2 3", "symbol_markup": "log_{2} 3", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 1.585, the number of bits in a trit; transcendental by Gelfond–Schneider, since 2 raised to it is 3.", "more": "log₂ 3 ≈ 1.58496 is the exponent that turns 2 into 3, the information in one three-way choice. It is irrational because a positive power of 2 is always even and a positive power of 3 always odd, so 2ᵃ = 3ᵇ is impossible for positive integers, and transcendental by Gelfond–Schneider: if it were algebraic and irrational, 2 raised to it would be transcendental, but 2^(log₂ 3) = 3.", "cites": [ "r-4.9" ], "links_to_cards": [ "gelfschn" ] }, { "id": "sqrt2sqrt2", "url": "https://www.teclamairme.com/#g-sqrt2sqrt2", "name": "root two to the root two", "symbol": "√2√2", "symbol_markup": "√2^{√2}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 1.6325, a number of the kind Hilbert asked about in 1900; his own examples were 2^√2 and e^π; transcendental by Gelfond–Schneider.", "more": "√2^√2 ≈ 1.63253 is a number of the kind Hilbert’s seventh problem of 1900 asked about, an algebraic irrational raised to an algebraic irrational, and it is transcendental by the Gelfond–Schneider theorem of 1934. It is also the number in the classic proof that an irrational to an irrational power can be rational, which needs only to know that (√2^√2)^√2 = 2, without knowing which of two cases holds: either √2^√2 is itself rational, or it is irrational and raising it to √2 gives the rational 2.", "cites": [ "r-4.19", "r-4.9" ], "links_to_cards": [ "epi", "gelfond", "gelfschn", "hilbert7", "sqrt2" ] }, { "id": "komornik", "url": "https://www.teclamairme.com/#g-komornik", "name": "the Komornik–Loreti constant", "symbol": "qKL", "symbol_markup": "q_{KL}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 1.7872, the smallest base in which 1 has only one expansion using the digits 0 and 1; transcendental (Allouche and Cosnard, 2000).", "more": "The Komornik–Loreti constant q ≈ 1.787231650 is the smallest number greater than 1 such that 1 has a unique expansion Σ tₖ/qᵏ in digits 0 and 1, where the digits tₖ are the Thue–Morse sequence; Vilmos Komornik and Paola Loreti found it in 1998. Jean-Paul Allouche and Michel Cosnard proved it transcendental in 2000 in the American Mathematical Monthly. Built by hand, from the same sequence as the Thue–Morse constant beside it.", "cites": [ "r-4.10" ], "links_to_cards": [ "tm" ] }, { "id": "paperfold", "url": "https://www.teclamairme.com/#g-paperfold", "name": "the paperfolding constant", "symbol": "Pfold", "symbol_markup": "P_{fold}", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 0.8507, the number whose binary digits are the creases of a strip of paper folded in half again and again; transcendental.", "more": "The paperfolding constant 0.85073618… has for its binary digits the regular paperfolding sequence 1101100111001001…, the pattern of hill and valley creases when a strip is folded in half repeatedly and unfolded, described by Martin Gardner in 1967 and first analyzed in print by Chandler Davis and Donald Knuth in 1970. The sequence is automatic and not periodic, and Boris Adamczewski and Yann Bugeaud proved in 2007 that every irrational number with an automatic expansion is transcendental. Built by hand.", "cites": [ "r-4.10", "r-4.33" ], "links_to_cards": [] }, { "id": "gamma14", "url": "https://www.teclamairme.com/#g-gamma14", "name": "gamma of one quarter", "symbol": "Γ(¼)", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 3.6256, the value that ties the gamma function to the lemniscate; transcendental (Chudnovsky, ~1976).", "more": "Γ(¼) ≈ 3.62561 is the gamma function of Chapter ᚐ(A) at a quarter, and it carries the lemniscate: ϖ = Γ(¼)²/(2√(2π)). Gregory Chudnovsky proved in the 1970s that Γ(¼) and π are algebraically independent; the result appeared in the Doklady of the Ukrainian Academy of Sciences in 1976, was presented at the ICM in Helsinki in 1978, and was collected in his 1984 monograph Contributions to the Theory of Transcendental Numbers, which is the spread the tilde marks. So Γ(¼) is transcendental; ϖ and Gauss’s constant beside it had already been proved transcendental, by Siegel in 1932 and by Schneider in 1941; Γ(⅓) is transcendental by the same work, and Γ(½) = √π is transcendental because π is. Whether Γ(⅕) is transcendental is open.", "cites": [ "r-4.12", "r-4.54" ], "links_to_cards": [ "gammafn", "gammahalf", "gaussc", "lemn", "pi" ] }, { "id": "knuth34", "url": "https://www.teclamairme.com/#g-knuth34", "name": "three tetrated to four", "symbol": "3↑↑4", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "3^(3^(3^3)) = 3^7625597484987, a number with about 3.6 trillion digits.", "more": "3↑↑4 = 3^(3↑↑3) = 3^7625597484987, a tower of four threes, has about 3.6 trillion decimal digits: more than any prime ever found, fewer than the googolplex. It is the last rung of the tower a reader can still describe in words, and the next arrow, 3↑↑↑3, is a tower of 3↑↑3 threes. An integer.", "cites": [ "r-4.5" ], "links_to_cards": [ "googolplex", "knuth3", "knuth5" ] }, { "id": "knuth5", "url": "https://www.teclamairme.com/#g-knuth5", "name": "three pentated to three, Conway’s 3→3→3", "symbol": "3↑↑↑3", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "A tower of 7,625,597,484,987 threes; in Conway’s chained-arrow notation it is 3→3→3.", "more": "3↑↑↑3 = 3↑↑(3↑↑3) is a power tower of 7,625,597,484,987 threes, a value of pentation, the hyperoperation after tetration; Reuben Goodstein named tetration, pentation, and hexation in 1947, following Ackermann’s function of 1928. In John Conway’s chained-arrow notation it is 3→3→3, since a→b→c is a with c arrows b; the chains run past Knuth’s arrows, and Graham’s number lies between 3→3→64→2 and 3→3→65→2. Larger than Skewes’ number. An integer.", "cites": [ "r-4.32", "r-4.5" ], "links_to_cards": [ "ackfn", "chain", "g64", "knuth3", "skewes", "uparrow" ] }, { "id": "loader", "url": "https://www.teclamairme.com/#g-loader", "name": "Loader’s number", "symbol": "Loader’s", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The output of Ralph Loader’s 2001 program, the winning entry in a contest for the largest number a C program of at most 512 characters could return; often said to exceed SSCG(3), a comparison this book has not been able to source.", "more": "Loader’s number is the output of a C program of at most 512 characters written by Ralph Loader in December 2001 for the Bignum Bakeoff, a contest for the largest number a tiny program can compute. The program runs every well-formed term of the calculus of constructions up to a bound, and its output, D⁵(99), is usually said to exceed TREE(3) and SSCG(3), a comparison this book repeats without a primary source; it waits on this chapter’s workbench before Chapter ᚃ(F) leaves computability behind. An integer.", "cites": [ "r-4.32", "r-4.5" ], "links_to_cards": [ "sscg3", "tree3" ] }, { "id": "uparrow", "url": "https://www.teclamairme.com/#g-uparrow", "name": "Knuth’s up-arrow notation", "symbol": "a↑ⁿb", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "One arrow is a power, two a tower, three a tower of towers: Knuth’s 1976 notation for the hyperoperations, the language the giants are written in.", "more": "Donald Knuth introduced up-arrow notation in Mathematics and Computer Science: Coping with Finiteness, Science, 1976: a↑b = a^b, a↑↑b = a↑(a↑(…)) with b copies, and each extra arrow iterates the operation before it. 2↑↑4 = 65536, 3↑↑3 = 7,625,597,484,987, 3↑↑↑3 is a tower of 3↑↑3 threes, and Graham’s number is 3 with a number of arrows that is itself a giant. A notation, so a tool of the framework rather than a value.", "cites": [ "r-4.35", "r-4.5" ], "links_to_cards": [ "g64", "hyperop", "knuth3", "knuth5" ] }, { "id": "chain", "url": "https://www.teclamairme.com/#g-chain", "name": "Conway’s chained-arrow notation", "symbol": "a→b→c", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "John Conway’s 1996 notation that runs past Knuth’s arrows: a→b→c is a with c arrows b, and longer chains grow faster than any fixed number of arrows.", "more": "Conway chained-arrow notation, from The Book of Numbers by John Conway and Richard Guy, 1996, writes a→b→c for a↑ᶜb and defines longer chains recursively, so that 3→3→3 = 3↑↑↑3 and 3→3→64→2 < Graham’s number < 3→3→65→2. Four-element chains already outrun Knuth’s arrows. A notation, a tool of the framework.", "cites": [ "r-4.29", "r-4.5" ], "links_to_cards": [ "g64", "knuth5", "uparrow" ] }, { "id": "hyperop", "url": "https://www.teclamairme.com/#g-hyperop", "name": "the hyperoperation sequence", "symbol": "Hn(a, b)", "symbol_markup": "H_{n}(a, b)", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Succession, addition, multiplication, exponentiation, tetration, pentation, hexation: each operation repeats the one before, in Goodstein’s 1947 naming.", "more": "The hyperoperations are the ladder of repeated operations: addition repeats succession, multiplication repeats addition, exponentiation repeats multiplication, and tetration, pentation, and hexation continue upward. Wilhelm Ackermann’s three-argument function of 1928 was an early version, and Reuben Goodstein cleaned it up and coined the Greek names in 1947. Knuth’s arrows and Conway’s chains are notations for it; the Ackermann function of Chapter ᚇ(D) is a version of it. A concept, a tool of the framework.", "cites": [ "r-4.31", "r-4.5" ], "links_to_cards": [ "chain", "uparrow" ] }, { "id": "fgh", "url": "https://www.teclamairme.com/#g-fgh", "name": "the fast-growing hierarchy", "symbol": "fα(n)", "symbol_markup": "f_{α}(n)", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Löb and Wainer’s 1970 ladder of functions indexed by ordinals, the ruler that says how big a giant really is: Graham’s number is about fω+1(64), TREE(3) is beyond fΓ₀.", "quick_markup": "Löb and Wainer’s 1970 ladder of functions indexed by ordinals, the ruler that says how big a giant really is: Graham’s number is about f_{ω+1}(64), TREE(3) is beyond f_{Γ₀}.", "more": "The fast-growing hierarchy of Martin Löb and Stanley Wainer, 1970, defines f₀(n) = n + 1, f_{α+1}(n) = f_{α} applied n times to n, and for a limit ordinal takes the diagonal of the functions below it. It is the ruler for giants: Knuth’s k arrows live at f_{k+1}, Graham’s number below f_{ω+1}(64), and TREE(3) beyond f_{Γ₀}; the Busy Beaver function of Chapter ᚃ(F) outgrows the whole computable hierarchy. A concept, a tool of the framework.", "cites": [ "r-4.30", "r-4.5" ], "links_to_cards": [ "busybeaver", "g64", "tree3" ] }, { "id": "nonconstructible", "url": "https://www.teclamairme.com/#g-nonconstructible", "name": "a non-constructible number", "symbol": "∛2, π", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "A real number that cannot be reached from a unit length using compass and unmarked straightedge.", "more": "Constructibility means an exact construction from a unit length. Arithmetic and successive square roots reach every constructible real number; a non-constructible number lies outside that collection. For example, ∛2 has degree 3 over the rationals, so it cannot be reached this way. Every constructible number is algebraic, which also rules out transcendental numbers such as π. The algebraic denominator groups only the non-constructible algebraic examples; the transcendental examples have their own section. Being non-constructible does not mean being uncomputable: these numbers can still be approximated.", "cites": [ "r-4.20", "r-4.9" ], "links_to_cards": [ "cbrt2", "constructible", "pi", "uncomputable" ] }, { "id": "constructible", "url": "https://www.teclamairme.com/#g-constructible", "name": "a constructible number", "symbol": "ℚ→√", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Reachable from a unit length by compass and unmarked straightedge, using arithmetic and successive square roots.", "more": "A length is constructible with compass and straightedge exactly when it lies at the top of a finite tower of quadratic extensions of the rationals, so its degree [ℚ(γ):ℚ] is 2ʳ for some r. That degree condition is necessary but is not sufficient on its own. Pierre Wantzel proved the tower criterion in 1837 and read the two famous impossibilities straight off it: ∛2 has minimal polynomial x³ − 2, of degree 3, so the cube cannot be doubled, and cos(π/9) has minimal polynomial 8x³ − 6x − 1, also of degree 3, so the general angle cannot be trisected. The same criterion says which regular polygons can be drawn—the 17-gon can, the heptagon cannot. This is the line the chapter’s algebraic brace is drawn along: above it the numbers a Greek could have built, below it the ones no straightedge reaches. A criterion, not a number, so uncolored.", "cites": [ "r-4.20", "r-4.8" ], "links_to_cards": [ "cbrt2" ] }, { "id": "lindweier", "url": "https://www.teclamairme.com/#g-lindweier", "name": "the Lindemann–Weierstrass theorem", "symbol": "L–W", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Raising e to any nonzero algebraic power gives a transcendental number (Lindemann, 1882; Weierstrass proved the general form in 1885), so e and π are both transcendental.", "more": "The theorem that makes most of this chapter’s transcendental brace possible. Lindemann proved in 1882 that e^α is transcendental for every nonzero algebraic α; Weierstrass proved the general independence statement in 1885: if α₁, …, αₙ are algebraic and linearly independent over ℚ, then e^α₁, …, e^αₙ are algebraically independent over ℚ. Two corollaries fall out at once. Taking α = 1 gives e. And if π were algebraic then so would πi be, which would make e^(iπ) = −1 transcendental—so π is not algebraic either, which is why the circle cannot be squared. Everything in the chapter that is transcendental “by Hermite and Lindemann” is transcendental by this. A theorem, not a number, so uncolored.", "cites": [ "r-4.9" ], "links_to_cards": [] }, { "id": "gelfschn", "url": "https://www.teclamairme.com/#g-gelfschn", "name": "the Gelfond–Schneider theorem", "symbol": "G–S", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "If a is algebraic and not 0 or 1, and b is algebraic and irrational, then a^b is transcendental.", "more": "Proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider, answering the seventh of the problems Hilbert set in 1900. It is the theorem behind 2^√2, e^π (which is (−1)^(−i)), i^i, and √2^√2—the second brace of transcendentals in this chapter, the ones that are transcendental for a reason different from e and π’s. The hypotheses are exactly as tight as they look: 1^√2 = 1 is why a ≠ 1 is needed, and 2² is why b must be irrational. A theorem, not a number, so uncolored.", "cites": [ "r-4.31", "r-4.9" ], "links_to_cards": [ "epi", "sqrt2" ] }, { "id": "cantorcount", "url": "https://www.teclamairme.com/#g-cantorcount", "name": "Cantor’s counting argument", "symbol": "ℵ₀ < 𝕔", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "The real algebraic numbers are countable and the reals are not, so almost every real number is transcendental.", "more": "Cantor proved in 1874, in Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, that the real algebraic numbers can be put in one-to-one correspondence with the positive integers, and that an interval of reals cannot. The two halves together put a transcendental number in every interval, and Cantor’s argument, run on a list of the algebraic numbers, actually builds one: the transcendentals are not a scarce curiosity but everything left over. Liouville had proved in 1844 that transcendental numbers exist and built one by hand in 1851; Cantor’s counting showed that they are almost all of them.", "cites": [ "r-4.21" ], "links_to_cards": [ "bijection", "reals" ] }, { "id": "mihailescu", "url": "https://www.teclamairme.com/#g-mihailescu", "name": "Catalan’s conjecture, Mihăilescu’s theorem", "symbol": "3² − 2³ − 1", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "8 and 9 are the only consecutive perfect powers, so 3² − 2³ − 1 = 0; posed by Catalan in a note published in 1844 and proved by Mihăilescu in 2002.", "more": "Catalan asked, in a note published in 1844, whether 2³ = 8 and 3² = 9 are the only two perfect powers that differ by 1. Preda Mihăilescu proved that they are in 2002. Written as 3² − 2³ − 1 the theorem is exactly 0, a numerator in waiting, left on the workbench because a brace can only be so wide. An integer.", "cites": [ "r-4.22" ], "links_to_cards": [] }, { "id": "narc153", "url": "https://www.teclamairme.com/#g-narc153", "name": "a narcissistic number", "symbol": "153", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "1³ + 5³ + 3³ = 153: a number equal to the sum of the cubes of its own digits.", "more": "A number that equals the sum of its own digits, each raised to the power of how many digits it has, is called narcissistic; 153 = 1 + 125 + 27. An integer.", "cites": [ "r-4.23" ], "links_to_cards": [] }, { "id": "lcm2520", "url": "https://www.teclamairme.com/#g-lcm2520", "name": "the least common multiple of 1 through 10", "symbol": "2520", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "The smallest number that each of 1 through 10 divides.", "more": "2520 = 2³ · 3² · 5 · 7, each prime taken to the highest power that appears among the numbers 1 to 10. An integer.", "cites": [ "r-4.23" ], "links_to_cards": [] }, { "id": "cannonball", "url": "https://www.teclamairme.com/#g-cannonball", "name": "the cannonball number", "symbol": "4900", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "Apart from 1, the only perfect square that is also a square pyramidal number: 1² + 2² + … + 24² = 70².", "more": "Stack cannonballs in a square pyramid with a 24-by-24 base and the 4900 balls can be relaid as a single 70-by-70 square; apart from the one-ball pile, no other square pyramid does this. An integer.", "cites": [ "r-4.22" ], "links_to_cards": [] }, { "id": "wheat", "url": "https://www.teclamairme.com/#g-wheat", "name": "the wheat and chessboard", "symbol": "2⁶⁴ − 1", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "One grain on the first square, doubling on each of the 64: 18,446,744,073,709,551,615 grains in all.", "more": "The total is 1 + 2 + 4 + … + 2⁶³ = 2⁶⁴ − 1, because every sum of doublings is one short of the next power of two. An integer.", "cites": [ "r-4.23" ], "links_to_cards": [] }, { "id": "centillion", "url": "https://www.teclamairme.com/#g-centillion", "name": "centillion", "symbol": "10³⁰³", "chapter": 4, "type": "int", "badge": "Integer or rational", "quick": "10³⁰³ on the short scale and 10⁶⁰⁰ on the long: the highest -illion name the major English dictionaries include.", "more": "Larger -illions can be built by the same naming rules, but the dictionaries stop here. An integer.", "cites": [ "r-4.23" ], "links_to_cards": [] }, { "id": "sqrt3", "url": "https://www.teclamairme.com/#g-sqrt3", "name": "Theodorus’ constant", "symbol": "√3", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.7320508, the square root of 3, which Theodorus of Cyrene proved irrational, as Plato reports in the Theaetetus.", "more": "In Plato’s dialogue, Theodorus shows that the square roots of 3, 5, and the other non-squares up to 17 are irrational, and stops there; how he proved it is not known. √3 is constructible: it is the height of an equilateral triangle with side 2. Algebraic of degree 2.", "cites": [ "r-4.23" ], "links_to_cards": [ "constructible" ] }, { "id": "supergolden", "url": "https://www.teclamairme.com/#g-supergolden", "name": "supergolden ratio", "symbol": "ψ", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.4655712, the real root of x³ = x² + 1.", "more": "A cousin of the golden ratio: where φ satisfies x² = x + 1, ψ satisfies x³ = x² + 1. It is algebraic of degree 3, and 3 is not a power of 2, so compass and straightedge cannot construct it.", "cites": [ "r-4.20", "r-4.23" ], "links_to_cards": [ "phi" ] }, { "id": "semitone", "url": "https://www.teclamairme.com/#g-semitone", "name": "the twelfth root of two", "symbol": "¹²√2", "chapter": 4, "type": "alg", "badge": "Algebraic irrational", "quick": "About 1.059463, the frequency ratio of one semitone in twelve-tone equal temperament.", "more": "Twelve equal semitones make an octave, a doubling of frequency, so each one multiplies the frequency by 2^(1/12). Algebraic of degree 12, a root of x¹² = 2.", "cites": [ "r-4.23" ], "links_to_cards": [] }, { "id": "tau", "url": "https://www.teclamairme.com/#g-tau", "name": "tau", "symbol": "τ", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "τ = 2π ≈ 6.283185, one full turn in radians; transcendental because π is.", "more": "Michael Hartl’s Tau Manifesto of 2010 argued that the full turn, not the half turn, deserves its own letter. A nonzero algebraic multiple of π is transcendental. Not to be confused with the tribonacci constant τ₃ of the algebraic basket.", "cites": [ "r-4.23" ], "links_to_cards": [ "pi", "tribonacci" ] }, { "id": "goldenangle", "url": "https://www.teclamairme.com/#g-goldenangle", "name": "golden angle", "symbol": "2π/φ²", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "About 137.5077°, or 2π/φ² radians: the angle that separates successive florets on a sunflower head.", "more": "Turn by this angle before placing each new seed and the seeds never line up into spokes, which leaves each one room to grow; Vogel’s sunflower model of 1979 uses exactly this rule. In radians it is π times the algebraic number 2(2 − φ), so it is transcendental.", "cites": [ "r-4.23" ], "links_to_cards": [ "phi" ] }, { "id": "magicangle", "url": "https://www.teclamairme.com/#g-magicangle", "name": "the magic angle", "symbol": "θₘ", "chapter": 4, "type": "tr", "badge": "Proven transcendental", "quick": "arccos(1/√3) ≈ 54.7356°, the tilt used in magic-angle spinning NMR.", "more": "Spinning a solid sample about an axis tilted at this angle to the magnetic field averages away the anisotropic interactions that blur its spectrum. In radians it is the arccosine of an algebraic number, so it is transcendental by the Lindemann–Weierstrass theorem.", "cites": [ "r-4.23" ], "links_to_cards": [ "lindweier", "sqrt3" ] }, { "id": "kruskal", "url": "https://www.teclamairme.com/#g-kruskal", "name": "Kruskal’s tree theorem", "symbol": "Tᵢ ≤ Tⱼ", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "Any infinite sequence of finite trees contains an earlier tree that embeds into a later one, a 1960 theorem whose finite version defines the explosively fast-growing function behind TREE(3).", "more": "Joseph Kruskal proved in 1960 that the finite trees, ordered by whether one embeds homeomorphically into another, form a well-quasi-order: no infinite sequence of them can avoid having an earlier tree embed in a later one. Harvey Friedman later gave a finite, miniaturized version of the same statement, and the length of the longest sequence it allows, for trees labeled from a set of three colors with the i-th tree having at most i vertices, is TREE(3), one of the giant integers on this chapter’s ladder. A theorem, uncolored.", "cites": [ "r-4.37" ], "links_to_cards": [ "tree3" ] }, { "id": "hilbert7", "url": "https://www.teclamairme.com/#g-hilbert7", "name": "Hilbert’s seventh problem", "symbol": "ab", "symbol_markup": "a^{b}", "chapter": 4, "type": "fn", "badge": "Function or operator", "quick": "David Hilbert’s 1900 question of whether a to the power b is always transcendental when a is algebraic and not 0 or 1 and b is algebraic and irrational, answered yes by the Gelfond-Schneider theorem in 1934.", "more": "Hilbert set the transcendence of a^b as the seventh of his twenty-three problems, naming 2^√2 and e^π, which is (−1)^(−i), as his own test cases. Aleksandr Gelfond and Theodor Schneider settled it independently in 1934, proving that a^b is transcendental under exactly those hypotheses; the theorem now bears both their names and covers every case this chapter’s second brace of transcendentals needs, from 2^√2 itself to i^i and √2^√2. A problem, resolved, uncolored.", "cites": [ "r-4.38" ], "links_to_cards": [ "epi", "gelfond" ] }, { "id": "cope", "url": "https://www.teclamairme.com/#g-cope", "name": "Copeland–Erdős constant", "symbol": "Copeland–Erdős", "chapter": 5, "type": "open", "badge": "Status open", "quick": "0.235711131719…, the primes concatenated; normal in base 10, hence irrational, transcendence unknown.", "more": "The Copeland–Erdős constant 0.235711131719… is the primes written one after another as decimals. Arthur Copeland and Paul Erdős proved in 1946 that it is normal in base 10, and a normal number is automatically irrational, since a rational’s digits repeat. Whether it is transcendental is open, which is why it is tinted as status open and why it stands among the unresolved constants in Chapter ᚃ(F); it also appears in Chapter ᚂ(L) as the hand-built cousin under Champernowne’s constant.", "cites": [ "r-5.4" ], "links_to_cards": [ "champ", "normalnum" ] }, { "id": "legendre", "url": "https://www.teclamairme.com/#g-legendre", "name": "Legendre’s constant", "symbol": "Legendre’s", "chapter": 5, "type": "int", "badge": "Integer or rational", "quick": "Conjectured to be about 1.08366 in 1808 and proven to be exactly 1 in 1899.", "more": "In 1808 Legendre proposed that the number of primes up to x is about x/(ln x − B) with B ≈ 1.08366, and B was called Legendre’s constant. In 1899 de la Vallée Poussin proved that the best value of B is exactly 1. That is why Legendre’s constant minus 1 is the zero of the unresolved-constants column.", "cites": [ "r-5.1" ], "links_to_cards": [ "primecount" ] }, { "id": "gamma", "url": "https://www.teclamairme.com/#g-gamma", "name": "Euler–Mascheroni constant", "symbol": "γ", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.5772, and not even known to be irrational, one of the embarrassments of number theory.", "more": "The Euler–Mascheroni constant γ ≈ 0.5772 is the limit of 1 + ½ + … + 1/n − ln n, introduced by Euler in 1734. Lorenzo Mascheroni computed it to 32 decimal places in 1790, of which only the first 19 were right; the error was caught by von Soldner in 1809. It appears throughout analysis and number theory, and it is not known whether it is irrational, let alone transcendental; Aptekarev observed in 2009 that at least one of γ and the Gompertz constant is irrational, and Rivoal in 2012 that at least one is transcendental. The flagship of the not-even-known-irrational brace.", "cites": [ "r-5.30", "r-5.6" ], "links_to_cards": [ "gompertz" ] }, { "id": "feig", "url": "https://www.teclamairme.com/#g-feig", "name": "Feigenbaum’s δ", "symbol": "δ", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 4.6692, the universal period-doubling ratio of chaos; irrationality unproven.", "more": "Feigenbaum’s constant δ ≈ 4.6692 is the universal ratio at which period-doubling bifurcations approach chaos, found numerically by Mitchell Feigenbaum in 1975 and proven universal by Lanford in 1982. It is the same for every one-dimensional map with a single quadratic maximum, which is why it counts as a constant of nature within mathematics. Whether it is irrational is unknown.", "cites": [ "r-5.30", "r-5.7" ], "links_to_cards": [] }, { "id": "feigA", "url": "https://www.teclamairme.com/#g-feigA", "name": "Feigenbaum’s α", "symbol": "αF", "symbol_markup": "α_{F}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "Its magnitude is about 2.5029; the signed convention is approximately −2.5029. Its irrationality remains open.", "more": "Feigenbaum’s second constant describes the spatial scaling of successive period-doubling forks. A common signed convention gives α_{F} ≈ −2.502907875; many expositions quote its magnitude. This book displays |α_{F}| when a positive denominator value is intended. The universality statement concerns one-dimensional maps with a single quadratic maximum, the same class named on the δ card; Lanford’s 1982 computer-assisted proof established it for that class. Its irrationality is unknown.", "cites": [ "r-5.30", "r-5.7" ], "links_to_cards": [ "feig" ] }, { "id": "catalan", "url": "https://www.teclamairme.com/#g-catalan", "name": "Catalan’s constant", "symbol": "G", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.9160; whether it is even irrational is unknown.", "more": "Catalan’s constant G = 1 − 1/9 + 1/25 − 1/49 + … ≈ 0.9160, named after Eugène Catalan, who studied it in 1865. It appears in combinatorics and in the volumes of hyperbolic solids, and it is not known whether it is irrational. This is the G of Chapter ᚃ(F); Gauss’s constant, also written G and proven transcendental, is in Chapter ᚂ(L).", "cites": [ "r-5.30", "r-5.8" ], "links_to_cards": [ "gaussc" ] }, { "id": "khin", "url": "https://www.teclamairme.com/#g-khin", "name": "Khinchin’s constant", "symbol": "K", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 2.6854, the geometric mean of almost every continued fraction’s terms; status open.", "more": "Khinchin’s constant K ≈ 2.6854 is the geometric mean of the partial quotients of the continued fraction of almost every real number, proven by Aleksandr Khinchin in ~1934 to exist and to be the same for almost all numbers. It is not known to be irrational; it is not even known whether K’s own continued fraction has this property.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "contfrac" ] }, { "id": "glaisher", "url": "https://www.teclamairme.com/#g-glaisher", "name": "Glaisher–Kinkelin constant", "symbol": "A", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.2824; status open.", "more": "The Glaisher–Kinkelin constant A ≈ 1.2824 appears in the asymptotics of the hyperfactorial 1¹ · 2² · 3³ · … · nⁿ, studied by Hermann Kinkelin in 1860 and James Glaisher in 1878, and it is connected to the derivative of the zeta function at −1. Its irrationality is open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "zeta" ] }, { "id": "brun", "url": "https://www.teclamairme.com/#g-brun", "name": "Brun’s constant", "symbol": "B2", "symbol_markup": "B_{2}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "The sum of reciprocals of twin primes, estimated at about 1.9022; rational if there are finitely many twin primes.", "more": "Brun’s constant B₂ ≈ 1.9022 is the sum of the reciprocals of the twin primes, (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + …. Viggo Brun proved in 1919 that the sum converges, so twin primes are sparse even if infinite. If there are finitely many twin primes it is a rational number, so its status is tied to the twin prime conjecture; its computation by Nicely in 1994 famously exposed the Pentium division bug. Its familiar digits are an extrapolation from the twin primes below 10^{16}; what is proved is only that it lies between 1.84 and 2.29. This B₂ is Brun’s constant, not the Bernoulli number B₂ = ⅙ of Chapter ᚂ(L).", "cites": [ "r-5.10", "r-5.30" ], "links_to_cards": [ "bern2" ] }, { "id": "eb", "url": "https://www.teclamairme.com/#g-eb", "name": "Erdős–Borwein constant", "symbol": "EEB", "symbol_markup": "E_{EB}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.6067, proven irrational in 1948; transcendence open.", "more": "The Erdős–Borwein constant E_{EB} = Σ 1/(2ⁿ − 1) ≈ 1.6067, the sum of the reciprocals of the Mersenne numbers. Paul Erdős proved it irrational in 1948, and Peter Borwein gave a second proof in 1992; whether it is transcendental is open, so it stands in the irrational-but-unclassified brace.", "cites": [ "r-5.2" ], "links_to_cards": [ "mersnum" ] }, { "id": "gompertz", "url": "https://www.teclamairme.com/#g-gompertz", "name": "Gompertz constant", "symbol": "Gompertz", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.5963; at least one of it and γ is transcendental (Rivoal, 2012), and nobody knows which.", "more": "The Gompertz constant ≈ 0.5963 is the integral of e^(−x)/(1 + x) from 0 to ∞, named by Le Lionnais for its role in survival analysis, after Benjamin Gompertz, whose 1825 law of human mortality gives the Gompertz curve, and connected to Euler’s divergent series 1 − 1 + 2 − 6 + 24 − …. Aptekarev (2009) observed that at least one of γ and this constant is irrational, and Rivoal (2012) that at least one is transcendental; nobody knows which.", "cites": [ "r-5.31", "r-5.6" ], "links_to_cards": [ "gamma" ] }, { "id": "mills", "url": "https://www.teclamairme.com/#g-mills", "name": "Mills’ constant", "symbol": "Mills’", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.3064, the smallest θ with ⌊θ^(3ⁿ)⌋ prime for every n; proved irrational by Kota Saito in 2024, transcendence open; its digits are known only under the Riemann Hypothesis.", "more": "Mills’ constant θ ≈ 1.3064 is the smallest real number such that the integer part of θ^(3ⁿ) is prime for every n, from William Mills’ 1947 theorem that such a number exists. Its digits are known only under the Riemann Hypothesis, which is needed to guarantee a prime between consecutive cubes; the number itself is defined without it, and Kota Saito proved it irrational in 2024 with no such assumption (Mathematika 71, 2025). Whether it is transcendental is open: Saito showed that either θ is transcendental or some power θ^(3^m) is a cubic Pisot number, and that transcendence would follow from Cramér’s conjecture or from the Density Hypothesis. Status open.", "cites": [ "r-5.11", "r-5.31" ], "links_to_cards": [ "rh" ] }, { "id": "pie", "url": "https://www.teclamairme.com/#g-pie", "name": "the unclassified combinations of π and e", "symbol": "π + e, π·e, πe", "symbol_markup": "π + e, π·e, π^{e}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "π + e ≈ 5.86, π·e ≈ 8.54, and π^e ≈ 22.46; the irrationality of each is open, though at least one of π + e and π·e must be transcendental.", "more": "π + e ≈ 5.8599, π·e ≈ 8.5397, and π^e ≈ 22.459 are the simplest combinations of the two most famous transcendentals whose status is unknown: none is known to be irrational. Because π and e are both roots of x² − (π + e)x + πe, at least one of π + e and πe must be transcendental, but which one is open. Contrast e^π, which Gelfond–Schneider settles and which lives in Chapter ᚂ(L).", "cites": [ "r-5.12" ], "links_to_cards": [ "epi", "gelfschn" ] }, { "id": "twinprime", "url": "https://www.teclamairme.com/#g-twinprime", "name": "twin prime constant", "symbol": "C2", "symbol_markup": "C_{2}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.6602, the constant in the Hardy–Littlewood twin prime conjecture; status open.", "more": "The twin prime constant C₂ = Π (1 − 1/(p − 1)²) over odd primes ≈ 0.6602, the constant in Hardy and Littlewood’s 1923 conjecture for how many twin primes lie below x. Its irrationality is open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "mm", "url": "https://www.teclamairme.com/#g-mm", "name": "Meissel–Mertens constant", "symbol": "Meissel–Mertens", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.2615, the γ of the primes; status open.", "more": "The Meissel–Mertens constant M ≈ 0.2615 is the limit of the sum of 1/p over primes up to n minus ln ln n, the primes’ own version of γ, established by Meissel in 1866 and Mertens in ~1874. Not known to be irrational.", "cites": [ "r-5.31", "r-5.6" ], "links_to_cards": [ "gamma" ] }, { "id": "recipfib", "url": "https://www.teclamairme.com/#g-recipfib", "name": "reciprocal Fibonacci constant", "symbol": "ψ", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 3.3599, proven irrational in 1989; transcendence open.", "more": "The reciprocal Fibonacci constant ψ = 1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + … ≈ 3.3599. Richard André-Jeannin proved it irrational in 1989; whether it is transcendental is open, so it stands beside the Erdős–Borwein constant in the irrational-but-unclassified brace.", "cites": [ "r-5.3" ], "links_to_cards": [ "eb" ] }, { "id": "gd", "url": "https://www.teclamairme.com/#g-gd", "name": "Golomb–Dickman constant", "symbol": "Golomb–Dickman", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.6243, the expected share of a random permutation taken up by its longest cycle, as the permutation grows; status open.", "more": "The Golomb–Dickman constant λ ≈ 0.6243 is the expected fraction of n taken up by the longest cycle of a random permutation of n objects, as n grows, and equally the expected size of the largest prime factor of a random integer on a logarithmic scale; found by Golomb in 1964 and, in the number-theoretic form, by Dickman in 1930. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "lr", "url": "https://www.teclamairme.com/#g-lr", "name": "Landau–Ramanujan constant", "symbol": "Landau–Ramanujan", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.7642, the constant in the count of numbers that are sums of two squares; status open.", "more": "The Landau–Ramanujan constant K ≈ 0.7642 is the constant in the count of integers up to x that are sums of two squares, about K·x/√(ln x), proven by Landau in 1908 and independently found by Ramanujan. Its irrationality is open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "apery", "url": "https://www.teclamairme.com/#g-apery", "name": "Apéry’s constant", "symbol": "ζ(3)", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.2021, the sum of the reciprocal cubes; Apéry proved it irrational in 1978, and whether it is transcendental is open.", "more": "Apéry’s constant ζ(3) = 1 + 1/8 + 1/27 + … ≈ 1.2020569, the zeta function of Chapter ᚑ(O) at 3. Euler found ζ(2) = π²/6 in 1734 by the usual account and read it to the St Petersburg Academy in December 1735, the year other accounts give for the result; no closed form for ζ(3) has ever been found; Roger Apéry stunned the field in 1978 by proving it irrational with methods that had been available for 200 years. Its transcendence is open, which puts it in the irrational-but-unclassified brace here as well as in Chapter ᚑ(O)’s denominator. Status open.", "cites": [ "r-5.30", "r-5.5" ], "links_to_cards": [ "zeta", "zeta2" ] }, { "id": "zeta5", "url": "https://www.teclamairme.com/#g-zeta5", "name": "zeta of five", "symbol": "ζ(5)", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.0369, the sum of the reciprocal fifth powers; not known to be irrational, though at least one of ζ(5), ζ(7), ζ(9), ζ(11) is (Zudilin, 2001).", "more": "ζ(5) = 1 + 1/32 + 1/243 + … ≈ 1.0369278. Apéry’s method does not reach it: Rivoal proved in 2000 that infinitely many odd zeta values are irrational, and Wadim Zudilin in 2001 that at least one of ζ(5), ζ(7), ζ(9), and ζ(11) is, but no single one of them has been settled. The even values are all rational multiples of powers of π, and so transcendental; the odd ones are the open frontier. Status open.", "cites": [ "r-5.13", "r-5.31" ], "links_to_cards": [] }, { "id": "pipi", "url": "https://www.teclamairme.com/#g-pipi", "name": "pi to the pi and e to the e", "symbol": "ππ + ee", "symbol_markup": "π^{π} + e^{e}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 36.46 and 15.15; neither is known to be irrational, though Schanuel’s conjecture would make both transcendental.", "more": "π^π ≈ 36.4622 and e^e ≈ 15.1543 are the two simplest self-powers of the famous transcendentals, and neither is known to be irrational; Gelfond–Schneider needs an algebraic base, and π and e are not. Schanuel’s conjecture of the 1960s would settle these, π + e, π·e, and π^e in one stroke, and remains open. Status open.", "cites": [ "r-5.12", "r-5.24", "r-5.31" ], "links_to_cards": [ "gelfschn", "pie", "schanuel" ] }, { "id": "levy", "url": "https://www.teclamairme.com/#g-levy", "name": "Lévy’s constant", "symbol": "γL", "symbol_markup": "γ_{L}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 3.2758, e^(π²/(12 ln 2)), the growth rate of the denominators of almost every continued fraction; irrationality open.", "more": "Lévy’s constant e^(π²/(12 ln 2)) ≈ 3.275823 is the limit of the nth root of the nth convergent denominator of the continued fraction of almost every real number. Khinchin proved in ~1935 that the limit exists and is the same for almost all numbers, and Paul Lévy found the closed form in 1936; Khinchin’s own constant beside it is the geometric mean of the partial quotients. Nothing is known about whether it is irrational. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "contfrac", "khin" ] }, { "id": "fransen", "url": "https://www.teclamairme.com/#g-fransen", "name": "the Fransén–Robinson constant", "symbol": "F", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 2.8078, the area under 1/Γ(x) from 0 to ∞; irrationality open.", "more": "The Fransén–Robinson constant F = ∫₀^∞ dx/Γ(x) ≈ 2.8077702, the area under the reciprocal gamma function of Chapter ᚐ(A), computed to high precision by Herman P. Robinson, who reached 36 places by Newton–Cotes quadrature, and by Arne Fransén, who reached 65 by Euler–Maclaurin summation. It is the continuous analogue of e = Σ 1/n!, and it has no known closed form and no known irrationality proof. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "e", "gammafn" ] }, { "id": "bernstein", "url": "https://www.teclamairme.com/#g-bernstein", "name": "Bernstein’s constant", "symbol": "β", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.2802, the limit of how badly polynomials of degree n must miss |x| on [−1, 1], times n; irrationality open.", "more": "Bernstein’s constant β ≈ 0.2801694 governs how well |x| can be approximated by polynomials on [−1, 1]: Sergei Bernstein showed in 1913 or 1914 that the best degree-n error times n tends to a limit, and computed it to a few digits; Varga and Carpenter found in 1985 that it is not 1/(2√π), which Bernstein had suspected. Its irrationality is open. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "viswanath", "url": "https://www.teclamairme.com/#g-viswanath", "name": "Viswanath’s constant", "symbol": "1.1319…", "chapter": 5, "type": "open", "badge": "Status open", "quick": "The growth rate of a Fibonacci sequence in which each step adds or subtracts at random; found by Divakar Viswanath in 1999, status open.", "more": "Viswanath’s constant ≈ 1.1319882 is the rate at which the absolute values of a random Fibonacci sequence, xₙ = ±xₙ₋₁ ± xₙ₋₂ with the signs chosen by coin flips, grow almost surely; Divakar Viswanath computed it in 1999, using the Stern–Brocot tree and a fractal measure. Nothing is known about its irrationality. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "niven", "url": "https://www.teclamairme.com/#g-niven", "name": "Niven’s constant", "symbol": "cN", "symbol_markup": "c_{N}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.7052, the average over all integers of the largest exponent in their factorization; irrationality open.", "more": "Niven’s constant ≈ 1.7052111 is the average, over the integers up to x as x grows, of the largest exponent appearing in the prime factorization; Ivan Niven found it in 1969 as 1 + Σ (1 − 1/ζ(k)) over k ≥ 2, so it is built from the zeta function of Chapter ᚑ(O). Its irrationality is open. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "zeta" ] }, { "id": "artin", "url": "https://www.teclamairme.com/#g-artin", "name": "Artin’s constant", "symbol": "CArtin", "symbol_markup": "C_{Artin}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.3740, the conjectured density of primes for which 2 is a primitive root; irrationality open.", "more": "Artin’s constant Π (1 − 1/(p(p − 1))) over primes ≈ 0.3739558 is the density, conjectured by Emil Artin in 1927, of primes p for which 2 generates every nonzero residue mod p, as in the cyclic number 142857 of Chapter ᚂ(L), where 10 is such a root of 7. Hooley proved the conjecture in 1967 assuming the generalized Riemann Hypothesis; unconditionally it is open, and so is the constant’s irrationality. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "cyclic", "rh" ] }, { "id": "sierpinski", "url": "https://www.teclamairme.com/#g-sierpinski", "name": "Sierpiński’s constant", "symbol": "KS", "symbol_markup": "K_{S}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 2.5850, the limit of a sum counting the ways integers up to n are sums of two squares, once π ln n is taken away; irrationality open.", "more": "Sierpiński’s constant K_{S} ≈ 2.5849817 is the limit of the sum Σ r₂(k)/k over k ≤ n minus π ln n, where r₂ counts representations as a sum of two squares; Wacław Sierpiński found it in 1907, and it is built from π, γ, and the Gamma function at ¼. It is not known to be irrational. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "gamma", "gammafn", "pi", "twosq" ] }, { "id": "mrb", "url": "https://www.teclamairme.com/#g-mrb", "name": "the MRB constant", "symbol": "MRB", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.1879, the limit of the alternating sums of nth roots of n; found by Marvin Ray Burns in 1999, status open.", "more": "The MRB constant ≈ 0.1878596 is the limit of the even partial sums of Σ (−1)ᵏ k^(1/k), the sums of the first 2n terms; the full series diverges, since its terms tend to ±1, but the even partial sums settle down, and their limit is the constant. Marvin Ray Burns found it in 1999 and it has since been computed to millions of digits without any closed form or irrationality proof. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "somos", "url": "https://www.teclamairme.com/#g-somos", "name": "Somos’ quadratic recurrence constant", "symbol": "σ", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.6617, the limit of √(1·√(2·√(3·…))), found by Michael Somos in 1999; irrationality open.", "more": "Somos’ quadratic recurrence constant σ = √(1·√(2·√(3·√(4·…)))) ≈ 1.6616879, the infinite nested radical Michael Somos studied in 1999; it equals Π n^(1/2ⁿ) and has expressions through the polylogarithm. It is not known to be irrational. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [ "polylog" ] }, { "id": "keplerb", "url": "https://www.teclamairme.com/#g-keplerb", "name": "the Kepler–Bouwkamp constant", "symbol": "ρKB", "symbol_markup": "ρ_{KB}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 0.1149, the radius of the circle left after inscribing a triangle, a square, a pentagon, and every polygon after them inside a unit circle, each inside the last; irrationality open.", "more": "The Kepler–Bouwkamp constant ≈ 0.1149420 is the radius of the circle that remains when a unit circle has a triangle inscribed in it, a circle inscribed in the triangle, a square in that circle, a circle in the square, a pentagon, and so on forever: the product of cos(π/n) over n ≥ 3. Kepler drew the first steps in 1596 when modeling the planets, in the Mysterium Cosmographicum, and Christoffel Jacob Bouwkamp computed the limit in 1965. Not known to be irrational. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "backhouse", "url": "https://www.teclamairme.com/#g-backhouse", "name": "Backhouse’s constant", "symbol": "BN", "symbol_markup": "B_{N}", "chapter": 5, "type": "open", "badge": "Status open", "quick": "About 1.4561, the growth rate hidden in the power series whose coefficients are the primes; found by Nigel Backhouse in 1995, status open.", "more": "Backhouse’s constant ≈ 1.4560749 is the limit of the ratios of successive coefficients of 1/P(x), where P(x) = 1 + 2x + 3x² + 5x³ + 7x⁴ + … has the primes as coefficients; Nigel Backhouse noticed the limit in 1995 and Philippe Flajolet proved it exists. It has no irrationality proof. Status open.", "cites": [ "r-5.26", "r-5.9" ], "links_to_cards": [] }, { "id": "schanuel", "url": "https://www.teclamairme.com/#g-schanuel", "name": "Schanuel’s conjecture", "symbol": "Schanuel", "chapter": 5, "type": "open", "badge": "Status open", "quick": "Stephen Schanuel’s unproved conjecture on exponentials, which would settle π + e, π·e, π to the e, π to the π, and e to the e in one stroke.", "more": "Schanuel’s conjecture says that if z₁, …, zₙ are complex numbers linearly independent over the rationals, then the field ℚ(z₁, …, zₙ, e^z₁, …, e^zₙ) has transcendence degree at least n over ℚ. Stephen Schanuel formulated it and Serge Lang published it in 1966. It would make e and π algebraically independent, and so settle π + e, π·e, π^e, π^π, and e^e—every unclassified combination in this chapter’s second brace—in one stroke. It remains one of the major unsolved problems of transcendental number theory, and nothing in this chapter assumes it. Status open.", "cites": [ "r-5.24" ], "links_to_cards": [ "pie", "pipi" ] }, { "id": "normalnum", "url": "https://www.teclamairme.com/#g-normalnum", "name": "a normal number", "symbol": "freq(b) = 1/10^|b|", "chapter": 5, "type": "open", "badge": "Status open", "quick": "Every block of digits turns up as often as chance says it should. Almost every real is normal, and almost no specific one is known to be.", "more": "A number is normal in a base when every finite block of digits appears in its expansion with exactly the frequency chance would assign it, and absolutely normal when that is true in every base at once. Émile Borel introduced the idea in 1909 and proved that almost every real number is normal—the exceptions form a set of measure zero—which is the classic case of knowing a property holds almost everywhere while being unable to certify it anywhere in particular. Sierpiński managed to specify an absolutely normal number in 1917, and Becher and Figueira produced a computable one in 2002, but whether π, e, or √2 is normal is open. The one thing normality does settle at once is irrationality: a rational expansion is eventually periodic, and a periodic expansion cannot give every block its share. Status open.", "cites": [ "r-5.28" ], "links_to_cards": [ "aeverywhere", "e", "pi", "sqrt2" ] }, { "id": "mersnum", "url": "https://www.teclamairme.com/#g-mersnum", "name": "Mersenne numbers", "symbol": "2ⁿ − 1", "chapter": 5, "type": "fn", "badge": "Function or operator", "quick": "A Mersenne number is any number of the form 2ⁿ − 1, named for Marin Mersenne, and it can only be prime when n itself is prime, which makes hunting for Mersenne primes one of the oldest games in number theory.", "more": "Mersenne numbers are 2ⁿ − 1 for n = 1, 2, 3, …; primality of n is necessary but not sufficient for 2ⁿ − 1 to be prime. The Erdős–Borwein constant is the sum of the reciprocals of all Mersenne numbers, Σ 1/(2ⁿ − 1), which Paul Erdős proved irrational in 1948. The largest known prime of any kind is almost always a Mersenne prime, such as the record-holder 2¹³⁶²⁷⁹⁸⁴¹ − 1, because a fast primality test, the Lucas–Lehmer test, exists just for this family.", "cites": [ "r-5.40" ], "links_to_cards": [ "eb" ] }, { "id": "uncomputable", "url": "https://www.teclamairme.com/#g-uncomputable", "name": "uncomputable", "symbol": "uncomputable", "chapter": 5, "type": "unc", "badge": "Function or property", "quick": "A precisely defined quantity or function for which no single algorithm can produce answers to every requested accuracy or input.", "more": "In this book, uncomputability means that no general method exists, as for the Busy Beaver function or the digits of a fixed Chaitin constant. It does not mean that no individual finite value can ever be written down. The x in Rose’s number is a stronger case: it is not even definable, so no formula names it and no program prints it.", "cites": [ "r-1.1", "r-5.25" ], "links_to_cards": [ "busybeaver", "omega" ] }, { "id": "infeasibility", "url": "https://www.teclamairme.com/#g-infeasibility", "name": "infeasibility", "symbol": "infeasibility", "chapter": 5, "type": "unc", "badge": "Author clarification", "quick": "The condition of being impracticable: possible in principle, but beyond a stated practical limit such as available time, space, or displayable size.", "more": "Author’s clarification: this is a glossary term used to name the book’s distinction between an exact finite quantity and a full calculation or expansion that cannot feasibly be carried out. It is not a new kind of number and does not mean mathematical impossibility.", "cites": [ "r-5.25" ], "links_to_cards": [] }, { "id": "bb6", "url": "https://www.teclamairme.com/#g-bb6", "name": "sixth Busy Beaver number", "symbol": "BB(6)", "chapter": 5, "type": "int", "badge": "Integer or rational", "quick": "Radó’s 1962 function at 6: a fixed finite integer whose exact value remains unknown; the function n ↦ BB(n), not this one value, is uncomputable.", "more": "The Busy Beaver function BB(n), defined by Tibor Radó in 1962, is the largest number of steps a halting Turing machine with n states and two symbols can make when started on a blank tape. No algorithm computes BB(n) for every n. BB(5) = 47,176,870 was formally proved in 2024; BB(6) remains unresolved, and a machine found in June 2025 shows it exceeds 2↑↑↑5 — a tower of twos whose height is itself a tower of twos. The distinction matters: BB as a function is uncomputable, but BB(6) is one fixed natural number and therefore computable in the standard existential sense, even though nobody presently knows its digits. Type: integer, status unresolved.", "cites": [ "r-5.19", "r-5.21", "r-9.5" ], "links_to_cards": [ "bb5", "busybeaver", "turingm", "uncomputable", "uparrow" ] }, { "id": "rayo", "url": "https://www.teclamairme.com/#g-rayo", "name": "Rayo’s number", "symbol": "Rayo’s", "chapter": 5, "type": "int", "badge": "Integer or rational", "quick": "The smallest integer larger than every integer nameable in at most a googol symbols of set theory; the winner of the 2007 big-number duel at MIT, claimed at the time as the largest named number.", "more": "Rayo’s number was defined by Agustín Rayo on 26 January 2007, in a big-number duel against Adam Elga at MIT, as the smallest positive integer larger than every finite integer denoted by a qualifying expression of first-order set theory using at most a googol symbols. The exact formal-language conditions are essential. It exceeds TREE(3), which has a much shorter description. Like every fixed finite integer it is computable in the technical sense, although no feasible computation could write it out. Type: integer beyond feasible calculation.", "cites": [ "r-5.15" ], "links_to_cards": [ "googol", "tree3" ] }, { "id": "inf", "url": "https://www.teclamairme.com/#g-inf", "name": "positive infinity of the extended reals", "symbol": "∞", "chapter": 5, "type": "inf", "badge": "Infinity or limit device", "quick": "The limit target in n → ∞; infinity labels the process and is not used as an ordinary exponent.", "more": "Positive infinity on the extended real line is a limit device, not an ordinary real number. In Chapter ᚃ(F) the formula uses it only in n → ∞ for the limit of x^{n} when 0 < x < 1; that limit is 0.", "cites": [ "r-5.17" ], "links_to_cards": [] }, { "id": "omega", "url": "https://www.teclamairme.com/#g-omega", "name": "Chaitin’s constant", "symbol": "Ω", "chapter": 5, "type": "unc", "badge": "Definable but uncomputable", "quick": "The halting probability: definable, uncomputable, and transcendental.", "more": "Fix an optimal universal prefix-free computer. Prefix-free means that no valid program is the beginning of another valid program; these programs delimit themselves. Its halting probability is Ω = Σ 2^{−|p|}, summed over its halting programs p. The value depends on the chosen computer. Gregory Chaitin defined the halting probability in 1975. It is a definite real number between 0 and 1 whose first n bits would decide the halting problem for all programs up to about n bits long, so no algorithm can compute it; it is also known to be transcendental, and algorithmically random, which is why it is normal. Its type is uncomputable, which outranks transcendental in this book’s color scheme. This Ω is Chaitin’s, not the Lambert omega constant W(1) of Chapter ᚐ(A).", "cites": [ "r-5.18" ], "links_to_cards": [ "W1", "halting", "kolmogorov", "uncomputable" ] }, { "id": "limop", "url": "https://www.teclamairme.com/#g-limop", "name": "the limit operator", "symbol": "lim", "chapter": 5, "type": "fn", "badge": "Function or operator", "quick": "The value approached by a sequence or function; here n tends to infinity while an ordinary power tends to zero.", "more": "The limit notation used in Chapter ᚃ(F). For every real x with 0 < x < 1, lim_{n→∞}x^{n} = 0. Newton used limiting arguments in the seventeenth century; Cauchy gave the subject systematic definitions in 1821 and Weierstrass supplied the modern epsilon language. In this formula infinity labels a process and is never treated as an ordinary exponent. A function of the framework.", "cites": [ "r-5.17", "r-5.31" ], "links_to_cards": [ "inf" ] }, { "id": "collatz", "url": "https://www.teclamairme.com/#g-collatz", "name": "the Collatz problem", "symbol": "n/2 or 3n + 1", "chapter": 5, "type": "open", "badge": "Status open", "quick": "Halve it if it is even, triple and add one if it is odd. Does every start reach 1? Verified for every start up to 2.36 × 10²¹, but never proved in general.", "more": "Lothar Collatz introduced the rule in the 1930s: halve an even number, replace an odd n by 3n + 1, repeat. The conjecture is that every positive integer eventually reaches 1. The orbits wander up and down so wildly that they are called hailstone sequences, and the conjecture has been checked for every start up to 2.36 × 10²¹ without a general proof appearing. It matters in this chapter because it is not only unsolved but obstructive: the machines that block the sixth Busy Beaver value are Collatz-like, so a problem nobody can settle about arithmetic becomes a problem nobody can settle about a particular six-state machine. Status open.", "cites": [ "r-5.29" ], "links_to_cards": [ "bb6" ] }, { "id": "wilson", "url": "https://www.teclamairme.com/#g-wilson", "name": "Wilson’s theorem", "symbol": "(12! + 1) mod 13", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "(p − 1)! + 1 is divisible by p exactly when p is prime; 479001601 = 13 × 36846277.", "more": "Wilson’s theorem states that (p − 1)! ≡ −1 (mod p) if and only if p is prime. Ibn al-Haytham knew it around 1000; Leibniz, the evidence suggests, knew it a century before Wilson and never published; John Wilson found it and Edward Waring printed it without proof in 1770, later calling it a most elegant property of primes; Lagrange gave the first proof soon after. Here p = 13, and 12! + 1 = 479,001,601 = 13 × 36,846,277, so the remainder is 0. The remainder of an integer division is an integer, which is why it carries the integer tag and hands 0 home directly.", "cites": [ "r-6.1", "r-6.2", "r-6.23", "r-6.38" ], "links_to_cards": [] }, { "id": "totient", "url": "https://www.teclamairme.com/#g-totient", "name": "Euler’s totient of 13", "symbol": "φ(13)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "Twelve numbers below 13 share no factor with it; this φ is not the golden ratio.", "more": "Euler’s totient φ(n), introduced by Euler in 1763, counts the integers from 1 to n that share no factor with n; the symbol φ is Gauss’s, from the Disquisitiones Arithmeticae of 1801. For a prime it is p − 1, so φ(13) = 12. It is the size of the multiplicative group of remainders and the modulus for the two exponents in RSA; the symbol φ is reused in this book, and here it is not the golden ratio of Chapter ᚂ(L).", "cites": [ "r-6.29", "r-6.3" ], "links_to_cards": [ "eulerthm", "phi", "rsa" ] }, { "id": "fermatlittle", "url": "https://www.teclamairme.com/#g-fermatlittle", "name": "Fermat’s little theorem", "symbol": "212 mod 13", "symbol_markup": "2^{12} mod 13", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "For a prime p that does not divide a, a^(p−1) leaves remainder 1 on division by p, so 4096 mod 13 = 1.", "more": "Fermat’s little theorem, stated in a letter to Frénicle de Bessy in 1640 and first proved in print by Euler in 1736, says that a^(p−1) ≡ 1 (mod p) whenever p is prime and does not divide a. So 2¹² = 4096 = 315 × 13 + 1 leaves remainder 1. It is the basis of the Fermat primality test and one of the ideas behind public-key cryptography, though a working cryptosystem needs a great deal more; it is an integer because it is a remainder.", "cites": [ "r-6.32", "r-6.4" ], "links_to_cards": [] }, { "id": "mod4", "url": "https://www.teclamairme.com/#g-mod4", "name": "thirteen is one more than a multiple of four", "symbol": "13 mod 4", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "So 13 = 2² + 3² by Fermat’s theorem on sums of two squares.", "more": "An odd prime leaves remainder 1 or 3 on division by 4, and Fermat’s theorem on sums of two squares, stated in 1640 and proved by Euler in 1749, says the odd primes that are sums of two squares are exactly those with remainder 1. 13 = 4 × 3 + 1, and 13 = 2² + 3². The remainder is an integer; it is placed here so the denominator asks the same prime a third question.", "cites": [ "r-6.34", "r-6.5" ], "links_to_cards": [ "fermat2sq" ] }, { "id": "legsym", "url": "https://www.teclamairme.com/#g-legsym", "name": "Legendre symbol", "symbol": "(3/13)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "Asks whether 3 is a perfect square in the world of remainders mod 13; it is, because 4² = 16 leaves remainder 3, so the symbol is 1.", "more": "The Legendre symbol (a/p), introduced by Legendre in 1797 or 1798, is 1 if a is a nonzero square modulo p, −1 if it is not, and 0 if p divides a. Since 4² = 16 ≡ 3 (mod 13), (3/13) = 1. Gauss’s law of quadratic reciprocity, which he proved in 1796 and called the golden theorem, tells you how (p/q) and (q/p) are related; the symbol is always an integer, and for a nonzero square it is exactly 1.", "cites": [ "r-6.34", "r-6.6" ], "links_to_cards": [ "qrecip" ] }, { "id": "ord13", "url": "https://www.teclamairme.com/#g-ord13", "name": "the order of 2 modulo 13", "symbol": "ord13(2)", "symbol_markup": "ord_{13}(2)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "The smallest k with 2^k ≡ 1 (mod 13) is 12, so 2 is a primitive root modulo 13.", "more": "The multiplicative order of a modulo n is the least k ≥ 1 with a^k ≡ 1 (mod n); by Fermat’s little theorem it divides p − 1 for a prime p. For a = 2 and p = 13 the powers 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1 run through every nonzero remainder before returning to 1, so the order is 12 and 2 is a primitive root, a notion Euler named; Gauss proved in 1801 that every prime has one. The Carmichael function λ(13) = 12 is the largest order that occurs. An order is a positive integer.", "cites": [ "r-6.29", "r-6.7" ], "links_to_cards": [ "fermatlittle" ] }, { "id": "rsa", "url": "https://www.teclamairme.com/#g-rsa", "name": "RSA", "symbol": "m^(ed) ≡ m (mod n)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Exponentiate modulo the product of two primes; the encrypting and decrypting exponents are inverses modulo the totient, so doing both returns the message.", "more": "RSA takes a modulus n = pq for two large primes, a public exponent e, and a private exponent d chosen so that ed ≡ 1 modulo the totient of n; encryption is m ↦ m^e mod n and decryption is the same operation with d, and it works because raising to ed returns the message — Euler’s theorem doing for a composite modulus what Fermat’s little theorem does for a prime one. Rivest, Shamir, and Adleman described it publicly in 1977, in Scientific American’s Mathematical Games column that August; Clifford Cocks had found an equivalent system secretly at GCHQ in 1973, and it stayed classified until 1997. The chapter mentions it to make the opposite point to the usual one: the totient identity is the arithmetic skeleton, and a working cryptosystem needs a great deal more than a skeleton. A scheme, uncolored.", "cites": [ "r-6.36" ], "links_to_cards": [ "eulerthm", "fermatlittle" ] }, { "id": "mobius", "url": "https://www.teclamairme.com/#g-mobius", "name": "the Möbius function at Kaprekar’s constant", "symbol": "μ(6174)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "6174 = 2·3²·7³ has a repeated prime factor, so μ is 0; this μ is not the Ramanujan–Soldner constant of Chapter ᚐ(A).", "more": "The Möbius function μ(n), defined by August Ferdinand Möbius in 1832, is 1 for n = 1, (−1)^k when n is a product of k distinct primes, and 0 when n has a repeated prime factor. Kaprekar’s constant 6174 = 2 × 3² × 7³ has two repeated primes, so μ(6174) = 0. The function inverts sums over divisors, and a bound on the growth of its partial sums is equivalent to the Riemann Hypothesis; its values are integers, hence the integer tag. It is not the Ramanujan–Soldner constant μ of Chapter ᚐ(A).", "cites": [ "r-6.26", "r-6.8" ], "links_to_cards": [ "kaprekar", "mertensM", "mu", "rh" ] }, { "id": "gcdlcm", "url": "https://www.teclamairme.com/#g-gcdlcm", "name": "the gcd–lcm identity", "symbol": "gcd(12, 18)·lcm(12, 18) − 12·18", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "0: 6 × 36 = 216 = 12 × 18. For any two positive integers the greatest common divisor times the least common multiple is their product.", "more": "Every prime power appears in gcd(a, b) with the smaller of its two exponents and in lcm(a, b) with the larger, so together they carry exactly the exponents of a·b. Euclid’s algorithm (Elements VII.2, about 300 BC) computes the gcd by repeated division and is, in Knuth’s phrase, the oldest nontrivial algorithm that has survived to the present day; the identity then gives the lcm for free. An integer identity and the carrier for divisibility.", "cites": [ "r-6.25", "r-6.37", "r-6.9" ], "links_to_cards": [ "euclidalg" ] }, { "id": "gcd1218", "url": "https://www.teclamairme.com/#g-gcd1218", "name": "the greatest common divisor of 12 and 18", "symbol": "gcd(12, 18)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "6, found by Euclid’s algorithm in two lines: 18 = 12 + 6, 12 = 2·6.", "more": "The largest integer dividing both 12 and 18. Euclid’s algorithm replaces the pair by the smaller number and the remainder until the remainder is 0; the last nonzero remainder is the gcd. Lamé proved in 1844 that the number of steps is at most five times the number of decimal digits of the smaller number. An integer.", "cites": [ "r-6.10", "r-6.34", "r-6.9" ], "links_to_cards": [ "euclidalg" ] }, { "id": "lcm1218", "url": "https://www.teclamairme.com/#g-lcm1218", "name": "the least common multiple of 12 and 18", "symbol": "lcm(12, 18)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "36, the first number that both 12 and 18 divide.", "more": "The smallest positive integer that both numbers divide; here 36 = 2²·3². Together with the gcd it satisfies gcd·lcm = 12·18. An integer.", "cites": [ "r-6.9" ], "links_to_cards": [ "gcdlcm" ] }, { "id": "euclidalg", "url": "https://www.teclamairme.com/#g-euclidalg", "name": "the Euclidean algorithm", "symbol": "a = qb + r", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Divide, keep the remainder, repeat: the gcd is the last nonzero one. Knuth calls it the oldest nontrivial algorithm that has survived to the present day.", "more": "To find the greatest common divisor of a and b, divide the larger by the smaller and keep the remainder, then repeat with the smaller and the remainder. The remainders strictly decrease and cannot go below zero, so the process stops, and the last nonzero remainder is the gcd. It is in Book VII of the Elements. For 12 and 18 it takes two lines: 18 = 12 + 6 and 12 = 2 × 6, so the answer is 6. Knuth’s phrase for it is “the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day”. An algorithm, uncolored.", "cites": [ "r-6.10", "r-6.25" ], "links_to_cards": [] }, { "id": "partition5", "url": "https://www.teclamairme.com/#g-partition5", "name": "Ramanujan’s partition congruence, at nine", "symbol": "p(9) mod 5", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "p(9) = 30, so the remainder is 0: the n = 1 case of Ramanujan’s congruence p(5n+4) ≡ 0 (mod 5), which holds for every n.", "more": "Let p(n) count the ways to write n as an unordered sum of positive integers: p(4) = 5 and p(9) = 30. Ramanujan proved in 1919 that p(5n+4) ≡ 0 (mod 5) for every n ≥ 0, together with a companion congruence modulo 7; the third, modulo 11, is his as well, but its proof reached print only when Hardy drew it out of an unpublished manuscript in 1921. The formula shows the n = 1 instance, 30 mod 5 = 0. It is a celebrated doorway from elementary counting into q-series and modular forms. The displayed remainder is exactly the integer 0.", "cites": [ "r-6.11", "r-6.12", "r-6.31" ], "links_to_cards": [ "p4" ] }, { "id": "p4", "url": "https://www.teclamairme.com/#g-p4", "name": "the partitions of four", "symbol": "p(4)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "5: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1, the n = 0 case of Ramanujan’s congruence, since 5 is itself a multiple of 5.", "more": "The partition function p(n) counts the ways to write n as an unordered sum of positive integers: p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, and p(100) = 190,569,292. Euler found its generating function in the 1740s; Hardy and Ramanujan found its asymptotic size in 1918, and Rademacher an exact series in 1937. An integer.", "cites": [ "r-6.11", "r-6.12", "r-6.34" ], "links_to_cards": [] }, { "id": "ramanujancong", "url": "https://www.teclamairme.com/#g-ramanujancong", "name": "Ramanujan’s partition congruences", "symbol": "p(5n+4)≡0(5), p(7n+5)≡0(7), p(11n+6)≡0(11)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Srinivasa Ramanujan’s 1919 discovery that the partition function is always divisible by 5, 7, or 11 along three specific arithmetic progressions of n.", "more": "Ramanujan found that p(5n+4) is always a multiple of 5, p(7n+5) always a multiple of 7, and p(11n+6) always a multiple of 11, for every nonnegative n, congruences he announced in 1919. Ken Ono and others have since extended the pattern to further prime moduli. The chapter’s own card is the single mod-5 instance at n = 1, p(9) mod 5 = 0; this is the general family it belongs to. A theorem, uncolored.", "cites": [ "r-6.42" ], "links_to_cards": [ "partition5" ] }, { "id": "binomial", "url": "https://www.teclamairme.com/#g-binomial", "name": "two choose five", "symbol": "C(2, 5)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "There are no ways to choose five things from two, so the binomial coefficient is 0.", "more": "The binomial coefficient C(n, k), the number of ways to choose k things from n, appears in Pascal’s triangle, written in 1654 and published in 1665 after his death, but known in India, Persia, and China centuries earlier. When k exceeds n there is nothing to choose, so C(2, 5) = 0. Counting results are integers, hence the tag; the display keeps the (n, k) order used in the formula.", "cites": [ "r-6.13", "r-6.34" ], "links_to_cards": [ "permfact" ] }, { "id": "ramsey33", "url": "https://www.teclamairme.com/#g-ramsey33", "name": "the smallest nontrivial Ramsey number", "symbol": "R(3, 3)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "6: with acquaintance treated as mutual, any six people contain three mutual acquaintances or three mutual strangers; five can avoid both.", "more": "Frank Ramsey proved in 1930 that complete disorder is impossible: for any r and s there is a least R(r, s) such that every two-coloring of the edges of a complete graph on that many vertices contains a red clique of size r or a blue one of size s. Model the people as vertices of a complete graph, coloring each edge red for mutual acquaintance and blue for mutual strangers. For six vertices, choose v. By the pigeonhole principle at least three of its five incident edges have the same color; exchange the colors if necessary and call it red. Let a, b, and c be their other endpoints. If any edge among a, b, and c is red, it forms a red triangle with v. Otherwise all three edges among a, b, and c are blue, so they form a blue triangle. Hence six always suffice. For five vertices arranged around a pentagon, color the five sides red and the five diagonals blue. Each color forms a five-cycle, and neither contains a triangle. Thus five do not suffice. Together these bounds prove R(3, 3) = 6. Greenwood and Gleason computed R(3, 4) = 9 and R(4, 4) = 18 in 1955, and R(5, 5) is still only known to lie between 43 and 46; Erdős’s story is that if aliens demanded R(5, 5) we should compute it, but if they demanded R(6, 6) we should attack. A count of vertices, hence the integer tag.", "cites": [ "r-6.14", "r-6.34" ], "links_to_cards": [ "pigeon", "ramsey44" ] }, { "id": "catalan3", "url": "https://www.teclamairme.com/#g-catalan3", "name": "the third Catalan number", "symbol": "C3", "symbol_markup": "C_{3}", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "5: the number of ways to bracket four factors, to triangulate a pentagon, or to match three pairs of parentheses.", "more": "The Catalan numbers 1, 1, 2, 5, 14, 42, … are C_n = (2n)!/((n + 1)! n!); Euler counted triangulations of polygons with them in 1751, Eugène Catalan attached his name to them through bracketing problems, and Ming Antu had found them in China around 1730. They count more than two hundred families of objects in Stanley’s catalog. Not to be confused with Catalan’s constant G in Chapter ᚃ(F). An ordinary count, hence the integer tag.", "cites": [ "r-6.15", "r-6.34", "r-6.37" ], "links_to_cards": [ "catalan" ] }, { "id": "bell3", "url": "https://www.teclamairme.com/#g-bell3", "name": "the third Bell number", "symbol": "B₃", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "5: the ways to split three things into nonempty groups, {abc}, {ab}{c}, {ac}{b}, {bc}{a}, {a}{b}{c}.", "more": "The Bell numbers 1, 1, 2, 5, 15, 52, … count the partitions of a set of n things into nonempty groups. The earliest known exhaustive enumeration was a medieval Japanese parlor game, genjikō, on the 52 partitions of five incense sticks; Dobiński gave a formula in 1877; Eric Temple Bell wrote about them in 1934 and 1938, saying they had been rediscovered many times, and Becker and Riordan named them for him in 1948. B₃ = 5, the same 5 as the Catalan number beside it, by coincidence. A count, hence an integer.", "cites": [ "r-6.16", "r-6.34" ], "links_to_cards": [ "catalan3" ] }, { "id": "derange4", "url": "https://www.teclamairme.com/#g-derange4", "name": "the derangements of four", "symbol": "!4", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "9: the ways to return four hats to four people so that nobody gets their own.", "more": "A derangement is a permutation with no fixed point; !n counts them, and !4 = 9 (the sequence runs 1, 0, 1, 2, 9, 44, …). Pierre Rémond de Montmort posed the problem in his Essay d’analyse sur les jeux de hazard in 1708 and solved it in 1713, as did Nicolaus Bernoulli; the name subfactorial is William Allen Whitworth’s. The fraction !n/n! tends to 1/e; the secretary constant of Chapter ᚋ(M) is the same number. A count, hence an integer.", "cites": [ "r-6.17", "r-6.34" ], "links_to_cards": [ "secretary" ] }, { "id": "binomthm", "url": "https://www.teclamairme.com/#g-binomthm", "name": "the binomial theorem", "symbol": "(x+y)ⁿ = Σ C(n, k) xᵏ yⁿ⁻ᵏ", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "The binomial theorem expands (x + y)ⁿ into a sum of terms C(n, k) xᵏ yⁿ⁻ᵏ, turning the binomial coefficients into the coefficients of an algebraic expansion.", "more": "Each term’s coefficient C(n, k), the number of ways to choose k things from n, is exactly the entry Pascal’s triangle puts in that position, so the triangle is a row-by-row picture of the theorem. Isaac Newton generalized it in 1665 to non-integer exponents, turning the finite sum into an infinite series. The convention 0! = 1, which makes C(n, 0) = C(n, n) = 1, is what makes the expansion’s first and last terms come out right.", "cites": [ "r-6.45" ], "links_to_cards": [ "zerofact" ] }, { "id": "chrom", "url": "https://www.teclamairme.com/#g-chrom", "name": "the chromatic polynomial of a triangle at two colors", "symbol": "P(K₃, 2)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "A triangle cannot be 2-colored, so the count of proper colorings is 0.", "more": "The chromatic polynomial P(G, k), introduced for planar graphs by George Birkhoff in 1912 during work on the four-color problem and since extended to all graphs, counts the proper colorings of a graph G with k colors. For the triangle K₃ it is k(k − 1)(k − 2), which is 0 at k = 2: three mutually adjacent vertices need three colors. A count of colorings is an integer.", "cites": [ "r-6.18", "r-6.30" ], "links_to_cards": [] }, { "id": "handshake", "url": "https://www.teclamairme.com/#g-handshake", "name": "the handshaking lemma, for the Petersen graph", "symbol": "10·3 − 2·15", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "0: the Petersen graph has ten vertices of degree three and fifteen edges, and the degrees add to twice the edges, as they do in every finite graph.", "more": "For a finite undirected graph G = (V, E), adding the degrees of all vertices counts each edge twice, once at each endpoint, so Σ_{v∈V}deg(v) = 2|E|. The Petersen graph (Julius Petersen, 1898), ten vertices each of degree three and fifteen edges, gives 10·3 − 2·15 = 0. The lemma implies that every graph has an even number of odd-degree vertices and sits behind Euler trails and the Königsberg bridges. An integer identity and the carrier for graph theory.", "cites": [ "r-6.19" ], "links_to_cards": [ "konigsberg", "petersenE", "petersengraph" ] }, { "id": "petersenE", "url": "https://www.teclamairme.com/#g-petersenE", "name": "the edges of the Petersen graph", "symbol": "|E(Petersen)|", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "15: the ten-vertex, degree-three graph named for Petersen’s paper of 1898, a counterexample to many optimistic predictions about graphs in general.", "more": "Julius Petersen drew the graph in 1898 as the smallest bridgeless cubic graph with no three-edge coloring of its edges, though the same drawing had appeared twelve years earlier in a paper of A. B. Kempe, as the ten lines of the Desargues configuration; it has ten vertices, fifteen edges, girth five, is not planar, has no Hamiltonian cycle, and turns up as the smallest counterexample so often that a whole book (Holton and Sheehan, 1993) is devoted to it. An integer.", "cites": [ "r-6.19", "r-6.24", "r-6.37", "r-6.38" ], "links_to_cards": [] }, { "id": "chromK33", "url": "https://www.teclamairme.com/#g-chromK33", "name": "the chromatic polynomial of a triangle at three colors", "symbol": "P(K₃, 3)", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "6: with three colors a triangle can be properly colored in 3·2·1 ways, the same count as its symmetries.", "more": "The chromatic polynomial of the triangle is k(k − 1)(k − 2): the first vertex has k choices, the next k − 1, the last k − 2. At k = 2 it is 0 (the numerator’s zero); at k = 3 it is 6. Birkhoff introduced the polynomial in 1912 in the hope of proving the four-color theorem by showing P(G, 4) > 0 for every planar graph; Appel and Haken’s proof of 1976 went instead through reducible configurations. An integer.", "cites": [ "r-6.18", "r-6.30" ], "links_to_cards": [ "chrom" ] }, { "id": "congr", "url": "https://www.teclamairme.com/#g-congr", "name": "congruence notation", "symbol": "a ≡ b (mod n)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Gauss’s 1801 notation for two integers that leave the same remainder on division by n; ≡ reads as congruent.", "more": "Carl Friedrich Gauss introduced the congruence a ≡ b (mod n), meaning n divides a − b, at the opening of the Disquisitiones Arithmeticae (1801), choosing ≡ deliberately to look like equality because it behaves like it: congruences can be added, subtracted, and multiplied. Every symbol in this chapter’s modular-arithmetic brace is a statement in that notation. It is a relation, not a value, hence the operator tag.", "cites": [ "r-6.20", "r-6.29" ], "links_to_cards": [] }, { "id": "zmod13", "url": "https://www.teclamairme.com/#g-zmod13", "name": "the integers modulo 13", "symbol": "ℤ/13ℤ", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "The thirteen remainders 0 to 12 with clock arithmetic; because 13 is prime, every nonzero remainder has a reciprocal, so it is a field.", "more": "ℤ/nℤ is the ring of remainders modulo n, written ℤ_n by some authors; it has n elements, and it is a field exactly when n is prime, because then every nonzero remainder has a multiplicative inverse (the pairing trick in Wilson’s proof). ℤ/13ℤ is the world every term of this chapter’s modular-arithmetic brace is computed in. Finite fields of every prime-power order exist and are unique, a theorem of E. H. Moore (1893). It is a structure rather than a value, so it is tagged as a tool.", "cites": [ "r-6.20", "r-6.34" ], "links_to_cards": [ "wilson" ] }, { "id": "crt", "url": "https://www.teclamairme.com/#g-crt", "name": "the Chinese remainder theorem", "symbol": "x ≡ 2 (mod 3), x ≡ 3 (mod 5) ⇒ x ≡ 8 (mod 15)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Congruences to coprime moduli can always be solved together, with a unique answer modulo the product.", "more": "The Chinese remainder theorem says that if the moduli n₁, …, n_k are pairwise coprime, any system x ≡ a_i (mod n_i) has exactly one solution modulo n₁⋯n_k. The oldest known instance is in Sunzi’s Suanjing (3rd to 5th century AD): a number that leaves 2 on division by 3, 3 by 5, and 2 by 7 is 23. Qin Jiushao gave the general algorithm in 1247, and Gauss restated it in 1801. It is the reason ℤ/15ℤ behaves like ℤ/3ℤ × ℤ/5ℤ and the workhorse of RSA decryption. A theorem, hence the tool tag.", "cites": [ "r-6.21", "r-6.37" ], "links_to_cards": [ "congr", "rsa" ] }, { "id": "qrecip", "url": "https://www.teclamairme.com/#g-qrecip", "name": "quadratic reciprocity", "symbol": "(p/q)(q/p) = (−1)^((p−1)/2 · (q−1)/2)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Gauss’s golden theorem: whether p is a square modulo q decides whether q is a square modulo p, up to a sign that depends only on the two primes modulo 4.", "more": "The law of quadratic reciprocity, conjectured by Euler and Legendre and first proved by Gauss in 1796 (published 1801; he gave eight proofs, six of them in his lifetime, and privately called it the golden theorem), relates the Legendre symbols (p/q) and (q/p) for odd primes: their product is −1 exactly when both p and q leave remainder 3 on division by 4. With p = 3 and q = 13, (13/3) = (1/3) = 1 and the exponent is 1 × 6 = 6, which is even, so the law predicts (3/13) = 1, which agrees with 4² ≡ 3. It is the start of class field theory. A law, hence the tool tag.", "cites": [ "r-6.34", "r-6.6" ], "links_to_cards": [ "legsym" ] }, { "id": "eulerthm", "url": "https://www.teclamairme.com/#g-eulerthm", "name": "Euler’s theorem", "symbol": "a^φ(n) ≡ 1 (mod n)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Fermat’s little theorem for any modulus: raise a number coprime to n to the totient of n and the remainder is 1.", "more": "Euler’s theorem, published by Euler in 1763, in a paper presented in 1758 that uses the totient without yet naming it, generalizes Fermat’s little theorem from primes to every modulus: if gcd(a, n) = 1 then a^φ(n) ≡ 1 (mod n). With n = 13 it is exactly the bracket 2¹² ≡ 1 in the denominator; with n = 9 it says 2⁶ = 64 ≡ 1 (mod 9). It is the theorem that makes RSA decryption undo encryption. A theorem, hence the tool tag.", "cites": [ "r-6.3", "r-6.37", "r-6.4" ], "links_to_cards": [ "fermatlittle", "rsa" ] }, { "id": "floorceil", "url": "https://www.teclamairme.com/#g-floorceil", "name": "floor and ceiling", "symbol": "⌊13/4⌋, ⌈13/4⌉", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Round down and round up: ⌊13/4⌋ = 3 and ⌈13/4⌉ = 4, and 13 mod 4 is 13 − 4⌊13/4⌋ = 1.", "more": "The floor ⌊x⌋ is the largest integer not above x and the ceiling ⌈x⌉ the smallest not below it. Gauss wrote [x] for the floor in his third proof of quadratic reciprocity (1808); Kenneth Iverson named floor and ceiling and drew the brackets in A Programming Language (1962), the same book that introduced the notation behind Chapter ᚆ(H)’s Iverson bracket. Every remainder in this chapter is a floor in disguise: a mod n = a − n⌊a/n⌋, so 13 mod 4 = 13 − 4 × 3 = 1. A notation rather than a value, hence the tool tag.", "cites": [ "r-6.20", "r-6.37" ], "links_to_cards": [ "iverson", "mod4", "qrecip" ] }, { "id": "pigeon", "url": "https://www.teclamairme.com/#g-pigeon", "name": "the pigeonhole principle", "symbol": "n + 1 objects into n boxes", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Put more things than boxes and some box holds two: the simplest counting argument, used by Dirichlet in 1842.", "more": "If n + 1 objects are placed in n boxes, some box contains at least two. Dirichlet used it in 1842 to approximate irrationals by rationals, and later called it the Schubfachprinzip, and it is the seed of Ramsey theory: R(3, 3) = 6 is proved by a pigeonhole argument. Trivial to state and endlessly useful, it is a principle rather than a value, hence the tool tag.", "cites": [ "r-6.14", "r-6.38" ], "links_to_cards": [ "ramsey33" ] }, { "id": "permfact", "url": "https://www.teclamairme.com/#g-permfact", "name": "counting notation", "symbol": "n!, P(n, k), C(n, k)", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Factorials count orderings, P(n, k) counts ordered choices, C(n, k) counts unordered ones; 5! = 120 is the size of S₅ in Chapter ᚌ(G).", "more": "n! = n(n − 1)⋯1 counts the orderings of n things (the notation is Christian Kramp’s, 1808), P(n, k) = n!/(n − k)! counts ordered selections, and C(n, k) = n!/(k!(n − k)!) counts unordered ones; 0! = 1 by the empty-product convention. The binomial theorem (x + y)ⁿ = Σ C(n, k) xᵏ yⁿ⁻ᵏ ties them to algebra, and Stirling’s approximation n! ≈ √(2πn)(n/e)ⁿ, from Stirling’s work of 1729 and 1730, ties them to the √(2π) of Chapter ᚋ(M). Notation, hence the tool tag.", "cites": [ "r-6.22", "r-6.37" ], "links_to_cards": [ "binomthm", "s5", "sqrt2pi", "zerofact" ] }, { "id": "stirling42", "url": "https://www.teclamairme.com/#g-stirling42", "name": "a Stirling number of the second kind", "symbol": "{4 2} = 7", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "The ways to split four things into exactly two nonempty groups: 7. Summing the Stirling numbers of a row gives the Bell number.", "more": "The Stirling numbers of the second kind S(n, k) count the ways to split a set of n things into exactly k nonempty groups. They are written {n k} in a brace notation that Jovan Karamata used in 1935 and Marx and Salmeri in 1962, and that Knuth adopted in 1968; James Stirling introduced them in Methodus Differentialis (1730). S(4, 2) = 7, and S(4, 1) + S(4, 2) + S(4, 3) + S(4, 4) = 1 + 7 + 6 + 1 = 15 = B₄. The numbers of the first kind count permutations by cycles. Key entry only; an integer.", "cites": [ "r-6.16", "r-6.38" ], "links_to_cards": [ "bell3" ] }, { "id": "ramsey44", "url": "https://www.teclamairme.com/#g-ramsey44", "name": "the next Ramsey number", "symbol": "R(4, 4) = 18", "chapter": 6, "type": "int", "badge": "Integer or rational", "quick": "Among eighteen people there are four mutual acquaintances or four mutual strangers, and seventeen are not enough; R(5, 5) is unknown, somewhere from 43 to 46.", "more": "Greenwood and Gleason proved R(4, 4) = 18 in 1955; the witness that seventeen are not enough is a two-coloring of the complete graph on seventeen vertices with no monochromatic four-clique. Beyond that the subject collapses: R(5, 5) is only known to lie between 43 (Exoo 1989) and 46 (Angeltveit and McKay 2024), and Erdős’s story is that if aliens demanded R(5, 5) we should compute it, but if they demanded R(6, 6) we should attack. Key entry beside R(3, 3) = 6 in the denominator; an integer.", "cites": [ "r-6.14" ], "links_to_cards": [ "ramsey33" ] }, { "id": "konigsberg", "url": "https://www.teclamairme.com/#g-konigsberg", "name": "the seven bridges of Königsberg", "symbol": "7 bridges", "chapter": 6, "type": "fn", "badge": "Function or operator", "quick": "Could a walk cross each of Königsberg’s seven bridges exactly once? Euler answered no in 1736, and that paper is where graph theory begins.", "more": "Königsberg had seven bridges over the Pregel, and the question was whether a walk could cross each one exactly once. Euler’s answer was no, and the reason is the whole of the method: throw away the map. What the walk needs is not distances or shapes but which land masses touch which bridges, and once you have that, a walk that enters a land mass must also leave it, so every land mass except the two endpoints needs an even number of bridges. Königsberg had four with an odd number, so there is no such walk. The paper is Solutio problematis ad geometriam situs pertinentis, and it is the beginning of graph theory. A theorem, uncolored.", "cites": [ "r-6.27", "r-6.33" ], "links_to_cards": [] }, { "id": "petersengraph", "url": "https://www.teclamairme.com/#g-petersengraph", "name": "the Petersen graph", "symbol": "10 vertices, 15 edges", "chapter": 6, "type": "obj", "badge": "Non-real object", "quick": "A ten-vertex, degree-three graph built from a pentagon and a pentagram, the standard small counterexample that disproves many optimistic conjectures in graph theory.", "more": "The Petersen graph joins the five points of a pentagon to the five points of a pentagram, one to one, giving ten vertices each of degree three. Julius Petersen published it in 1898 as the smallest bridgeless cubic graph that cannot be properly edge-colored with three colors, though A. B. Kempe had already drawn the same graph in 1886. It has since served as graph theory’s standard counterexample to optimistic conjectures. An object, uncolored.", "cites": [ "r-6.39" ], "links_to_cards": [] }, { "id": "nim356", "url": "https://www.teclamairme.com/#g-nim356", "name": "a losing Nim position", "symbol": "3 ⊕ 5 ⊕ 6", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "011 ⊕ 101 ⊕ 110 = 000; whoever must move from here loses against correct play.", "more": "Heaps of 3, 5, and 6 are 011, 101, and 110 in binary; each column holds two ones, so the nim-sum is 000. By Bouton’s theorem of 1901 the player who must move from this position loses against correct play, whichever move they choose. It is not the smallest losing position with three heaps, which is 1, 2, 3; it was chosen because all three binary columns cancel visibly. The nim-sum is an integer, so the numerator hands 0 home directly.", "cites": [ "r-7.1", "r-7.2", "r-7.3" ], "links_to_cards": [ "nimsum" ] }, { "id": "nim124", "url": "https://www.teclamairme.com/#g-nim124", "name": "a winning Nim position", "symbol": "1 ⊕ 2 ⊕ 4", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "Equal to 7, so the mover can win: reduce the heap of 4 to 3 and the nim-sum becomes 0.", "more": "Heaps of 1, 2, and 4 are 001, 010, and 100, and their nim-sum is 111, which is 7. A nonzero nim-sum means the mover can win: the winning move is to reduce the heap of 4 to 3, because 1 ⊕ 2 ⊕ 3 = 0, and from then on the mover answers every move by restoring a zero nim-sum. The value 7 is a nonzero integer, which is what a denominator needs.", "cites": [ "r-7.2", "r-7.4" ], "links_to_cards": [ "nimsum" ] }, { "id": "nimprod", "url": "https://www.teclamairme.com/#g-nimprod", "name": "nimber multiplication", "symbol": "2 ⊗ 2", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "In Conway’s nimber field 2 ⊗ 2 = 3; the denominator is 7 + 3 = 10, because each term comes home as an integer and the + is ordinary addition.", "more": "John Conway, in On Numbers and Games (1976), defined a multiplication ⊗ that together with the nim-sum makes the nonnegative integers a field of characteristic 2, in which every number is its own negative: the nimbers. The rule is that a ⊗ b is the smallest number not of the form (a′ ⊗ b) ⊕ (a ⊗ b′) ⊕ (a′ ⊗ b′) for a′ < a and b′ < b, and it gives 2 ⊗ 2 = 3. The nimbers below 2^(2^n) form finite fields, so a product of nonzero nimbers is never zero. The value 3 is an integer, and it joins the nim-sum 7 as an ordinary sum: the denominator of this brace is 7 + 3 = 10, not a further nim-sum.", "cites": [ "r-7.18", "r-7.7", "r-7.8" ], "links_to_cards": [ "nimsum" ] }, { "id": "mpvalue", "url": "https://www.teclamairme.com/#g-mpvalue", "name": "the value of matching pennies", "symbol": "v(matching pennies)", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "0: in the zero-sum game where one player wins a penny if the coins match and loses one if they differ, the best either player can guarantee, by flipping fairly, is to break even.", "more": "Matching pennies is a smallest zero-sum game with no pure-strategy solution: each player shows heads or tails, one wins the penny on a match, the other on a mismatch. Von Neumann’s minimax theorem of 1928 says every finite two-player zero-sum game has a value, the payoff a player can guarantee with a mixed strategy, and here it is 0, reached when both players flip fairly. Nash’s 1950 theorem extended the existence of equilibrium to every finite game. An integer, and the carrier for classical game theory.", "cites": [ "r-7.10", "r-7.11", "r-7.12", "r-7.20" ], "links_to_cards": [ "minimax", "nasheq" ] }, { "id": "mixhalf", "url": "https://www.teclamairme.com/#g-mixhalf", "name": "the matching-pennies mixed strategy", "symbol": "½", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "One half: show heads with probability ½, and the opponent can no longer exploit you. When both players do this, that is the only Nash equilibrium of the game.", "more": "A mixed strategy is a probability distribution over the moves. In matching pennies any bias toward heads or tails can be punished, so the unique equilibrium, in the sense Nash defined in 1950, is for both players to mix fifty-fifty; then each move earns the same expected payoff and neither player can gain by changing. A rational number.", "cites": [ "r-7.11", "r-7.12" ], "links_to_cards": [ "nasheq" ] }, { "id": "pdpayoff", "url": "https://www.teclamairme.com/#g-pdpayoff", "name": "the prisoner’s dilemma equilibrium payoff", "symbol": "u(D, D)", "chapter": 7, "type": "int", "badge": "Integer or rational", "quick": "1: with the usual payoffs 5, 3, 1, 0, both prisoners defect and each gets 1, though both would have got 3 by cooperating.", "more": "Merrill Flood and Melvin Dresher devised the game at RAND in 1950 and Albert Tucker gave it the prisoners, and with them the name. With temptation 5, reward 3, punishment 1, and sucker’s payoff 0, defecting is better whatever the other player does, so mutual defection is the only Nash equilibrium and pays each player 1; mutual cooperation would pay 3. It is the standard example of an equilibrium that is not the best outcome for either player. An integer.", "cites": [ "r-7.13", "r-7.14" ], "links_to_cards": [ "nasheq" ] }, { "id": "nimsum", "url": "https://www.teclamairme.com/#g-nimsum", "name": "nim-sum", "symbol": "⊕", "chapter": 7, "type": "fn", "badge": "Function or operator", "quick": "Bitwise exclusive-or of heap sizes; a position is lost for the mover exactly when it is 0, under the last-move-wins rule (Bouton 1901).", "more": "The nim-sum ⊕ is bitwise exclusive-or: write the heap sizes in binary and add each column without carrying, which is exactly what a binary adder computes for its sum bit: A ⊕ B in a half adder, A ⊕ B ⊕ C_in in a full one. Charles Bouton proved in 1901 that a Nim position in normal play is a loss for the player about to move exactly when the nim-sum of the heaps is 0, the first complete solution of a combinatorial game in general form, with any number of heaps of any size. The game’s own origin is unknown: variants have been played since ancient times, a Chinese origin has been proposed and disputed, and Bouton himself said he had not been able to discover much concerning its history; the name is usually traced to the German imperative nimm, take. The Sprague–Grundy theorem (1935, 1939) extends it to every finite impartial game under the same convention. It is an operation rather than a value, hence the operator tag.", "cites": [ "r-7.1", "r-7.17", "r-7.2", "r-7.5", "r-7.6", "r-7.9" ], "links_to_cards": [ "sg" ] }, { "id": "sg", "url": "https://www.teclamairme.com/#g-sg", "name": "the Sprague–Grundy value", "symbol": "𝒢(G) = mex", "chapter": 7, "type": "fn", "badge": "Function or operator", "quick": "Every finite impartial game under normal play is worth exactly one Nim heap, of size given by the mex, the least number not among the values of the positions one move away.", "more": "Roland Sprague (1935) and Patrick Grundy (1939) independently proved that any finite impartial game under normal play is equivalent to a single Nim heap, whose size 𝒢(G) is computed recursively as the mex, the minimum excluded value, of the Grundy values of the positions reachable in one move. A position is a loss for the mover exactly when 𝒢 = 0, and the value of a sum of games is the nim-sum of the values, which is why ⊕ and Bouton’s theorem decide every such game and not only Nim. A function on games, hence the tool tag.", "cites": [ "r-7.2", "r-7.9" ], "links_to_cards": [ "nimsum" ] }, { "id": "minimax", "url": "https://www.teclamairme.com/#g-minimax", "name": "the minimax theorem", "symbol": "max min = min max", "chapter": 7, "type": "fn", "badge": "Function or operator", "quick": "Every finite two-player zero-sum game has a value: a payoff each player can guarantee by mixing moves at random in the right proportions. Von Neumann, 1928.", "more": "John von Neumann proved it in Zur Theorie der Gesellschaftsspiele (Mathematische Annalen 100, 1928, pages 295–320): in a finite two-player zero-sum game the most the first player can guarantee against the worst reply equals the least the second player can be held to, max_x min_y f(x, y) = min_y max_x f(x, y), once both are allowed mixed strategies; the two players agree on one number, and that number is the value of the game. Every hypothesis earns its place: two players, finitely many strategies each, strictly opposed payoffs, and mixing. Without mixing there may be no solution at all: matching pennies has no pure-strategy equilibrium, and under mixing its value is 0. Nash’s theorem of 1950 drops the zero-sum condition, but what survives the generalization is the existence of an equilibrium rather than a single guaranteed value.", "cites": [ "r-7.10", "r-7.11", "r-7.12" ], "links_to_cards": [ "nasheq" ] }, { "id": "nasheq", "url": "https://www.teclamairme.com/#g-nasheq", "name": "Nash’s theorem", "symbol": "Nash", "chapter": 7, "type": "fn", "badge": "Function or operator", "quick": "Every finite game—finitely many players, finitely many strategies each—has at least one equilibrium in mixed strategies. John Nash, 1950.", "more": "In Equilibrium points in n-person games (Proceedings of the National Academy of Sciences 36, 1950, pages 48–49) John Nash proved that a game with finitely many players, each having finitely many strategies, always has at least one profile of mixed strategies from which no player can gain by changing their own strategy alone. Finiteness is the hypothesis popular statements drop. The theorem is existence only: it does not say how many equilibria a game has, which one gets played, or that an equilibrium is good for anybody. Matching pennies has exactly one, and it is a mixture rather than a move. The prisoner’s dilemma’s only equilibrium pays both players less than cooperating would. Von Neumann’s minimax theorem of 1928 is the two-player zero-sum case, where existence comes with a single number both players can guarantee.", "cites": [ "r-7.10", "r-7.11", "r-7.12", "r-7.14" ], "links_to_cards": [ "minimax" ] }, { "id": "fermat", "url": "https://www.teclamairme.com/#g-fermat", "name": "Fermat’s Last Theorem as a bit", "symbol": "[∃ n, x, y, z ∈ ℤ⁺, n > 2 : xⁿ + yⁿ = zⁿ]", "chapter": 8, "type": "int", "badge": "Integer or rational", "quick": "Asserted false in a margin about 1637; shown false when Wiles proved the theorem (announced 1993, completed with Richard Taylor in 1994, published 1995), so the bracket is 0 after 358 years.", "more": "Fermat’s Last Theorem says xⁿ + yⁿ = zⁿ has no solution in positive integers for any n > 2. Fermat claimed it in a margin about 1637; Euler settled n = 3, Sophie Germain the first case for a whole class of primes, Kummer the regular primes in ~1847; the tilde is there because the two standard histories date it differently: MacTutor has the paper reaching Dirichlet and the Berlin Academy in September 1847, while Wikipedia’s Regular prime gives 1850, the year of the published general proof in Crelle 40. Both are cited at 8.5. Andrew Wiles announced a proof in 1993 and closed a gap in it in September 1994 with his former student Richard Taylor; it appeared in 1995 as two papers in the Annals of Mathematics, one by Wiles and one by Taylor and Wiles. The bracket contains the negation, the existence of a solution, which is false, so its value is 0. A truth value is an integer, hence the tag.", "cites": [ "r-8.4", "r-8.5", "r-8.6" ], "links_to_cards": [] }, { "id": "truths", "url": "https://www.teclamairme.com/#g-truths", "name": "four true statements", "symbol": "[1+1=2] + [√2 ∉ ℚ] + [∞ many primes] + [ℵ₀ < 𝔠]", "chapter": 8, "type": "int", "badge": "Integer or rational", "quick": "Each bracket is 1, so these four sum to 4, and the law of excluded middle beside them makes the denominator 5; only established statements are placed below the line, which keeps the denominator provably nonzero.", "more": "Four theorems, each with an old proof: 1 + 1 = 2 (a consequence of the Peano axioms of 1889, and the famous proposition *54.43 on page 379 of the first volume of Whitehead and Russell’s Principia Mathematica in 1910, which remarks only that it will follow once addition has been defined; the proof itself arrives in the second volume, in 1912), √2 is irrational (attributed to the Pythagoreans, before 400 BC), there are infinitely many primes (Euclid, Elements IX.20, about 300 BC), and ℵ₀ < 𝔠 (Cantor, 1874 and the diagonal argument of 1891). Each bracket is 1, so the sum is 4. Only statements with proven truth values are placed in the denominator, which keeps it provably nonzero.", "cites": [ "r-8.10", "r-8.11", "r-8.14", "r-8.8", "r-8.9" ], "links_to_cards": [ "aleph0", "axioms", "cont", "diagonal", "euclidprimes", "excluded", "russell", "sqrt2" ] }, { "id": "boolean", "url": "https://www.teclamairme.com/#g-boolean", "name": "the law of non-contradiction", "symbol": "x ∧ ¬x", "chapter": 8, "type": "int", "badge": "Integer or rational", "quick": "In Boolean algebra a statement x and its negation ¬x, joined by AND, give 0, whatever x is.", "more": "In the Boolean algebra that George Boole published in The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854), ∧ is AND, ¬ is NOT, and x ∧ ¬x = 0 for every x: a statement and its negation cannot both hold. Aristotle called this principle, now known as the law of non-contradiction, the firmest of all principles. The value is the Boolean zero, which comes home as the integer 0, so it carries the integer tag.", "cites": [ "r-8.2", "r-8.7" ], "links_to_cards": [ "boolealg" ] }, { "id": "excluded", "url": "https://www.teclamairme.com/#g-excluded", "name": "the law of excluded middle", "symbol": "x ∨ ¬x", "chapter": 8, "type": "int", "badge": "Integer or rational", "quick": "A statement or its negation is always true, so in Boolean algebra x ∨ ¬x = 1 for every x; the partner of the numerator’s zero.", "more": "The law of excluded middle, tertium non datur, is the third of the classical laws of thought and in Boole’s algebra (1847, 1854) reads x ∨ ¬x = 1: every statement is true or its negation is. It is the complement of the numerator’s x ∧ ¬x = 0, and in Boolean algebra the two laws are De Morgan duals (Augustus De Morgan, 1847). Intuitionists after Brouwer (1908) decline to assume it, which is why constructive mathematics will not prove that something exists by showing that its absence is contradictory. Its value is the Boolean 1, which comes home as the integer 1.", "cites": [ "r-8.16", "r-8.20", "r-8.7" ], "links_to_cards": [ "boolealg", "boolean", "demorgan" ] }, { "id": "iverson", "url": "https://www.teclamairme.com/#g-iverson", "name": "Iverson bracket", "symbol": "[ · ]", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "1 if the statement inside is true and 0 if it is false (Iverson 1962, popularized by Knuth).", "more": "The Iverson bracket [P] equals 1 when the statement P is true and 0 when it is false. Kenneth Iverson introduced the notation in his 1962 book A Programming Language, restricted to a single relation in parentheses; the square brackets and the generality are Donald Knuth’s, who campaigned for them in a 1992 article, Two Notes on Notation. It is a function from statements to integers rather than a number, so it is tagged as an operator, and the chapters that use it (this one, D, G, and the epilogue) treat its output as an ordinary integer.", "cites": [ "r-8.1" ], "links_to_cards": [] }, { "id": "logicsyms", "url": "https://www.teclamairme.com/#g-logicsyms", "name": "the symbols of logic", "symbol": "∀ ∃ ¬ ∧ ∨ ⇒ ⇔", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "For all, there exists, not, and, or, implies, if and only if: the seven signs from which logic builds its statements.", "more": "The quantifiers ∀ (for all) and ∃ (there exists) were introduced by Gerhard Gentzen in 1935 and Giuseppe Peano in 1897 respectively, Gentzen’s by analogy with Peano’s; Frege’s Begriffsschrift (1879) had already invented quantification, in a two-dimensional notation of which only the turnstile survives. The connective ∨ is Russell’s, from 1908, and ¬ and ∧ are Heyting’s, from 1930; ⇒ and ⇔ are Bourbaki’s, from 1954. The Fermat bracket in the numerator is read ∃ n, x, y, z ∈ ℤ⁺ (n > 2 ∧ xⁿ + yⁿ = zⁿ). They are notation, hence the tool tag.", "cites": [ "r-8.15" ], "links_to_cards": [ "turnstile" ] }, { "id": "godel", "url": "https://www.teclamairme.com/#g-godel", "name": "Gödel’s incompleteness theorems", "symbol": "Con(ZFC)", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "Any consistent, effectively axiomatized system strong enough for arithmetic has true statements it cannot prove, and cannot prove its own consistency; so some brackets have a value that system itself can never read.", "more": "Kurt Gödel proved in 1931 that any consistent, effectively axiomatized theory containing arithmetic is incomplete (first theorem) and cannot prove the statement Con(T) expressing its own consistency (second theorem). Con(ZFC) is therefore a definite statement whose Iverson bracket has a value, 1 if set theory is consistent, that ZFC itself can never establish. It is the reason Chapter ᚆ(H) separates truth from proof within a formal system: a bracket can have a value that a given system cannot establish. A theorem, hence the tool tag.", "cites": [ "r-8.12" ], "links_to_cards": [ "axioms", "iverson", "zfc" ] }, { "id": "chindep", "url": "https://www.teclamairme.com/#g-chindep", "name": "the continuum hypothesis, undecided", "symbol": "[CH]", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "Cantor’s guess that no infinity sits between the integers and the reals is neither provable nor refutable from the usual axioms, if those axioms are consistent (Gödel 1938, in full 1940; Cohen 1963).", "more": "The continuum hypothesis, Cantor’s 1878 conjecture that 2^ℵ₀ = ℵ₁, was the first of Hilbert’s 1900 problems. Gödel showed in 1940 that, assuming the usual consistency background, CH cannot be refuted from ZFC; Cohen showed in 1963 that it cannot be proved from ZFC. Thus ZFC decides neither CH nor its negation. Each model of ZFC nevertheless makes CH either true or false (Gödel’s model satisfies it, Cohen’s does not); independence means only that ZFC’s axioms do not decide which. Chapter ᚚ(P) carries ℵ₁ and 𝔠 as objects. An independent statement, hence the tool tag.", "cites": [ "r-27.22", "r-8.11", "r-8.13" ], "links_to_cards": [ "aleph1", "axioms", "ch", "cont", "hilbert23", "ints", "reals", "turnstile", "zfc" ] }, { "id": "axioms", "url": "https://www.teclamairme.com/#g-axioms", "name": "the axioms", "symbol": "PA, ZFC", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "Peano’s axioms (1889) for the natural numbers and Zermelo–Fraenkel set theory with choice (1908, 1922) for everything else: the ground the brackets stand on.", "more": "Giuseppe Peano published his axioms for the natural numbers in 1889, building on Dedekind’s 1888 analysis: zero, successor, and induction. Ernst Zermelo axiomatized set theory in 1908 and Abraham Fraenkel and Thoralf Skolem repaired it in 1922, giving ZF; with the axiom of choice it is ZFC, the default foundation of modern mathematics, in which 1 + 1 = 2 is a theorem and the truths in this chapter’s denominator are provable. Axioms are tools rather than values.", "cites": [ "r-8.14", "r-8.20" ], "links_to_cards": [ "ac", "nats", "turnstile", "zfc" ] }, { "id": "turnstile", "url": "https://www.teclamairme.com/#g-turnstile", "name": "the turnstiles: provable and true", "symbol": "⊢, ⊨", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "⊢ φ says φ has a proof, ⊨ φ says φ is true in every model; Gödel’s completeness theorem of 1929 says the two agree for first-order logic.", "more": "Frege drew the turnstile ⊢ in the Begriffsschrift (1879) as his judgment stroke, and it now reads as syntactic consequence: Γ ⊢ φ means φ can be derived from Γ by the rules. The double turnstile ⊨ is semantic consequence: every model of Γ makes φ true. Gödel’s completeness theorem (1929) proves ⊢ and ⊨ coincide for first-order logic, and his incompleteness theorems of 1931 show that truth in arithmetic still outruns proof. The Iverson bracket of this chapter records whether a statement is true; the denominator’s four theorems are also provable (⊢). A relation, hence the tool tag.", "cites": [ "r-8.15", "r-8.20" ], "links_to_cards": [ "iverson" ] }, { "id": "demorgan", "url": "https://www.teclamairme.com/#g-demorgan", "name": "De Morgan’s laws", "symbol": "¬(x ∧ y) = ¬x ∨ ¬y", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "Negation swaps AND with OR: not (both) is (not one) or (not the other), and not (either) is (not one) and (not the other).", "more": "Augustus De Morgan stated the laws in modern form in Formal Logic (1847), though William of Ockham and Jean Buridan had them in the fourteenth century and Aristotle noticed cases. They are why this chapter’s x ∧ ¬x = 0, in the numerator, and x ∨ ¬x = 1, the fifth term of the denominator, are mirror images: negate one law and you get the other. In set theory they read (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ, and in circuit design they let NAND gates build everything. A pair of identities, hence the tool tag.", "cites": [ "r-8.16" ], "links_to_cards": [ "boolean", "excluded" ] }, { "id": "qedsyms", "url": "https://www.teclamairme.com/#g-qedsyms", "name": "therefore, and the end of a proof", "symbol": "∴, ∎", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "∴ is therefore, ∵ is because, and the tombstone ∎ marks the end of a proof where older books wrote Q.E.D.", "more": "The three-dot therefore ∴ appears in Johann Rahn’s Teutsche Algebra (1659) and its inverse ∵ was reserved for because only in the nineteenth century. The tombstone or halmos ∎, closing a proof, was introduced, on his own account, by Paul Halmos, who took it from magazine typography; before it, proofs ended with Q.E.D., quod erat demonstrandum, Euclid’s ὅπερ ἔδει δεῖξαι. The four theorems in this chapter’s denominator each end with one. Punctuation of the framework, hence the tool tag.", "cites": [ "r-8.17" ], "links_to_cards": [] }, { "id": "boolealg", "url": "https://www.teclamairme.com/#g-boolealg", "name": "Boolean algebra", "symbol": "0, 1, ∧, ∨, ¬", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "The algebra of truth values George Boole published in 1847 and 1854, in which statements are 0 and 1 and can be added and multiplied; Shannon showed in 1937 that it is also the arithmetic of switching circuits.", "more": "George Boole gave logic an algebra in The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854): a statement carries the value 0 or 1, and the connectives can be written arithmetically—in the two-element algebra x ∧ y is the product xy, ¬x is 1 − x, and x ∨ y is x + y − xy. That is what lets this chapter add and multiply brackets at all. Its two constants are the chapter’s numerator and the fifth term of its denominator: x ∧ ¬x = 0 and x ∨ ¬x = 1. Claude Shannon’s master’s thesis of 1937 showed that the same algebra describes relay and switching circuits.", "cites": [ "r-8.2", "r-8.3" ], "links_to_cards": [ "boolean", "excluded" ] }, { "id": "euclidprimes", "url": "https://www.teclamairme.com/#g-euclidprimes", "name": "Euclid’s theorem", "symbol": "∞ many primes", "chapter": 8, "type": "fn", "badge": "Function or operator", "quick": "There are infinitely many primes; the proof is Elements IX.20, about 300 BC, and it is the third of the four brackets in this chapter’s denominator.", "more": "Euclid’s proof that the primes do not run out stands at Elements IX.20 and is conventionally dated to about 300 BC. Take any finite list of primes, multiply them together and add 1. No prime on the list divides the result, because each of them divides the product and would then have to divide 1 as well; so the result has a prime factor the list does not contain. The constructed number need not itself be prime, and the argument needs no contradiction, though it is often reported as one. The bracket around the statement is 1.", "cites": [ "r-8.10" ], "links_to_cards": [] }, { "id": "church", "url": "https://www.teclamairme.com/#g-church", "name": "Church zero, decoded", "symbol": "(λf.λx.x) S 0", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "The Church numeral for zero applies a function zero times, so applied to the successor and 0 it returns the number 0 itself: two beta reductions and a computation lands on zero.", "more": "Alonzo Church introduced the lambda calculus in 1932 and the Church numerals soon after, encoding the number n as a function that applies its argument n times: 0 is λf.λx.x, 1 is λf.λx.f x, and so on; Stephen Kleene later found how to λ-define the predecessor function. Decoding is standard: feed the numeral the successor function S and the ordinary number 0, and beta reduction hands back the ordinary number n. For zero that is two reductions, (λf.λx.x) S 0 → (λx.x) 0 → 0. The bare numeral is only an encoding; the decoding is a computation that halts holding the integer 0, hence the integer tag.", "cites": [ "r-9.1", "r-9.19" ], "links_to_cards": [ "lambdacalc" ] }, { "id": "halting", "url": "https://www.teclamairme.com/#g-halting", "name": "the halting problem, as a bit", "symbol": "[∃ a halting decider]", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "Turing proved in 1936 that no program can decide, for every program, whether it halts, so the bracket is 0.", "more": "Alan Turing proved in his 1936 paper On Computable Numbers that no Turing machine can decide, for every machine and input, whether that machine halts; the proof builds a machine that consults the supposed decider about itself and does the opposite. The bracket contains the false statement that such a decider exists, so it is 0. Earlier in 1936, before Turing’s paper was received on 28 May, Church had published a proof, in the language of functions rather than machines, that no procedure decides the Entscheidungsproblem; Turing’s appendix shows that the two describe the same class. The bracket is an integer because a truth value is one.", "cites": [ "r-9.19", "r-9.2", "r-9.3" ], "links_to_cards": [ "entscheidung", "turingm" ] }, { "id": "bb5", "url": "https://www.teclamairme.com/#g-bb5", "name": "the fifth Busy Beaver number", "symbol": "BB(5)", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "47,176,870, proven in 2024 after decades of search; the largest Busy Beaver value actually known.", "more": "Tibor Radó defined the Busy Beaver function in 1962: BB(n) is the maximum number of steps an n-state, two-symbol Turing machine started on a blank tape can run before halting. BB(1) = 1, BB(2) = 6, BB(3) = 21, BB(4) = 107. The five-state champion running 47,176,870 steps was found by Heiner Marxen and Jürgen Buntrock in 1989, and it was proved to be the true BB(5) by the Busy Beaver Challenge collaboration on 2 July 2024, with a proof checked in the Coq (now Rocq) proof assistant. The function is uncomputable as a whole, because knowing it would decide the halting problem, but each known value is a plain integer, hence the tag.", "cites": [ "r-9.20", "r-9.5", "r-9.6", "r-9.7", "r-9.8" ], "links_to_cards": [ "busybeaver", "rice", "turingm", "uncomputable" ] }, { "id": "ack", "url": "https://www.teclamairme.com/#g-ack", "name": "Ackermann’s function at three and three", "symbol": "A(3, 3)", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "61, from the function that grows faster than any primitive recursion.", "more": "Wilhelm Ackermann published his function in 1928 as an example of something computable that is not primitive recursive, that is, not reachable by loops whose lengths are fixed in advance. In the common Péter form A(0, n) = n + 1, A(m, 0) = A(m − 1, 1), A(m, n) = A(m − 1, A(m, n − 1)), and A(3, 3) = 61; A(4, 2) has 19,729 digits and A(4, 3) cannot be written out in full. A(3, 3) is an ordinary integer, hence the tag.", "cites": [ "r-9.10", "r-9.9" ], "links_to_cards": [] }, { "id": "bbsmall", "url": "https://www.teclamairme.com/#g-bbsmall", "name": "the four smaller Busy Beaver numbers", "symbol": "BB(1) + BB(2) + BB(3) + BB(4)", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "1 + 6 + 21 + 107 = 135: every Busy Beaver value below BB(5), each a plain count of steps.", "more": "BB(1) = 1 and BB(2) = 6 were settled in the early 1960s, BB(3) = 21 was proved by Lin and Radó in 1965, and BB(4) = 107 by Brady in 1983. Each is the longest a halting two-symbol Turing machine with that many states can run on a blank tape, and each is an ordinary integer once proved, which is why they sit in the denominator beside BB(5) while BB(6), still unknown, sits in Chapter ᚃ(F). Sum 135.", "cites": [ "r-9.19", "r-9.5", "r-9.6" ], "links_to_cards": [ "bb5", "bb6", "turingm" ] }, { "id": "lambdacalc", "url": "https://www.teclamairme.com/#g-lambdacalc", "name": "the lambda calculus", "symbol": "λx.M", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Church’s 1932 model of computation: every term is a variable, a function λx.M, or one term applied to another, and its rule of computation is beta-reduction; it computes exactly what a Turing machine computes.", "more": "Alonzo Church introduced the lambda calculus in A set of postulates for the foundation of logic, published in 1932; it is a formal system built from abstraction and application, and beta-reduction, which applies a function to an argument, is its rule of computation. Numbers are encoded rather than built in: the Church numeral for n is a function that applies its argument n times, so zero is λf.λx.x, and Stephen Kleene found a λ-definition of the predecessor function. Untyped lambda calculus can simulate any Turing machine and a Turing machine can simulate it, which is why the three answers of the 1930s are one answer.", "cites": [ "r-9.1", "r-9.3" ], "links_to_cards": [ "ski", "turingm" ] }, { "id": "genrec", "url": "https://www.teclamairme.com/#g-genrec", "name": "general recursion", "symbol": "general recursive", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Herbrand and Gödel’s class of general recursive functions, set out in Gödel’s 1934 Princeton lectures on a suggestion of Jacques Herbrand; one of the decade’s three definitions of computable, and the same class as the other two.", "more": "Kurt Gödel, with Jacques Herbrand, formalized the definition of the class of general recursive functions, modifying his approach on Herbrand’s suggestion, and set the class out in his 1934 Princeton lectures. Church’s lambda-definable functions, these general recursive functions, and Turing’s machine-computable functions are extensionally the same class, which is what makes the Church–Turing thesis one thesis rather than three. The equivalence is a theorem about the three formal models; the further claim that this class is everything effectively calculable is not, because effective calculability has no agreed formal definition.", "cites": [ "r-9.3" ], "links_to_cards": [ "churchturing" ] }, { "id": "entscheidung", "url": "https://www.teclamairme.com/#g-entscheidung", "name": "the Entscheidungsproblem", "symbol": "Entscheidungsproblem", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Hilbert and Ackermann’s decision problem of 1928: is there an algorithm that says whether a given statement is valid in every structure? Church and Turing each showed in 1936 that there is none.", "more": "David Hilbert and Wilhelm Ackermann posed the Entscheidungsproblem in 1928: it asks for an algorithm that takes a statement and answers yes or no according to whether that statement is universally valid, meaning valid in every structure. In 1936 Alonzo Church and Alan Turing published independent papers showing that no general solution exists. Church’s was in print first; Turing’s was received on 28 May and printed on 30 November and 23 December, with an appendix showing that the lambda-definable and the machine-computable functions are the same class. Turing’s route to the answer, a machine that cannot decide whether another machine halts, is the one this chapter’s bracket records.", "cites": [ "r-9.21", "r-9.3" ], "links_to_cards": [ "fortytwo" ] }, { "id": "busybeaver", "url": "https://www.teclamairme.com/#g-busybeaver", "name": "the Busy Beaver function", "symbol": "BB(n)", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Radó’s 1962 function: the most steps a halting n-state, two-symbol Turing machine can make from a blank tape. It eventually dominates every computable function, and no algorithm computes it.", "more": "Tibor Radó introduced the busy beaver game in 1962; in the step-count convention used throughout this book, BB(n) is the maximum number of steps a halting n-state, two-symbol Turing machine makes when started on the all-zero tape. The function is uncomputable and eventually dominates every computable function, because an exact computation of its values would decide halting for machines of bounded size. Individual values are ordinary integers once proved: BB(1) = 1, BB(2) = 6, BB(3) = 21 by Lin and Radó in 1965, and BB(4) = 107 by Brady in 1983; BB(5) = 47,176,870, whose champion machine Marxen and Buntrock found in 1989, was proved in 2024. BB(6) is still unknown, and sits in Chapter ᚃ(F). Uncomputability is a property of the function, not of any one of its values.", "cites": [ "r-9.5", "r-9.6", "r-9.7", "r-9.8" ], "links_to_cards": [ "bb5", "bb6", "turingm", "uncomputable" ] }, { "id": "ackfn", "url": "https://www.teclamairme.com/#g-ackfn", "name": "Ackermann’s function", "symbol": "A(m, n)", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "A recursive function, published by Wilhelm Ackermann in 1928 and now usually given in two-argument form, that is total and computable yet grows faster than any primitive recursive function.", "more": "In the common Péter form, A(0, n) = n + 1, A(m, 0) = A(m − 1, 1), and A(m, n) = A(m − 1, A(m, n − 1)). The function is total and computable, but it is not primitive recursive: no loop whose bound is fixed in advance can compute it, because it grows faster than any primitive recursive function. Ackermann published it in 1928 to separate the computable functions from the primitive recursive ones; A(3, 3) = 61 is the one value with a card of its own, and A(4, 2) already runs to 19,729 decimal digits.", "cites": [ "r-9.25" ], "links_to_cards": [ "ack" ] }, { "id": "pnpsub", "url": "https://www.teclamairme.com/#g-pnpsub", "name": "the settled direction of P versus NP", "symbol": "[P ⊆ NP]", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "Every problem that can be solved quickly can be checked quickly, so this bracket is 1; the other direction is the epilogue’s million-dollar bit.", "more": "P is the class of decision problems a deterministic machine solves in polynomial time, NP the class whose yes-answers a machine can verify in polynomial time given a certificate; the polynomial-time formulation is Cobham’s and Edmonds’, both from 1965. P ⊆ NP is immediate, since a solver is a verifier that ignores its certificate, so the bracket is 1. Whether NP ⊆ P is the P versus NP problem, whose bracket [P = NP] closes the book in the epilogue and can never sit in a denominator. Cook (1971) and Levin (1973) proved that satisfiability is NP-complete, and Karp gave a list of 21 NP-complete problems in 1972. A known truth value, hence the integer tag.", "cites": [ "r-9.11", "r-9.12" ], "links_to_cards": [ "npcomplete", "pnp" ] }, { "id": "float64zero", "url": "https://www.teclamairme.com/#g-float64zero", "name": "a binary64 rounding zero", "symbol": "fl64(1+2−53)−1", "symbol_markup": "fl_{64}(1+2^{−53})−1", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0 under IEEE binary64 round-to-nearest, ties-to-even: the positive increment, half the gap between 1 and the next representable number, rounds away.", "more": "IEEE binary64 has 53 bits of significand precision. The next representable value above 1 is 1 + 2^{−52}, so 1 + 2^{−53} is exactly halfway. Under round-to-nearest, ties-to-even, the tie resolves to 1 because its final significand bit is even; subtracting 1 gives exactly 0. The rule and format are those standardized by IEEE 754. This is machine arithmetic, not an identity of the real numbers.", "cites": [ "r-9.13", "r-9.14" ], "links_to_cards": [ "ieee754", "reals" ] }, { "id": "eps64", "url": "https://www.teclamairme.com/#g-eps64", "name": "machine epsilon of binary64", "symbol": "2−52", "symbol_markup": "2^{−52}", "chapter": 9, "type": "int", "badge": "Integer or rational", "quick": "About 2.2 × 10⁻¹⁶: the gap between 1 and the next double-precision number, the resolution of the double, one of the two most common floating-point formats in use.", "more": "IEEE binary64, the double type on most systems since the 1985 standard, carries 53 bits of significand precision, so the numbers just above 1 are spaced 2^{−52} ≈ 2.220446 × 10^{−16} apart. Half of that gap is the largest relative rounding error a single operation can make, and the numerator’s zero sits exactly on it. A rational number. (That 0.1 + 0.2 ≠ 0.3 on a computer has a different cause: 0.1 and 0.2 have no finite binary expansion, so they are stored rounded.)", "cites": [ "r-9.13", "r-9.22" ], "links_to_cards": [ "ieee754" ] }, { "id": "kolmogorov", "url": "https://www.teclamairme.com/#g-kolmogorov", "name": "Kolmogorov complexity", "symbol": "K(x)", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "The length of the shortest program that prints x: Solomonoff 1960, Kolmogorov 1965, Chaitin 1966; not computable, by an argument modeled on Berry’s paradox.", "more": "The Kolmogorov complexity K(x) of a string is the length of the shortest program, in a fixed language, that outputs it; a string is random when no program is shorter than the string itself. Ray Solomonoff introduced the idea in 1960, Andrey Kolmogorov published it independently in 1965 and acknowledged Solomonoff’s priority when he learned of his work, and Gregory Chaitin’s paper followed in 1966. K is provably uncomputable, by a version of Berry’s paradox, and Chaitin’s Ω beside it is the halting probability of the same machine. A function, so a tool of the framework rather than a value.", "cites": [ "r-9.15" ], "links_to_cards": [ "berry", "omega", "uncomputable" ] }, { "id": "turingm", "url": "https://www.teclamairme.com/#g-turingm", "name": "Turing machine", "symbol": "TM", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Turing’s 1936 model of computation: a finite control, a tape, and a head that reads, writes, and moves; the Church–Turing thesis says everything computable is computable by one.", "more": "Alan Turing defined his machines in On Computable Numbers (1936): a finite set of states, an unbounded tape of cells, and a table saying, for each state and symbol, what to write, which way to move, and which state to enter. A universal machine can simulate any other from a description on its tape, which is the idea of the stored-program computer. Busy Beaver counts the steps of the most stubborn halting n-state machine; the halting bracket says no machine can predict them all. A model, hence the tool tag.", "cites": [ "r-9.3", "r-9.4" ], "links_to_cards": [ "churchturing" ] }, { "id": "churchturing", "url": "https://www.teclamairme.com/#g-churchturing", "name": "the Church–Turing thesis", "symbol": "Church–Turing thesis", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Every function a human could compute by a fixed procedure is computable by a Turing machine, equivalently by lambda calculus; a thesis, not a theorem, and never contradicted.", "more": "In 1936 Church’s lambda-definable functions, Herbrand and Gödel’s general recursive functions and Turing’s machine-computable functions were all shown to be the same class, and Church and Turing each proposed that this class is what computable means. The thesis cannot be proved, because effective procedure is an informal notion, and rigorous later models, including quantum Turing machines, have proved equivalent to it; only the theoretical models called hypercomputers go beyond it. It is the reason the halting bracket is a statement about all programs and not only Turing’s. A principle, hence the tool tag.", "cites": [ "r-9.3" ], "links_to_cards": [ "genrec", "lambdacalc", "turingm" ] }, { "id": "bigO", "url": "https://www.teclamairme.com/#g-bigO", "name": "big-O notation", "symbol": "O(n log n)", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "How fast a cost grows with the size of the input, ignoring constants: the language complexity classes are written in.", "more": "Paul Bachmann proposed the notation in 1894 and Edmund Landau extended it in 1909; Knuth fixed Ω and Θ for the family in 1976. f = O(g) means f is eventually bounded by a constant times g. Sorting by comparisons costs Θ(n log n), multiplying two n-digit numbers costs O(n log n) by Harvey and van der Hoeven, announced in 2019 and published in 2021, and P is the class of problems solvable in O(n^k) for some k. Notation, hence the tool tag.", "cites": [ "r-9.11" ], "links_to_cards": [] }, { "id": "rice", "url": "https://www.teclamairme.com/#g-rice", "name": "Rice’s theorem", "symbol": "Rice’s theorem", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Every non-trivial question about what a program computes is undecidable; the halting problem is only the first example.", "more": "Henry Gordon Rice proved in 1951 (published 1953) that for any property of the function a program computes, other than the trivial always-true and always-false ones, no algorithm can decide whether an arbitrary program has it. Does this program ever print 0, does it compute a total function, is it equivalent to that program: all undecidable. It turns the single halting bracket of the numerator into a whole class of zeros. A theorem, hence the tool tag.", "cites": [ "r-9.16" ], "links_to_cards": [ "halting" ] }, { "id": "ski", "url": "https://www.teclamairme.com/#g-ski", "name": "the combinators", "symbol": "S, K, I, Y", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "S, K, and I are the three building blocks from which every lambda term can be assembled; Y is the fixed-point combinator that gives recursion.", "more": "Moses Schönfinkel introduced combinatory logic in 1920, in a talk published in 1924, with S = λxyz.xz(yz) and K = λxy.x, from which I = SKK, and Haskell Curry rediscovered the combinators in 1927 and developed them in his 1930 thesis; every closed lambda term can be written with S and K alone. Curry’s fixed-point combinator Y = λf.(λx.f(xx))(λx.f(xx)) satisfies Y f = f(Y f), which is how a calculus with no loops or names still computes every recursive function, including the Church numerals’ arithmetic. Tools, hence the tag.", "cites": [ "r-9.17" ], "links_to_cards": [] }, { "id": "godelnum", "url": "https://www.teclamairme.com/#g-godelnum", "name": "Gödel numbering", "symbol": "⌜φ⌝", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "Encode every formula and proof as a natural number, so that arithmetic can talk about itself; the trick behind incompleteness and the universal machine.", "more": "Gödel’s 1931 proof assigns to each symbol, formula, and proof a natural number, using prime factorization so that the code can be decoded; the statement “this statement is unprovable” then becomes a statement about numbers. Turing’s universal machine of 1936 does the same for programs. The corner brackets ⌜φ⌝ are the standard notation for the number of the formula φ. A coding, hence the tool tag.", "cites": [ "r-9.18", "r-9.24", "r-9.4" ], "links_to_cards": [] }, { "id": "ieee754", "url": "https://www.teclamairme.com/#g-ieee754", "name": "IEEE 754", "symbol": "binary64", "chapter": 9, "type": "fn", "badge": "Function or operator", "quick": "The floating-point standard almost all modern computers follow, first published in 1985: its binary64 format keeps 53 bits of significand precision, and its default rounding is to the nearest representable number, ties to even.", "more": "The first IEEE standard for floating-point arithmetic, IEEE 754-1985, was published in 1985, and round to nearest, ties to even is its default rounding for binary formats. The binary64 format, the double type on most systems, carries 53 bits of significand precision with 52 of them explicitly stored, so the numbers just above 1 are spaced 2^{−52} ≈ 2.220446 × 10^{−16} apart. The tie rule is what makes this chapter’s numerator 0: 1 + 2^{−53} is exactly halfway to the next double, and of its two neighbors the one whose last significand bit is even is 1 itself. A decision made by a committee, not a theorem of arithmetic.", "cites": [ "r-9.13", "r-9.14", "r-9.22" ], "links_to_cards": [] }, { "id": "detA", "url": "https://www.teclamairme.com/#g-detA", "name": "determinant of A = [[1, 2], [2, 4]]", "symbol": "det A", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "Zero because the second row is twice the first; the matrix squashes the plane onto a line.", "more": "The determinant measures how a linear map scales area or volume. Seki Takakazu reached it in 1683 and Leibniz independently in a letter of 1693; Gauss coined its name in 1801 for something else, and Cauchy fixed the name on this meaning in 1812. For A with rows (1, 2) and (2, 4) it is 1·4 − 2·2 = 0: the second row is twice the first, so the plane is flattened onto the line through (1, 2), and A has no inverse. Integer entries give an integer determinant.", "cites": [ "r-10.20", "r-10.3", "r-10.4" ], "links_to_cards": [ "identity" ] }, { "id": "trA", "url": "https://www.teclamairme.com/#g-trA", "name": "trace of A", "symbol": "tr A", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "1 + 4 = 5.", "more": "The trace, the sum of the diagonal entries, is 1 + 4 = 5, and it equals the sum of the eigenvalues, 5 + 0. The name is a translation of the German Spur, which is why some older books write sp A. It is an integer because the entries are.", "cites": [ "r-10.4", "r-10.6" ], "links_to_cards": [] }, { "id": "frob", "url": "https://www.teclamairme.com/#g-frob", "name": "Frobenius norm of A", "symbol": "‖A‖F", "symbol_markup": "‖A‖_{F}", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "√(1 + 4 + 4 + 16) = 5.", "more": "The Frobenius norm, named for Ferdinand Georg Frobenius, is the square root of the sum of the squares of all entries: √(1 + 4 + 4 + 16) = √25 = 5. It also equals the square root of the sum of the squared singular values, and this matrix has one nonzero singular value, 5. The entries were chosen so that the norm is an integer.", "cites": [ "r-10.4", "r-10.7" ], "links_to_cards": [ "svd" ] }, { "id": "lmax", "url": "https://www.teclamairme.com/#g-lmax", "name": "largest eigenvalue of A", "symbol": "λmax(A)", "symbol_markup": "λ_{max}(A)", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "5; the other eigenvalue is 0.", "more": "The eigenvalues of A are the numbers λ for which A stretches some direction by exactly λ; they are the roots of λ² − 5λ = 0, so 5 and 0. The direction (1, 2) is stretched by 5 and the perpendicular direction (2, −1) is sent to zero. The word eigen is Hilbert’s, from about 1904; the idea is older, going back to Euler and Lagrange on rotating bodies. Here the largest eigenvalue is the integer 5.", "cites": [ "r-10.4", "r-10.8" ], "links_to_cards": [] }, { "id": "kronecker", "url": "https://www.teclamairme.com/#g-kronecker", "name": "the Kronecker delta off the diagonal", "symbol": "δ12", "symbol_markup": "δ_{12}", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "δᵢⱼ is 1 when i = j and 0 otherwise, so δ₁₂ = 0; this δ is not Feigenbaum’s.", "more": "Leopold Kronecker’s delta, from his work on bilinear forms of the 1860s, is δᵢⱼ = 1 when i = j and 0 otherwise. Off the diagonal it is 0, so δ₁₂ = 0, and the identity matrix has entries δᵢⱼ. It is an integer-valued symbol and is not Feigenbaum’s δ of Chapter ᚃ(F).", "cites": [ "r-10.5" ], "links_to_cards": [ "feig", "identity" ] }, { "id": "dot0", "url": "https://www.teclamairme.com/#g-dot0", "name": "two perpendicular vectors", "symbol": "(1, 2) · (2, −1)", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "The dot product 1·2 + 2·(−1) = 0: the two eigenvectors of A are at right angles, as a real symmetric matrix’s eigenvectors for distinct eigenvalues always are.", "more": "The dot product 𝐮 · 𝐯 = Σ uᵢvᵢ, given its dot by Gibbs in lecture notes privately printed in 1881, and reached independently by Heaviside, is zero for two nonzero vectors exactly when they are perpendicular; the name dot product arrived with Gibbs and Wilson’s Vector Analysis of 1901. Here (1, 2) and (2, −1) are the eigenvectors of A for the eigenvalues 5 and 0, and their dot product is 2 − 2 = 0, an instance of the spectral theorem: a real symmetric matrix has an orthonormal basis of eigenvectors, and eigenvectors belonging to distinct eigenvalues are always perpendicular. It is the same zero as the orthogonal states of Chapter ᚍ(K) in two real dimensions. An integer, hence the tag.", "cites": [ "r-10.16", "r-10.8" ], "links_to_cards": [ "spectralthm" ] }, { "id": "nilp", "url": "https://www.teclamairme.com/#g-nilp", "name": "a nilpotent matrix squared", "symbol": "N², N = [[0, 1], [0, 0]]", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "A nonzero matrix whose square is the zero matrix: N shifts the plane, and shifting twice sends everything to 0.", "more": "A matrix N is nilpotent when some power of it is the zero matrix; the smallest nonzero example is 2 × 2, N = [[0, 1], [0, 0]], with N² = 0. Its only eigenvalue is 0, its determinant and trace are 0, yet N itself is not 0: the algebra of matrices has zero divisors, unlike the reals, which is the same lesson the sedenions teach in Chapter ᚗ(V). Jordan’s normal form goes back to 1870, and with it every complex matrix is a diagonalizable part plus a nilpotent part that commutes with it, the Jordan–Chevalley decomposition. The zero matrix hands home the integer 0.", "cites": [ "r-10.9" ], "links_to_cards": [ "reals", "sedenions" ] }, { "id": "ranknull", "url": "https://www.teclamairme.com/#g-ranknull", "name": "the rank–nullity theorem", "symbol": "rank A + null A", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "1 + 1 = 2: the dimension of what A keeps plus the dimension of what it kills is the dimension of the plane.", "more": "The rank of a matrix is the dimension of its image, the nullity the dimension of its kernel, and the rank–nullity theorem says they add to the number of columns. A squashes the plane onto a line, so its rank is 1, and it kills the line through (2, −1), so its nullity is 1; the sum is 2. Sylvester named the nullity in 1884 and the word rank is Frobenius’s, from 1878 or 1879. A dimension count, hence the integer tag.", "cites": [ "r-10.10", "r-10.4" ], "links_to_cards": [] }, { "id": "cayham", "url": "https://www.teclamairme.com/#g-cayham", "name": "the Cayley–Hamilton theorem for A", "symbol": "A² − 5A", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "Every square matrix satisfies its own characteristic equation; for A that equation is λ² − 5λ = 0, so A² − 5A is the zero matrix.", "more": "The characteristic polynomial of A = [[1, 2], [2, 4]] is det(λI − A) = λ² − 5λ, and the Cayley–Hamilton theorem says a matrix is a root of its own characteristic polynomial: A² − 5A = 0, which one can check by hand since A² = [[5, 10], [10, 20]] = 5A. Hamilton proved a case for quaternions in 1853, Cayley stated it in 1858 and proved it only for 2 × 2 matrices, which is exactly this size, and Frobenius proved the general case in 1878. The value is the zero matrix, which hands home the integer 0.", "cites": [ "r-10.11", "r-10.4" ], "links_to_cards": [ "charpoly", "quaternions" ] }, { "id": "identity", "url": "https://www.teclamairme.com/#g-identity", "name": "identity, inverse, and transpose", "symbol": "I, A⁻¹, Aᵀ", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "I has δᵢⱼ as its entries and changes nothing; A⁻¹ undoes A and exists only when det A ≠ 0, so this A has none; Aᵀ swaps rows and columns.", "more": "The identity matrix I has ones on the diagonal and zeros elsewhere, the Kronecker delta as its entries, and I·A = A for every A. The inverse A⁻¹ satisfies A·A⁻¹ = I and exists exactly when det A ≠ 0; the matrix of this chapter has determinant 0 and so no inverse, which is what squashing the plane onto a line means. The transpose Aᵀ swaps rows and columns; A here is symmetric, Aᵀ = A, which is why its eigenvectors are perpendicular. Cayley published the first matrix inverse in 1853 and the first abstract definition of a matrix, with its algebra, in 1858; the word matrix itself is Sylvester’s, from 1850. Tools, hence the tag.", "cites": [ "r-10.12", "r-10.2", "r-10.21", "r-10.3" ], "links_to_cards": [ "detA" ] }, { "id": "svd", "url": "https://www.teclamairme.com/#g-svd", "name": "the largest singular value", "symbol": "σ₁(A)", "chapter": 10, "type": "int", "badge": "Integer or rational", "quick": "5, the same as the Frobenius norm here, because A has only one nonzero singular value; singular values measure how far any matrix stretches, eigenvalues how far a square one stretches the directions it keeps on their own line.", "more": "The singular values of a matrix are the square roots of the eigenvalues of AᵀA, and the singular value decomposition A = UΣVᵀ, developed by Beltrami and Jordan in 1873 and 1874 and made numerical by Golub and Kahan in 1965, factors any matrix into a rotation, a stretch, and a rotation. A has rank 1, so one nonzero singular value, σ₁ = 5, and the Frobenius norm √(Σσᵢ²) is the same 5. The SVD is used in least squares, principal components, and image compression. An integer here, hence the tag.", "cites": [ "r-10.13", "r-10.8" ], "links_to_cards": [ "frob", "leastsq" ] }, { "id": "cramer", "url": "https://www.teclamairme.com/#g-cramer", "name": "Cramer’s rule", "symbol": "x = det Aᵢ / det A", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "Solve a square system by determinants; it fails for this A because det A = 0, which is exactly when the system has no unique solution.", "more": "Colin Maclaurin was teaching the rule to his students by ~1729, and Gabriel Cramer published it in 1750: the i-th unknown of a system A𝐱 = 𝐛 is det Aᵢ divided by det A, where Aᵢ has its i-th column replaced by 𝐛. The rule divides by det A, so it breaks down exactly when the determinant is zero, which is also when the system has no unique solution: the system of this chapter’s A either has no solution or a whole line of them. That division is the reason for the rule of this book that a denominator must be provably nonzero. A rule, hence the tool tag.", "cites": [ "r-10.14", "r-10.23", "r-10.3" ], "links_to_cards": [ "detA" ] }, { "id": "vspace", "url": "https://www.teclamairme.com/#g-vspace", "name": "vector spaces, bases, and dimension", "symbol": "dim V, basis", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "A vector space is anything you can add and scale; a basis is a smallest spanning set, and every basis has the same size, the dimension.", "more": "Peano gave the axioms of a vector space in 1888, following Grassmann’s 1844 Ausdehnungslehre (Chapter ᚌ(G)), and the theorem that every basis of a space has the same number of elements is what makes dimension well defined; ℝ² has dimension 2, which is the 2 of the rank–nullity bracket, and the space of all polynomials has countably infinite dimension. Linear independence, span, subspaces, and linear maps are the vocabulary; matrices are what linear maps look like once a basis is chosen. Concepts, hence the tool tag.", "cites": [ "r-10.15" ], "links_to_cards": [] }, { "id": "tensor", "url": "https://www.teclamairme.com/#g-tensor", "name": "tensor and cross product", "symbol": "𝐮 ⊗ 𝐯, 𝐮 × 𝐯", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "The tensor product builds a matrix from two vectors, and A itself is (1, 2) ⊗ (1, 2); the cross product of two vectors in space is a vector perpendicular to both whose length is the area of their parallelogram, so it is 0 for parallel vectors.", "more": "The outer or tensor product 𝐮 ⊗ 𝐯 is the matrix with entries uᵢvⱼ; every rank-one matrix is one, and A = (1, 2) ⊗ (1, 2), which is why A squashes the plane. The cross product 𝐮 × 𝐯, Gibbs’s and Heaviside’s from the 1880s and Grassmann’s wedge in disguise (Chapter ᚌ(G)), is defined in three dimensions and vanishes for parallel vectors; (1, 2, 0) × (2, 4, 0) = 0 is the same collapse as det A = 0. Tensors of higher rank, Ricci and Levi-Civita’s 1900 calculus, are the language of general relativity. Operations, hence the tool tag.", "cites": [ "r-10.16" ], "links_to_cards": [ "cross7", "detA" ] }, { "id": "charpoly", "url": "https://www.teclamairme.com/#g-charpoly", "name": "the characteristic polynomial", "symbol": "p_A(λ) = det(λI − A)", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "The polynomial whose roots are the eigenvalues; for A it is λ² − 5λ = λ(λ − 5), so the eigenvalues are 0 and 5 and the trace 5 is the sum of the roots.", "more": "The characteristic polynomial p_A(λ) = det(λI − A) has the eigenvalues as roots; its coefficients are, up to sign, the trace and the determinant for a 2 × 2 matrix, so p_A(λ) = λ² − (tr A)λ + det A = λ² − 5λ + 0 here, which is why the trace 5 and the determinant 0 of this chapter reappear as the sum and product of the eigenvalues. Cauchy used the polynomial in 1829 and named the characteristic equation in 1840. A polynomial, keyed as a tool.", "cites": [ "r-10.17", "r-10.4", "r-10.6" ], "links_to_cards": [ "detA" ] }, { "id": "maps", "url": "https://www.teclamairme.com/#g-maps", "name": "maps, arrows, and composition", "symbol": "f: V → W, x ↦ Ax, g ∘ f", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "→ names a map between spaces, ↦ says where one element goes, and ∘ composes two maps; A ∘ A is A², and matrix multiplication is composition.", "more": "A linear map is written f: V → W, and x ↦ Ax says what it does to a vector. Composition g ∘ f means do f, then g, and the reason matrix multiplication is defined as it is (rows times columns) is that the product matrix is the matrix of the composed map. Here A ∘ A = A² = 5A by the Cayley–Hamilton entry. Notation, hence the tool tag.", "cites": [ "r-10.18" ], "links_to_cards": [ "cayham" ] }, { "id": "ninechapters", "url": "https://www.teclamairme.com/#g-ninechapters", "name": "the Nine Chapters on the Mathematical Art", "symbol": "Rectangular Arrays", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "The Chinese classic whose eighth chapter solves systems of simultaneous linear equations by reducing an array of coefficients with column operations: Gaussian elimination, two thousand years before Gauss.", "more": "Chapter 8, Rectangular Arrays, poses eighteen problems that reduce to systems of simultaneous linear equations, and solves them by writing the coefficients as an augmented array and reducing it to triangular form with elementary column operations—exactly the method now called Gaussian elimination. Most historians date the text to about 200 BC, with 206 BC to 50 AD covering its different parts, so the method predates the name by two millennia. Gauss met a six-by-six system in astronomy: from observations of the asteroid Pallas taken between 1803 and 1809 he obtained six linear equations in six unknowns, and in 1810 he devised the notation for symmetric elimination that nineteenth-century hand computers used on least-squares normal equations. It is the oldest ancestor of everything else in this chapter: the grid of coefficients that Sylvester would later name a matrix.", "cites": [ "r-10.1", "r-10.22" ], "links_to_cards": [ "gausselim" ] }, { "id": "orthproj", "url": "https://www.teclamairme.com/#g-orthproj", "name": "orthogonal projection", "symbol": "P² = P = Pᵀ", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "Drop each vector straight down onto a subspace. Doing it twice is the same as doing it once, and what is discarded is perpendicular to what is kept.", "more": "A projection is a linear map with P² = P—once you have landed on the subspace, landing again changes nothing—and it is orthogonal when the part thrown away is perpendicular to the part kept, equivalently when the matrix is also symmetric. That perpendicularity is what makes the projection the nearest point of the subspace, which is why least squares is a projection and not merely an analogy. This chapter’s matrix is five times such a map: A/5 sends the plane onto the line y = 2x and leaves nothing of the perpendicular direction, which is the whole reason its determinant is 0, its rank is 1, and its second eigenvalue is 0. A tool of the framework, uncolored.", "cites": [ "r-10.24" ], "links_to_cards": [ "leastsq" ] }, { "id": "spectralthm", "url": "https://www.teclamairme.com/#g-spectralthm", "name": "the spectral theorem", "symbol": "A = QΛQᵀ", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, so it is nothing but a set of stretches along perpendicular axes.", "more": "The spectral theorem says that a real symmetric matrix can be written QΛQᵀ with Λ diagonal and Q orthogonal: its eigenvalues are real, its eigenvectors for distinct eigenvalues are perpendicular, and it acts by stretching along a set of perpendicular axes and nothing else. Cauchy proved it for symmetric matrices, and von Neumann carried it to operators on a Hilbert space, where it became perhaps the most important result of operator theory, and where quantum mechanics uses it to attach probabilities to the values of a measurement. The chapter’s A is symmetric, which is exactly why its eigenvectors (1, 2) and (2, −1) meet at right angles and why the sentence about perpendicular eigenvectors is a theorem rather than a coincidence. A theorem, uncolored.", "cites": [ "r-10.25" ], "links_to_cards": [ "hilbert" ] }, { "id": "gausselim", "url": "https://www.teclamairme.com/#g-gausselim", "name": "Gaussian elimination", "symbol": "Ax = b → Ux = c", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "Reduce a system of linear equations to triangular form by row or column operations, then solve it by back-substitution.", "more": "The method appears in the eighth chapter of the Chinese Nine Chapters on the Mathematical Art, compiled over generations up to the first century AD, which solves systems of simultaneous equations by reducing an array of coefficients with column operations. Carl Friedrich Gauss used it in 1810 to solve six simultaneous equations arising from observations of the asteroid Pallas, devising a symmetric-elimination notation that nineteenth-century hand computers used for least-squares problems; his name attached to the method even though it predates him by nearly two thousand years. It underlies the row reduction used to compute rank, solve linear systems, and invert matrices.", "cites": [ "r-10.27" ], "links_to_cards": [ "ninechapters" ] }, { "id": "leastsq", "url": "https://www.teclamairme.com/#g-leastsq", "name": "least squares", "symbol": "min‖Ax−b‖²", "chapter": 10, "type": "fn", "badge": "Function or operator", "quick": "Least squares chooses the solution that minimizes the sum of the squared errors, the standard way to fit a line, curve, or overdetermined system when no exact solution exists.", "more": "For an inconsistent system Ax = b, the least-squares solution minimizes ‖Ax − b‖², and it is characterized by the normal equations AᵀAx = Aᵀb, exactly the condition that the residual be orthogonal to the column space of A, the same orthogonality that orthogonal projection formalizes. Adrien-Marie Legendre published the method in 1805 and Carl Friedrich Gauss claimed to have used it since 1795; historians have largely accepted Gauss’s claim, though Legendre keeps priority of publication. The singular value decomposition computes it more stably than the normal equations, as orthogonal methods do, because it avoids forming AᵀA.", "cites": [ "r-10.30" ], "links_to_cards": [ "orthproj" ] }, { "id": "zerodiv", "url": "https://www.teclamairme.com/#g-zerodiv", "name": "zero divisor", "symbol": "ab=0, a,b≠0", "chapter": 10, "type": "obj", "badge": "Non-real object", "quick": "A zero divisor is a ring element that multiplies with some nonzero element to give zero; a nonzero one breaks the rule, true in all the ordinary number systems, that a product is zero only when one of its factors is.", "more": "In the integers, the rationals, the reals, the complex numbers, the quaternions, and the octonions, ab = 0 forces a = 0 or b = 0; rings and algebras with zero divisors, such as the integers modulo 6, where 2 × 3 = 0, or the sedenions, lose that guarantee along with the ability to divide safely. Their presence is what keeps a ring from being an integral domain, and in the Cayley–Dickson ladder they first appear at the sedenions, the sixteen-dimensional algebra after the octonions, and persist in every algebra above them, as in the trigintaduonions. A nonzero matrix whose square is zero, such as the nilpotent N of Chapter ᚈ(T), is itself a zero divisor: multiplied by itself, it gives nothing.", "cites": [ "r-10.33" ], "links_to_cards": [ "ints", "nilp", "octonions", "quaternions", "reals", "sedenions", "trig" ] }, { "id": "ftc", "url": "https://www.teclamairme.com/#g-ftc", "name": "the fundamental theorem of calculus, for x²", "symbol": "∫012x dx − 1", "symbol_markup": "∫_{0}^{1}2x dx − 1", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "0: the integral of the derivative of x² from 0 to 1 is x² evaluated at the ends, 1 − 0, so the difference vanishes.", "more": "The fundamental theorem joins the two operations that created calculus. If f is continuously differentiable on [a, b] then ∫_{a}^{b}f′(x)dx = f(b) − f(a); for f(x) = x² on [0, 1] the integral of 2x is 1 and f(1) − f(0) is 1, so the residual is 0. Newton and Leibniz developed the reciprocal relationship independently in the late seventeenth century. The instance on the page needs only ordinary one-variable calculus.", "cites": [ "r-11.1", "r-11.23", "r-11.24" ], "links_to_cards": [] }, { "id": "dsq1", "url": "https://www.teclamairme.com/#g-dsq1", "name": "the derivative of x² at one", "symbol": "(x²)′(1)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "2: the slope of the parabola at x = 1, and the number the numerator’s integral is built from.", "more": "The derivative of x² is 2x, so at x = 1 it is 2. Newton called it a fluxion and Leibniz wrote dy/dx; Lagrange’s prime notation, f′, dates from 1797. An integer.", "cites": [ "r-11.1", "r-11.2" ], "links_to_cards": [] }, { "id": "heatzero", "url": "https://www.teclamairme.com/#g-heatzero", "name": "an exact heat-equation solution", "symbol": "(∂t−∂xx)[e−π²tsin(πx)]", "symbol_markup": "(∂_{t}−∂_{xx})[e^{−π²t}sin(πx)]", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: time decay and spatial curvature cancel for this solution of uₜ = uₓₓ.", "more": "Let u(x, t)=e^{−π²t}sin(πx). Then ∂_{t}u=−π²u and ∂_{xx}u=−π²u, so (∂_{t}−∂_{xx})u=0. Fourier used such separated modes in his 1807–1822 theory of heat.", "cites": [ "r-11.3" ], "links_to_cards": [] }, { "id": "decayrate", "url": "https://www.teclamairme.com/#g-decayrate", "name": "the decay rate of the first heat mode", "symbol": "π²", "chapter": 11, "type": "tr", "badge": "Proven transcendental", "quick": "About 9.8696: the mode sin(πx) of the heat equation on a unit rod decays like e^(−π²t), and π² is transcendental because π is.", "more": "Fourier’s method, from his 1807 memoir and his Théorie analytique de la chaleur of 1822, separates the heat equation u_{t} = u_{xx} into modes sin(nπx) that each decay at their own rate n²π²; the slowest, n = 1, sets how long a rod takes to cool. π² ≈ 9.8696 is transcendental by Lindemann’s 1882 theorem, since if π² were algebraic, its square root π would be too. It is also ζ(2)·6, which is how Chapter ᚑ(O) knows it.", "cites": [ "r-11.3", "r-11.4" ], "links_to_cards": [ "zeta2" ] }, { "id": "minimumzero", "url": "https://www.teclamairme.com/#g-minimumzero", "name": "the simplest convex minimum", "symbol": "minx∈ℝx²", "symbol_markup": "min_{x∈ℝ}x²", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0, attained uniquely at x = 0: the simplest problem optimization has, and the bottom of the bowl.", "more": "The real function x² is nonnegative and equals 0 only at x=0, so min_{x∈ℝ}x²=0. The example is elementary, but its notation opens the framework of mathematical optimization: objective functions, feasible sets, convexity, gradients, duality, and optimality conditions.", "cites": [ "r-11.5" ], "links_to_cards": [] }, { "id": "minplus1", "url": "https://www.teclamairme.com/#g-minplus1", "name": "the minimum of x squared plus one", "symbol": "minx∈ℝ(x² + 1)", "symbol_markup": "min_{x∈ℝ}(x² + 1)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "1, at x = 0: the same bowl as the numerator, lifted by one so that the least value is no longer nothing.", "more": "The function x² + 1 is convex, its derivative 2x vanishes only at x = 0, and its value there is 1. Fermat’s method for finding maxima and minima, setting the rate of change to zero, circulated in about 1636 and calculus-based rules for finding optima begin with it and with Lagrange. An integer.", "cites": [ "r-11.17", "r-11.5" ], "links_to_cards": [ "fermatmax" ] }, { "id": "fermatmax", "url": "https://www.teclamairme.com/#g-fermatmax", "name": "Fermat’s method of maxima and minima", "symbol": "f′(x) = 0", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "Locate an extremum by setting the rate of change to zero: Fermat’s method, from a Latin treatise of about 1636, before there was a differential calculus to put it in.", "more": "Pierre de Fermat’s Methodus ad disquirendam maximam et minimam circulated in about 1636. The procedure it describes, adequality, compares the value of a quantity at a point with its value at a neighboring point and then lets the difference vanish, which is in substance the rule that at a maximum or a minimum the rate of change is zero. It predates the differential calculus that would later justify it, and the calculus-based rules for finding optima begin with it and with Lagrange. A method, not a number.", "cites": [ "r-11.17" ], "links_to_cards": [] }, { "id": "divcurl", "url": "https://www.teclamairme.com/#g-divcurl", "name": "divergence of a curl", "symbol": "∇·(∇×𝐯)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "Identically 0 for any smooth field, because mixed partial derivatives commute.", "more": "The identity ∇·(∇×v) = 0 holds for every field whose second partial derivatives are continuous, by Clairaut’s theorem that mixed partials commute, proposed in 1740 and proved by Schwarz in 1873; it became a one-line identity once Heaviside and Gibbs put ∇ to work as grad, div, and curl in the 1880s, when Maxwell’s 1865 equations were rewritten in that notation. On all of ℝ³, or on a suitable region without holes, it is the compatibility condition for writing a divergence-free field such as the magnetic field as the curl of a vector potential; in the language of differential forms it is half of d² = 0. Its value is the number 0 at every point, an integer.", "cites": [ "r-11.13", "r-11.14", "r-11.18", "r-11.6" ], "links_to_cards": [ "clairaut", "ddzero", "maxwell", "vectorpot" ] }, { "id": "divr", "url": "https://www.teclamairme.com/#g-divr", "name": "divergence of the position field", "symbol": "∇·𝐫", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "3, one for each dimension.", "more": "The position field r = (x, y, z) has divergence ∂x/∂x + ∂y/∂y + ∂z/∂z = 3, one for each dimension; in n dimensions it is n. It is the simplest example of a field with a source everywhere, and it appears in the divergence theorem as the trick that turns a volume into a surface integral: the volume of a region is one third of the flux of r through its boundary. An integer, 3.", "cites": [ "r-11.8" ], "links_to_cards": [ "divthm", "third" ] }, { "id": "flux", "url": "https://www.teclamairme.com/#g-flux", "name": "Gauss’s flux integral", "symbol": "∮S² (𝐫̂/r²)·d𝐀", "symbol_markup": "∮_{S²} (𝐫̂/r²)·d𝐀", "chapter": 11, "type": "tr", "badge": "Proven transcendental", "quick": "4π, the solid angle of the whole sky.", "more": "Gauss’s law is the first of the lemmas Carl Friedrich Gauss proved in his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata, published in 1813 and written on the attraction of ellipsoids. Lagrange’s memoir of 1773 supplies the change of variables the proof runs on, but it contains no flux theorem: the memoir has been read end to end, and there is no closed surface in it, no flux, and no 4π. Gauss came back to inverse-square forces in general in his Allgemeine Lehrsätze of 1840. The law says the flux of an inverse-square field through any closed surface around its source is 4π times the source strength; for the unit field r̂/r² through a sphere it is exactly 4π, the solid angle of all directions. It is one of Maxwell’s equations, and it is essentially equivalent to the inverse-square laws of Coulomb and Newton. Transcendental, since π is, by Lindemann’s theorem of 1882.", "cites": [ "r-11.22", "r-11.25", "r-11.26", "r-11.9" ], "links_to_cards": [ "maxwell" ] }, { "id": "curlgrad", "url": "https://www.teclamairme.com/#g-curlgrad", "name": "curl of a gradient", "symbol": "∇×(∇f)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "Identically 0 when f has continuous second derivatives: a landscape of heights has no whirlpools, the twin of the divergence of a curl.", "more": "∇×(∇f) = 0 for any twice continuously differentiable f, again because mixed partials commute, which Clairaut proposed in 1740 and Schwarz proved in 1873. So a conservative force field, one that is a gradient of a potential, has no curl, and it does no net work around any loop; the identity is the other half of d² = 0 in the language of forms. The zero vector at every point, handed home as the integer 0.", "cites": [ "r-11.6" ], "links_to_cards": [ "ddzero", "divcurl" ] }, { "id": "laplace1r", "url": "https://www.teclamairme.com/#g-laplace1r", "name": "the 1/r potential is harmonic", "symbol": "∇²(1/r), r ≠ 0", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "The Laplacian of 1/r is 0 everywhere except at the origin: the potential of a point charge or mass satisfies Laplace’s equation in empty space.", "more": "Pierre-Simon Laplace put the equation at the center of potential theory in his memoir on the attraction of spheroids, in the Mémoires volume for 1782, writing it in polar rather than Cartesian coordinates; it was known before him, and it carries his name because he made it the working tool of the subject. 1/r solves it away from the origin, the distributional identity at the origin is ∇²(1/r) = −4πδ, and Siméon Denis Poisson showed in ~1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ instead, which is Gauss’s law again. The Laplacian ∇² = ∇·∇ is the sum of second derivatives. Exactly 0 for r ≠ 0, an integer.", "cites": [ "r-11.21", "r-11.7" ], "links_to_cards": [ "laplaceeq" ] }, { "id": "ddzero", "url": "https://www.teclamairme.com/#g-ddzero", "name": "the exterior derivative applied twice", "symbol": "d(dω)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "The exterior derivative applied twice is 0, d² = 0: the one identity from which div curl = 0 and curl grad = 0 both follow.", "more": "Élie Cartan formalized the exterior derivative in 1899; the identity d∘d = 0 holds because mixed partials commute, and in three dimensions it unpacks to curl grad = 0 and div curl = 0. Its dual statement is that the boundary of a boundary is empty, ∂∂ = 0, the axiom of homology in Chapter ᚊ(Q); Stokes’ theorem ∫_{M} dω = ∫_{∂M} ω links the two. The value is the zero form, handed home as the integer 0.", "cites": [ "r-11.16" ], "links_to_cards": [ "bdbd", "curlgrad", "divcurl", "homology", "stokes" ] }, { "id": "laprsq", "url": "https://www.teclamairme.com/#g-laprsq", "name": "the Laplacian of r squared", "symbol": "∇²(r²)", "chapter": 11, "type": "int", "badge": "Integer or rational", "quick": "6 in three dimensions: each of x², y², z² contributes a second derivative of 2.", "more": "For r² = x² + y² + z² the Laplacian is 2 + 2 + 2 = 6; in n dimensions it is 2n. It is the reason the mean value of a function over a small sphere exceeds its center value by ∇²f·r²/6, to leading order, in three dimensions. An integer.", "cites": [ "r-11.8" ], "links_to_cards": [] }, { "id": "greenarea", "url": "https://www.teclamairme.com/#g-greenarea", "name": "the area of the unit circle by Green’s theorem", "symbol": "½∮S¹(x dy − y dx)", "symbol_markup": "½∮_{S¹}(x dy − y dx)", "chapter": 11, "type": "tr", "badge": "Proven transcendental", "quick": "π: walking once round the unit circle and integrating measures the area inside, Green’s theorem.", "more": "George Green, a self-taught miller’s son from Nottingham, published a version of his theorem in 1828 in an essay he paid to print himself and sold to fifty-one subscribers, most of them members of the Nottingham Subscription Library; it went almost unnoticed until William Thomson, later Lord Kelvin, was given three copies in 1845, showed it to Liouville and Sturm in Paris that summer, and had it reprinted between 1850 and 1854. The theorem turns an integral round a boundary into an integral over the region inside, and the special case ½∮(x dy − y dx) gives the enclosed area, which for the unit circle is π. It is the two-dimensional case of Stokes’ theorem. Transcendental, because π is.", "cites": [ "r-11.11", "r-11.20", "r-11.9" ], "links_to_cards": [ "pi", "stokes" ] }, { "id": "nabla", "url": "https://www.teclamairme.com/#g-nabla", "name": "nabla", "symbol": "∇", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "The vector of partial derivatives (∂/∂x, ∂/∂y, ∂/∂z): dotted for divergence, crossed for curl, applied for gradient, squared for the Laplacian.", "more": "William Rowan Hamilton introduced the operator, writing it as a sideways ◁, and Peter Guthrie Tait fixed the upright ∇ as the standard symbol in his Elementary Treatise on Quaternions of 1867; the name nabla, after a Phoenician harp of that shape, by way of the Greek νάβλα, was proposed to Tait by William Robertson Smith in a letter of 10 November 1870. Josiah Willard Gibbs and Oliver Heaviside separated it from the quaternions in the 1880s into the modern grad, div, and curl. An operator, not a number.", "cites": [ "r-11.15", "r-11.26", "r-11.6", "r-22.15" ], "links_to_cards": [ "quaternions" ] }, { "id": "divthm", "url": "https://www.teclamairme.com/#g-divthm", "name": "the divergence theorem", "symbol": "∫V ∇·𝐅 dV = ∮∂V 𝐅·d𝐀", "symbol_markup": "∫_{V} ∇·𝐅 dV = ∮_{∂V} 𝐅·d𝐀", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "What flows out through a surface equals the total source inside; special cases Gauss ~1813, the general theorem Ostrogradsky 1826, and the attribution is contested.", "more": "Carl Friedrich Gauss proved three special cases in 1813, in a paper on the attraction of ellipsoids, and Mikhail Ostrogradsky gave the general theorem in 1826, published in 1831. It is the three-dimensional case of the general Stokes’ theorem and the reason Gauss’s flux integral in the denominator stays 4π through any closed surface around the source. A theorem, not a number.", "cites": [ "r-11.10" ], "links_to_cards": [ "divattrib", "flux", "stokes" ] }, { "id": "stokes", "url": "https://www.teclamairme.com/#g-stokes", "name": "Stokes’ theorem", "symbol": "∮∂S 𝐅·d𝐫 = ∫S (∇×𝐅)·d𝐀", "symbol_markup": "∮_{∂S} 𝐅·d𝐫 = ∫_{S} (∇×𝐅)·d𝐀", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "Circulation round a loop equals the curl through any surface it bounds; Kelvin sent it to Stokes in 1850 and Stokes set it as an exam question in 1854.", "more": "William Thomson (Lord Kelvin) wrote the theorem in a letter of 1850 and George Stokes put it on the Smith’s Prize examination at Cambridge in 1854, which is how it got his name; Maxwell sat that exam. The general form ∫_{M} dω = ∫_{∂M} ω is Élie Cartan’s, after earlier generalizations by Vito Volterra, Édouard Goursat, and Henri Poincaré; it contains Green’s theorem, the divergence theorem, and the fundamental theorem of calculus as special cases. A theorem, not a number.", "cites": [ "r-11.12" ], "links_to_cards": [ "divthm" ] }, { "id": "maxwell", "url": "https://www.teclamairme.com/#g-maxwell", "name": "Maxwell’s equations", "symbol": "∇·𝐄 = ρ/ε₀, ∇·𝐁 = 0, ∇×𝐄 = −∂𝐁/∂t, ∇×𝐁 = μ₀𝐉 + μ₀ε₀∂𝐄/∂t", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "The four equations of electromagnetism, 1865, in the vector form Heaviside gave them in 1884: the theory in which div and curl were first studied.", "more": "James Clerk Maxwell published twenty equations in 1865; Oliver Heaviside rewrote them as these four in 1884 using the new vector calculus. They predict light as an electromagnetic wave with speed 1/√(μ₀ε₀), as Maxwell had found in 1865. The second equation, ∇·𝐁 = 0, is the zero of Chapter ᚁ(B)’s laws brace, and it is consistent with this chapter’s numerator because 𝐁 is the curl of a vector potential. Four equations, not a number.", "cites": [ "r-11.13" ], "links_to_cards": [ "divB", "vectorpot" ] }, { "id": "clairaut", "url": "https://www.teclamairme.com/#g-clairaut", "name": "Clairaut’s theorem", "symbol": "∂²f/∂x∂y = ∂²f/∂y∂x", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "Mixed partial derivatives commute wherever the second partials are continuous; Alexis Clairaut proposed it in 1740 and H. A. Schwarz gave the first rigorous proof in 1873.", "more": "The brace’s two curl identities are this one theorem in disguise. Writing out ∇·(∇×v) or ∇×(∇f) gives terms that appear twice with opposite signs, once as ∂_{x}∂_{y} and once as ∂_{y}∂_{x}, so they cancel exactly rather than approximately. The hypothesis is not decoration: the equality is asserted where the second partial derivatives are continuous on a neighborhood of the point, which is why the chapter states the twin identity for functions whose second derivatives are continuous. Clairaut proposed the result in 1740; the first rigorous proof is Schwarz’s, in 1873. A theorem, not a number.", "cites": [ "r-11.6" ], "links_to_cards": [ "curlgrad" ] }, { "id": "laplaceeq", "url": "https://www.teclamairme.com/#g-laplaceeq", "name": "Laplace’s equation", "symbol": "∇²V = 0", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "The equation a potential satisfies in empty space; inside the matter itself Poisson’s correction of ~1813 replaces it with ∇²V = −4πρ.", "more": "The function 1/r solves ∇²V = 0 everywhere except at the origin, which is why the potential of a point charge or a point mass is harmonic in empty space; at the origin itself the distributional identity is ∇²(1/r) = −4πδ. Pierre-Simon Laplace made the equation the working tool of potential theory in his memoir on the attraction of spheroids, writing it in polar rather than Cartesian coordinates; the equation was known before him, and it carries his name because he made it the instrument of the subject. Siméon Denis Poisson showed in ~1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ instead, so that the whole of the missing divergence sits where the mass or charge is. An equation, not a number.", "cites": [ "r-11.21", "r-11.7" ], "links_to_cards": [ "laplace1r" ] }, { "id": "vectorpot", "url": "https://www.teclamairme.com/#g-vectorpot", "name": "the vector potential", "symbol": "B = ∇×A", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "A field whose curl is the field you started with: a smooth enough divergence-free field on the whole of space has one, but on a region with a point taken out it may not.", "more": "Because the divergence of a curl is 0, a field can be a curl only if its divergence vanishes. On all of ℝ³ that condition is also sufficient for a field that is twice continuously differentiable and decays at least as fast as 1/‖x‖. On a smaller region it depends on the shape of the region, and simple connectivity is not enough: ℝ³ with the origin removed is simply connected, yet the inverse-square field r̂/r² has zero divergence on it and no vector potential. What suffices is that the region be star-shaped or contractible, by the Poincaré lemma, and in general that its second de Rham cohomology vanish. The magnetic field is the physical instance, and in quantum mechanics A acts on a charged particle even where the magnetic field is zero, the Aharonov–Bohm effect. A field, not a number.", "cites": [ "r-11.14", "r-11.18" ], "links_to_cards": [ "cohomology", "divcurl" ] }, { "id": "divattrib", "url": "https://www.teclamairme.com/#g-divattrib", "name": "who the divergence theorem belongs to", "symbol": "1762 … 1839", "chapter": 11, "type": "fn", "badge": "Function or operator", "quick": "One theorem, six names and four dates: Gauss ~1813, Ostrogradsky 1826, Green 1828, and others before them, and this book takes the earliest it can source.", "more": "Six names are attached to this one identity and the reference works do not agree on which belongs on it. The Encyclopedia of Mathematics lists it under all six at once—Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula, and Gauss-Green-Ostrogradski formula—and adds that it is already present in the work of Euler and other eighteenth-century mathematicians. The dates in circulation are these. Lagrange introduced surface integrals in 1760 and is widely credited with the divergence theorem itself in 1762, and that credit does not survive being checked: his 1762 memoir integrates by parts and then disposes of the boundary terms, and Victor Katz’s history names him once, among those who used the fundamental theorem to reduce an integral by a dimension, and not as an author of the result. Katz’s bibliography does not list him at all. Gauss proved three special cases in 1813, while working on the gravitational attraction of an elliptical spheroid, and further special cases in 1833 and 1839. Mikhail Ostrogradsky submitted Demonstration d’un theoreme du calcul integral to the Paris Academy on 13 February 1826, stating and proving the general theorem; he presented it again in Paris on 6 August 1827 and in St Petersburg on 5 November 1828, and only the last of the three was published, in 1831, in Note sur la Theorie de la Chaleur. George Green proved special cases in 1828 in his Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Gauss’s law, which is the inverse-square statement rather than the general theorem, used to carry a pair of dates here too: Lagrange in 1773 and Gauss in 1835. Both were checked in September 2026 and both went. Lagrange’s 1773 memoir was read end to end and contains no flux theorem, no closed surface, and no 4π; what it supplies is the change of variables Gauss’s proof uses. Nothing at all could be found at 1835, and the citation hanging off that year in the reference work that prints it is the 1813 paper. Gauss’s law is Gauss’s, in 1813. That still leaves four dates for the theorem, in a chapter that gives Clairaut a year and Green a year and Poisson a year without anyone objecting. The disagreement is nobody’s mistake. A result that is stated for one special case, proved for another, generalized by a third man, presented three times, and published five years after its first presentation has no single date to be right about, and the names it collected record which paper each later school of physics happened to read first. This book does not adjudicate priority. Where the prose needs a date it uses the earliest it can source—Gauss ~1813 for the divergence theorem, Gauss 1813 for Gauss’s law—and the rest of the story lives here.", "cites": [ "r-11.10", "r-11.22", "r-11.25" ], "links_to_cards": [ "divthm" ] }, { "id": "chiT", "url": "https://www.teclamairme.com/#g-chiT", "name": "Euler characteristic of the torus", "symbol": "χ(T²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "V − E + F = 0 for any cutting of the torus into polygons.", "more": "The Euler characteristic χ = V − E + F was found by Euler in 1750 for convex polyhedra and generalized by Poincaré in 1895 to spaces of any dimension, the Euler–Poincaré characteristic; for a closed orientable surface of genus g it equals 2 − 2g, so the torus, with one handle, has χ = 0. A square with opposite edges glued, one vertex, two edges, and one face, gives 1 − 2 + 1 = 0 directly. The Klein bottle and Möbius band share the value. An integer.", "cites": [ "r-12.1", "r-12.21" ], "links_to_cards": [ "classsurf", "eulerchar", "kleinbottle" ] }, { "id": "chiS", "url": "https://www.teclamairme.com/#g-chiS", "name": "Euler characteristic of the sphere", "symbol": "χ(S²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "2, Euler’s polyhedron formula.", "more": "Euler’s polyhedron formula, V − E + F = 2, announced in 1750 and proved by Legendre in 1794 and Cauchy in 1811, holds for any polyhedron whose surface is a sphere; the cube gives 8 − 12 + 6 and the icosahedron 12 − 30 + 20. Descartes had a closely related result around 1630; his manuscript is lost, and it survives through a copy Leibniz made in 1676, itself lost until about 1860. Whether it amounts to the same formula is disputed. The 2 is the sphere’s Euler characteristic, and it is why exactly five Platonic solids exist. An integer.", "cites": [ "r-12.1", "r-12.21" ], "links_to_cards": [ "eulerchar", "platonic" ] }, { "id": "chiRP", "url": "https://www.teclamairme.com/#g-chiRP", "name": "Euler characteristic of the projective plane", "symbol": "χ(ℝP²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "1: half the sphere’s 2, because identifying antipodal points halves every count.", "more": "The real projective plane is the sphere with antipodal points identified, or a disk with its boundary glued to a Möbius band; it cannot be embedded in three-dimensional space without crossing itself. Its Euler characteristic is half the sphere’s, 1, because the identification halves every count. Nonorientable surfaces have χ = 2 − k for k cross-caps, so ℝP² is the k = 1 case. An integer.", "cites": [ "r-12.1" ], "links_to_cards": [ "classsurf", "eulerchar" ] }, { "id": "chiS2g", "url": "https://www.teclamairme.com/#g-chiS2g", "name": "Euler characteristic of the double torus", "symbol": "χ(Σ₂)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "−2, since χ = 2 − 2g.", "more": "The double torus, or genus-two surface, is a sphere with two handles, and χ = 2 − 2g gives −2; it is the first orientable surface whose Euler characteristic is negative, and by Gauss–Bonnet the first that must carry negative curvature on average, which is why its natural geometry is hyperbolic. The denominator uses its absolute value so that every term stays positive. An integer, −2, entered as 2.", "cites": [ "r-12.1" ], "links_to_cards": [ "eulerchar" ] }, { "id": "chipt", "url": "https://www.teclamairme.com/#g-chipt", "name": "Euler characteristic of a point", "symbol": "χ(•)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "1: one vertex, no edges, no faces.", "more": "A single point has one vertex, no edges, and no faces, so V − E + F = 1; it is the simplest space there is. Any contractible space, a disk or a solid ball, has the same value. An integer.", "cites": [ "r-12.1" ], "links_to_cards": [ "eulerchar" ] }, { "id": "b1S2", "url": "https://www.teclamairme.com/#g-b1S2", "name": "first Betti number of the sphere", "symbol": "b1(S²)", "symbol_markup": "b_{1}(S²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "0: every loop on a sphere contracts.", "more": "The first Betti number counts independent loops that cannot be shrunk to a point; on a sphere every loop slides off to a point, so b₁(S²) = 0, a fact Poincaré made rigorous in 1895. The sphere is simply connected as well, which is the stronger statement: a trivial fundamental group forces b₁ = 0, but b₁ = 0 does not force a trivial fundamental group, and in 1904 Poincaré himself built the counterexample, a closed 3-manifold with the homology of the 3-sphere and a fundamental group of 120 elements. That is why the Poincaré conjecture in the epilogue, that a simply connected closed 3-manifold is a sphere, is stated about loops and not about Betti numbers. An integer.", "cites": [ "r-12.17", "r-12.21", "r-12.3" ], "links_to_cards": [ "bettinum", "homology", "pi1", "poincare", "threemfld" ] }, { "id": "b02S2", "url": "https://www.teclamairme.com/#g-b02S2", "name": "one component and one cavity", "symbol": "b0(S²), b2(S²)", "symbol_markup": "b_{0}(S²), b_{2}(S²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "1 and 1: one connected piece and one enclosed cavity.", "more": "The sphere has one connected piece, b₀ = 1, and encloses one cavity, b₂ = 1, and these are the only nonzero Betti numbers it has; their alternating sum with b₁ = 0 gives χ = 2, recovering Euler’s polyhedron formula. Enrico Betti introduced these counts in his paper of 1871, and Poincaré, who named them after him, gave them a more precise definition in Analysis Situs in 1895. Integers, 1 and 1.", "cites": [ "r-12.2", "r-12.3" ], "links_to_cards": [ "b1S2", "bettinum", "chiS", "eulerchar" ] }, { "id": "b1T2", "url": "https://www.teclamairme.com/#g-b1T2", "name": "first Betti number of the torus", "symbol": "b1(T²)", "symbol_markup": "b_{1}(T²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "2 independent loops.", "more": "The torus has two independent loops that cannot be shrunk or slid into each other: one around the tube and one through the hole. So b₁(T²) = 2, and with b₀ = b₂ = 1 the alternating sum 1 − 2 + 1 = 0 is the torus’s Euler characteristic, the numerator’s zero seen from the other side. An integer.", "cites": [ "r-12.3" ], "links_to_cards": [ "eulerchar" ] }, { "id": "b1S2g", "url": "https://www.teclamairme.com/#g-b1S2g", "name": "first Betti number of the double torus", "symbol": "b1(Σ₂)", "symbol_markup": "b_{1}(Σ₂)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "4: two independent loops per handle, and the double torus has two handles.", "more": "A surface of genus g has b₁ = 2g, one around and one through each handle, so the double torus has four; the alternating sum 1 − 4 + 1 = −2 matches its Euler characteristic. These loops generate the fundamental group’s abelianization, which is how homology and homotopy meet. An integer.", "cites": [ "r-12.3" ], "links_to_cards": [ "eulerchar", "homology", "pi1" ] }, { "id": "chiK", "url": "https://www.teclamairme.com/#g-chiK", "name": "Euler characteristic of the Klein bottle", "symbol": "χ(Klein bottle)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "0, the same as the torus; the Klein bottle is glued from a square with one pair of edges joined straight and the other pair joined with a twist.", "more": "Felix Klein described the bottle in 1882; it is a closed nonorientable surface with two cross-caps, so χ = 2 − 2 = 0, and like the torus it is built from one square, V − E + F = 1 − 2 + 1. It cannot sit in three-dimensional space without passing through itself. An integer.", "cites": [ "r-12.1", "r-12.21" ], "links_to_cards": [ "eulerchar" ] }, { "id": "bdbd", "url": "https://www.teclamairme.com/#g-bdbd", "name": "the boundary of a boundary", "symbol": "∂(∂M)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "The boundary of a boundary is the empty chain: ∂∂ = 0. It is an operator identity, and this empty chain comes home as the integer 0.", "more": "The boundary of a region is a closed surface, and a closed surface has no boundary. Poincaré introduced homology in Analysis Situs (1895) and its complements (1899–1904), and ∂∘∂ = 0 is its foundation: a chain complex is exactly a sequence of maps whose successive compositions vanish, and homology measures the gap between cycles (∂c = 0) and boundaries (c = ∂d). It is the dual of d² = 0 in Chapter ᚉ(C). The value is the empty chain; as a mixed-type term, it comes home as the integer 0.", "cites": [ "r-12.15", "r-12.21", "r-12.4" ], "links_to_cards": [ "ddzero", "homology" ] }, { "id": "genusT", "url": "https://www.teclamairme.com/#g-genusT", "name": "the genus of the torus", "symbol": "g(T²)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "1: one handle, one hole, and χ = 2 − 2g gives the numerator’s zero.", "more": "The genus of a closed orientable surface is its number of handles; the classification of surfaces — begun by Möbius (1863, or 1861 by one specialist account) and Jordan (1866), carried on by von Dyck (1888) and by Dehn and Heegaard (1907), and proved rigorously by Brahana (1921) — says the genus determines the surface. The torus has genus 1, and the formula χ = 2 − 2g ties the whole chapter together. An integer.", "cites": [ "r-12.5" ], "links_to_cards": [ "classsurf", "eulerchar" ] }, { "id": "cohomology", "url": "https://www.teclamairme.com/#g-cohomology", "name": "cohomology", "symbol": "Hⁿ(X)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "Homology read backwards: instead of the cycles in a space, the functions on them—which is what lets cohomology classes be multiplied.", "more": "Where homology builds groups out of the cycles in a space, cohomology builds them out of cochains—functions on chains—and the reversal buys something homology does not have: a product. The cup product makes the cohomology of a space into a graded ring, so two classes can be multiplied and not merely counted. It arrived about forty years after homology: Alexander had a notion of cochain by 1930, he and Kolmogorov both introduced cohomology at a Moscow conference in 1935, and Whitney and Čech built the cup and cap products between 1936 and 1938. This is the language the epilogue’s Hodge question is asked in — a Hodge class is a cohomology class, and the conjecture asks whether every one of them comes from subvarieties. A tool of the framework, uncolored.", "cites": [ "r-12.20" ], "links_to_cards": [ "hodge", "hodgeclass", "homology" ] }, { "id": "bettinum", "url": "https://www.teclamairme.com/#g-bettinum", "name": "Betti numbers", "symbol": "bₙ(X)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "The ranks of a space’s homology groups, counting independent features of each dimension: connected pieces, independent loops, enclosed cavities, and so on.", "more": "Enrico Betti introduced these counts in a paper of 1871, Henri Poincaré named them after him and gave them a rigorous definition in Analysis situs of 1895, and Emmy Noether recast them in the 1920s as the ranks of the abelian homology groups Hₙ(X). For the sphere b₀ = 1, b₁ = 0, b₂ = 1; for the torus b₁ = 2; for a genus-two surface b₁ = 4. Their alternating sum is the Euler characteristic, χ = b₀ − b₁ + b₂ − ⋯, which is why every Euler characteristic in this chapter can be read off as a homology count. Specific values, such as the sphere’s and the tori’s first Betti numbers, have cards of their own; this card is the general notion.", "cites": [ "r-12.23" ], "links_to_cards": [ "b02S2", "b1S2", "eulerchar", "homology" ] }, { "id": "kleinbottle", "url": "https://www.teclamairme.com/#g-kleinbottle", "name": "the Klein bottle", "symbol": "N₂", "chapter": 12, "type": "obj", "badge": "Non-real object", "quick": "A closed nonorientable surface, glued from a square with one pair of edges joined straight and the other joined with a twist, that cannot be embedded in ordinary three-dimensional space without self-intersection.", "more": "Felix Klein first described the Klein bottle in 1882. Like the torus it has Euler characteristic 0, but unlike the torus it is one-sided: cut it in half and two Möbius bands fall out, and no amount of stretching turns the Klein bottle into the torus, since the torus is orientable and the Klein bottle is not. It sits in the classification of closed surfaces as N₂, the nonorientable surface with two cross-caps, alongside the projective plane N₁ and the orientable surfaces Σ_{g}; it can be pictured as a tube passed back through itself to join its two ends from the inside, which is why a physical model must let the surface pass through itself, something a true Klein bottle in four dimensions does not need to do.", "cites": [ "r-12.24" ], "links_to_cards": [ "eulerchar" ] }, { "id": "unknot", "url": "https://www.teclamairme.com/#g-unknot", "name": "crossing number of the unknot", "symbol": "c(○)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "The unknot can be drawn with no crossings at all.", "more": "The crossing number of a knot is the fewest crossings in any drawing of it; the unknot, a plain loop, can be drawn with none, so c(○) = 0, while the trefoil needs three and no knot needs one or two. Peter Guthrie Tait tabulated knots in the 1870s and 1880s for Lord Kelvin’s theory that atoms were knots in the ether; the theory was wrong and the tables outlived it. Whether a given tangle is the unknot is decidable, but no polynomial-time method is known. An integer.", "cites": [ "r-12.22", "r-12.6" ], "links_to_cards": [ "crossnum", "trefoil" ] }, { "id": "trefoil", "url": "https://www.teclamairme.com/#g-trefoil", "name": "crossing number of the trefoil", "symbol": "c(3₁)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "3: the simplest knot that is really knotted needs three crossings, and no knot needs one or two.", "more": "The trefoil is the first entry in Tait’s tables of the 1870s, the (2, 3) torus knot; its crossing number is 3, proved by tricoloring or by the Jones polynomial, and it is chiral, distinct from its mirror image, which Max Dehn showed in 1914. An integer.", "cites": [ "r-12.22", "r-12.6" ], "links_to_cards": [ "crossnum", "jonespoly" ] }, { "id": "jonesunknot", "url": "https://www.teclamairme.com/#g-jonesunknot", "name": "the Jones polynomial of the unknot", "symbol": "V○(t)", "symbol_markup": "V_{○}(t)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "1: the invariant Vaughan Jones found in 1984 gives the unknot the constant polynomial 1, and it is unknown whether any other knot does.", "more": "Vaughan Jones discovered his polynomial in 1984 while working on von Neumann algebras, and won the Fields Medal for it in 1990; it distinguishes the trefoil from its mirror image, which the older Alexander polynomial of 1928 could not. The unknot has V = 1, and whether the Jones polynomial detects the unknot, i.e. whether V = 1 forces a knot to be trivial, is an open problem. An integer, 1.", "cites": [ "r-12.22", "r-12.25", "r-12.7", "r-12.8" ], "links_to_cards": [ "jonespoly" ] }, { "id": "jonespoly", "url": "https://www.teclamairme.com/#g-jonespoly", "name": "the Jones polynomial", "symbol": "V(t)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "A knot invariant, a Laurent polynomial in t assigned to every knot or link, that distinguishes knots the older Alexander polynomial could not.", "more": "Vaughan Jones discovered the polynomial in 1984 while studying von Neumann algebras, and the discovery won him the 1990 Fields Medal. It is normalized so the unknot has V(t) = 1, and it can tell a knot from its mirror image—unlike the Alexander polynomial, it distinguishes the trefoil from its reflection. Whether any nontrivial knot can also have V(t) = 1, meaning the polynomial would fail to detect the unknot, remains an open problem. The unknot’s value has a card of its own; this card is the polynomial itself.", "cites": [ "r-12.25" ], "links_to_cards": [] }, { "id": "crossnum", "url": "https://www.teclamairme.com/#g-crossnum", "name": "crossing number", "symbol": "c(K)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "The fewest crossings in any diagram of a knot: 0 for the unknot, and at least 3 for every knot that is genuinely knotted.", "more": "The crossing number of a knot K is the minimum number of crossings over every possible planar diagram of it; the unknot achieves c(○) = 0, and no nontrivial knot can be drawn with fewer than three crossings, a bound the trefoil meets exactly. Peter Guthrie Tait tabulated knots by crossing number in the 1870s and 1880s while pursuing Lord Kelvin’s theory that atoms were knots in the ether, and the tables outlived the physics they were built for. Whether a given diagram already achieves the true minimum is decidable in principle but has no known fast general method. The unknot’s and the trefoil’s values have cards of their own; this card is the function itself.", "cites": [ "r-12.27" ], "links_to_cards": [ "unknot" ] }, { "id": "cylinder", "url": "https://www.teclamairme.com/#g-cylinder", "name": "Gaussian curvature of a cylinder", "symbol": "K(cylinder)", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "A cylinder is flat: it unrolls onto a plane, so its Gaussian curvature is 0 everywhere.", "more": "Gaussian curvature, defined by Gauss in his Disquisitiones generales circa superficies curvas of 1827, is the product of the two principal curvatures at a point; a cylinder curves one way and is straight the other, so the product is 0 everywhere, and it can be unrolled flat without stretching. Gauss’s Theorema Egregium says this curvature is intrinsic, measurable from within the surface, which is why a flat map of the round Earth must distort. An integer.", "cites": [ "r-12.9" ], "links_to_cards": [] }, { "id": "gaussbonnet", "url": "https://www.teclamairme.com/#g-gaussbonnet", "name": "Gauss–Bonnet for the sphere", "symbol": "∫S² K dA", "symbol_markup": "∫_{S²} K dA", "chapter": 12, "type": "tr", "badge": "Proven transcendental", "quick": "4π: total curvature equals 2π times the Euler characteristic, so geometry and topology agree on every closed surface.", "more": "The Gauss–Bonnet theorem says ∫K dA = 2πχ for any closed surface; the unit sphere has K = 1 and area 4π, and 2π·2 = 4π. Gauss had a version he never published and Pierre Ossian Bonnet published a special case in 1848, but the closed-surface statement is credited to Walther von Dyck in 1888. It is why the torus, with χ = 0, must be flat on average, and the double torus negatively curved on average. Transcendental, because π is.", "cites": [ "r-12.10", "r-12.16" ], "links_to_cards": [ "chiT", "eulerchar" ] }, { "id": "platonic", "url": "https://www.teclamairme.com/#g-platonic", "name": "the number of Platonic solids", "symbol": "5", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "Exactly five regular convex polyhedra exist, a fact Euclid’s Elements ends with; Euler’s formula caps the count at five in a line.", "more": "Book XIII of Euclid’s Elements, c. 300 BC, closes by proving that the five regular convex solids are the only ones, and appears to be based largely on an earlier treatise by Theaetetus; Euler’s polyhedron formula gives the modern proof, since V − E + F = 2 with p-gon faces and q faces at each vertex forces 1/p + 1/q > 1/2, which has five solutions. An integer.", "cites": [ "r-12.12", "r-12.22" ], "links_to_cards": [ "chiS", "eulerchar" ] }, { "id": "pythag345", "url": "https://www.teclamairme.com/#g-pythag345", "name": "the 3–4–5 right triangle", "symbol": "3²+4²−5²", "chapter": 12, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the most familiar integer instance of the Pythagorean theorem.", "more": "A right triangle with legs 3 and 4 has hypotenuse 5, so 3²+4²−5²=9+16−25=0. Euclid’s Elements I.47 gives the classical proof around 300 BC, while the theorem and integer triples were known in older mathematical cultures and attribution to Pythagoras is historically uncertain. The identity gives elementary Euclidean geometry a visible carrier.", "cites": [ "r-12.11", "r-12.12", "r-12.22" ], "links_to_cards": [] }, { "id": "anglesum", "url": "https://www.teclamairme.com/#g-anglesum", "name": "the angle sum of a triangle", "symbol": "∠A + ∠B + ∠C", "chapter": 12, "type": "tr", "badge": "Proven transcendental", "quick": "π, or 180 degrees: Euclid I.32, true in the flat plane and false on the sphere; on the unit sphere the excess over π is the triangle’s area.", "more": "Euclid’s Elements I.32 proves that the three angles of a triangle add to two right angles, using the parallel postulate. On a sphere of radius 1 the sum exceeds π by exactly the triangle’s area (Girard, ~1626: the triangle formula is in his 1626 corrected edition of Stevin, and his 1629 main work generalized it to spherical polygons), and on a saddle it falls short, which is why the angle sum decides which geometry you are in and why this chapter’s Gauss–Bonnet integral is the same fact for a whole surface. Transcendental, being π.", "cites": [ "r-12.12", "r-12.13" ], "links_to_cards": [ "gaussbonnet", "parallel" ] }, { "id": "homeo", "url": "https://www.teclamairme.com/#g-homeo", "name": "homeomorphism", "symbol": "X ≅ Y", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "Two spaces are the same to a topologist when one can be stretched into the other without cutting or gluing: the coffee cup and the doughnut.", "more": "A homeomorphism is a continuous bijection with continuous inverse; Poincaré’s Analysis Situs of 1895 made topology the study of what such maps preserve, and every number in the topology brace, χ, the Betti numbers, genus, is a homeomorphism invariant. A relation, not a number.", "cites": [ "r-12.2" ], "links_to_cards": [ "bettinum", "bijection", "eulerchar" ] }, { "id": "homology", "url": "https://www.teclamairme.com/#g-homology", "name": "homology groups", "symbol": "Hn(X)", "symbol_markup": "H_{n}(X)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "The groups that count n-dimensional holes; their ranks are the Betti numbers of the denominator.", "more": "Homology assigns to a space X a sequence of abelian groups H₀, H₁, H₂, …, built from the chain complex with ∂∂ = 0; Poincaré defined the idea in 1895, Emmy Noether recognized the groups behind Betti’s numbers in the 1920s, and b_{n} = rank H_{n}(X). For the torus H₁ ≅ ℤ², which is the 2 in the denominator. A functor, not a number.", "cites": [ "r-12.2", "r-12.3", "r-12.4" ], "links_to_cards": [ "b1T2", "bdbd", "bettinum" ] }, { "id": "pi1", "url": "https://www.teclamairme.com/#g-pi1", "name": "the fundamental group", "symbol": "π₁(X)", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "The loops in a space up to deformation; π₁(S²) is trivial, π₁(S¹) ≅ ℤ, and the Poincaré conjecture is a statement about π₁.", "more": "Poincaré defined the fundamental group in ~1894 as the group of loops based at a point, composed by concatenation, up to continuous deformation; its abelianization is H₁, so the first Betti number is its rank. The sphere’s π₁ is trivial, which is the numerator’s b₁(S²) = 0 seen through homotopy; the epilogue’s Poincaré conjecture says trivial π₁ characterizes the 3-sphere. A group, not a number.", "cites": [ "r-12.2", "r-12.3" ], "links_to_cards": [ "b1S2", "homology", "poincare" ] }, { "id": "classsurf", "url": "https://www.teclamairme.com/#g-classsurf", "name": "the classification of surfaces", "symbol": "Σg, Nk", "symbol_markup": "Σ_{g}, N_{k}", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "Every closed surface is a sphere with g handles (χ = 2 − 2g) or with k cross-caps (χ = 2 − k), and nothing else.", "more": "The classification theorem, stated by Möbius in 1863, or 1861 by one specialist account, carried further by Dehn and Heegaard in 1907, and first proved rigorously by Brahana in 1921, lists every closed surface once: orientable Σ_{g} with χ = 2 − 2g, nonorientable N_{k} with χ = 2 − k. Every surface in this chapter is on the list: S² = Σ₀, T² = Σ₁, Σ₂, ℝP² = N₁, the Klein bottle N₂. A theorem, not a number.", "cites": [ "r-12.1", "r-12.5" ], "links_to_cards": [ "kleinbottle" ] }, { "id": "eulerchar", "url": "https://www.teclamairme.com/#g-eulerchar", "name": "the Euler characteristic", "symbol": "χ", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "V − E + F for a polyhedron, and the alternating sum of the Betti numbers in general; it does not change under continuous deformation.", "more": "For a polyhedral decomposition, χ = V − E + F, and Euler announced the value 2 for convex polyhedra in a 1750 letter to Goldbach, publishing the details in two papers of ~1752—written in 1750 and 1751, and printed in the Novi Commentarii volume for 1752–53, which itself appeared only in 1758. The modern definition is the alternating sum of the Betti numbers, χ = b₀ − b₁ + b₂ − b₃ + ⋯. Because homology is a topological and even a homotopy invariant, so is χ—two spaces that can be deformed into one another share it. For a closed orientable surface of genus g it is 2 − 2g, which is why the Euler characteristics of this chapter’s orientable surfaces can be read off the number of handles. An invariant, not a number of this chapter’s own, so uncolored.", "cites": [ "r-12.1", "r-12.18" ], "links_to_cards": [ "bettinum", "chiS", "classsurf", "homology" ] }, { "id": "parallel", "url": "https://www.teclamairme.com/#g-parallel", "name": "the parallel postulate", "symbol": "Postulate 5", "chapter": 12, "type": "fn", "badge": "Function or operator", "quick": "Euclid’s fifth postulate, the assumption the angle sum of a triangle rests on; without it the sum need not be π: it rises above π on the sphere and falls below it on the saddle.", "more": "Euclid’s proof that the three angles of a triangle add to two right angles, Elements I.32, runs through I.31 and I.29 and so depends on the parallel postulate. That dependence is what makes the angle sum a test of which geometry you are standing in: on the unit sphere the sum exceeds π by exactly the triangle’s area, and a hyperbolic triangle’s angles add to less than π. The same seam, taken over a whole surface instead of one triangle, is this chapter’s Gauss–Bonnet integral. A postulate, not a number.", "cites": [ "r-12.12", "r-12.13" ], "links_to_cards": [ "anglesum", "gaussbonnet" ] }, { "id": "probR", "url": "https://www.teclamairme.com/#g-probR", "name": "the probability of drawing her exactly", "symbol": "P(X = ᚏ4.16)", "symbol_markup": "P(X = ᚏ_{4.16})", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "A point has no length, so under a continuous distribution such as the normal, the chance of one exact value is 0—a possible outcome with no chance, not an impossible one; Rose’s number crosses from the equation’s left side to its right, and measure theory (Chapter ᚓ(E)) is why.", "more": "Andrey Kolmogorov’s axioms of 1933 define a probability as a measure, and for any random variable with a density the probability of any single exact value is 0, since a point has zero length under the integral. This is the distinction between probability zero and impossibility: some value is always drawn, yet each particular one had chance 0. The number here is exactly 0, an integer, and it is the one place the left side’s undefinable number appears on the right, where measure theory (Chapter ᚓ(E)) gives it chance 0, though, like every value, it remains a possible draw.", "cites": [ "r-13.1", "r-13.2" ], "links_to_cards": [] }, { "id": "var0", "url": "https://www.teclamairme.com/#g-var0", "name": "the variance of a constant", "symbol": "Var(c)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "A quantity that never moves has no spread: every deviation from its mean is 0, so its variance is 0.", "more": "Variance is the average squared distance from the mean. For a constant c the mean is c and every distance is 0, so Var(c) = 0; it is also the constant case of Var(aX + b) = a²Var(X). Exactly 0, an integer.", "cites": [ "r-13.25", "r-13.3" ], "links_to_cards": [] }, { "id": "cov0", "url": "https://www.teclamairme.com/#g-cov0", "name": "the covariance of independent variables", "symbol": "Cov(X, Y), X ⟂ Y", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "Independent things do not move together, so their covariance is 0; the converse is famously false.", "more": "Covariance, E[(X − μₓ)(Y − μᵧ)], measures how two quantities move together. Independence forces E[XY] = E[X]E[Y] and so Cov = 0, but the converse fails: X and X² for X uniform on [−1, 1] have zero covariance and are dependent. Exactly 0 for independent variables with finite second moments, an integer.", "cites": [ "r-13.3" ], "links_to_cards": [] }, { "id": "sqrt2pi", "url": "https://www.teclamairme.com/#g-sqrt2pi", "name": "the bell curve’s normalizer", "symbol": "√(2π)", "chapter": 13, "type": "tr", "badge": "Proven transcendental", "quick": "The area under e^(−x²/2), the reason the normal distribution has a 2π in it; transcendental.", "more": "The integral of e^(−x²/2) over the whole line is √(2π). Laplace evaluated the probability integral in 1774 or 1782 (the sources differ); the familiar proof that squares the integral and changes to polar coordinates is credited to Poisson. The result supplies the normal density’s factor 1/√(2π), making its total probability 1. Transcendental, because π is: if √(2π) were algebraic, then 2π would be algebraic too.", "cites": [ "r-13.4" ], "links_to_cards": [] }, { "id": "secretary", "url": "https://www.teclamairme.com/#g-secretary", "name": "the secretary-problem constant", "symbol": "1/e", "chapter": 13, "type": "tr", "badge": "Proven transcendental", "quick": "With many candidates, pass over the first 37% and then take the first one better than all seen so far: both the skip fraction and the chance of success tend to 1/e; transcendental.", "more": "The secretary problem asks how to pick the best of n candidates interviewed in random order when each must be accepted or rejected on the spot; Martin Gardner popularized a related version in Scientific American in February 1960, and Dennis Lindley published the first solution of the classical secretary problem in 1961. Its optimal finite-n rule uses an integer cutoff near n/e and then takes the next candidate better than all seen so far. Both the cutoff fraction and the optimal success probability tend to 1/e ≈ 0.368 as n grows. Transcendental by Hermite’s 1873 theorem, since e is.", "cites": [ "r-13.28", "r-13.5" ], "links_to_cards": [] }, { "id": "pempty", "url": "https://www.teclamairme.com/#g-pempty", "name": "the probability of the impossible event", "symbol": "P(∅)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "0, by Kolmogorov’s axioms: the empty event contains no outcomes and therefore never happens.", "more": "Andrey Kolmogorov’s Grundbegriffe of 1933 made probability a measure on a σ-algebra with P(Ω) = 1, non-negativity, and countable additivity; P(∅) = 0 follows from additivity, since ∅ ∪ ∅ = ∅. It is the opposite of the numerator’s first zero: P(∅) = 0 because nothing is there, P(X = ᚏ) = 0 although something is. An integer.", "cites": [ "r-13.2" ], "links_to_cards": [ "pomega", "sigmaalg" ] }, { "id": "meandev", "url": "https://www.teclamairme.com/#g-meandev", "name": "the mean deviation from the mean", "symbol": "E[X − E[X]]", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "0: deviations from the average average out, which is why variance squares them first.", "more": "Expectation is linear, so E[X − μ] = E[X] − μ = 0 for any X with finite mean, and the deviations of a sample from its arithmetic mean sum to zero for the same reason. The signed deviations cancel exactly; measures of spread therefore use absolute or squared deviations instead. An integer, 0.", "cites": [ "r-13.13", "r-13.25", "r-13.3" ], "links_to_cards": [] }, { "id": "pomega", "url": "https://www.teclamairme.com/#g-pomega", "name": "the probability of the sure event", "symbol": "P(Ω)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "1: something happens. Kolmogorov’s second axiom, 1933.", "more": "The sample space Ω is the set of all outcomes and its probability is 1 by axiom; every probability lies between P(∅) = 0 and P(Ω) = 1. An integer.", "cites": [ "r-13.2" ], "links_to_cards": [ "pempty" ] }, { "id": "birthday", "url": "https://www.teclamairme.com/#g-birthday", "name": "the birthday problem", "symbol": "23", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "In a room of 23 people the odds are better than even that two share a birthday; the smallest such crowd.", "more": "The birthday problem, first published by Richard von Mises in 1939 and generally attributed before that to Harold Davenport, who did not publish it at the time, asks how many people make a shared birthday more likely than not; the chance that n people all differ is 365!/(365^{n}(365 − n)!), which first drops below ½ at n = 23, where it is about 49.3%, so the shared-birthday chance is about 50.7%. The same calculation bounds the security of hash functions, the birthday attack. An integer.", "cites": [ "r-13.7" ], "links_to_cards": [] }, { "id": "montyhall", "url": "https://www.teclamairme.com/#g-montyhall", "name": "the Monty Hall answer", "symbol": "2/3", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "Switching doors wins two times in three; the problem Steve Selvin posed in 1975 and Marilyn vos Savant made famous in 1990.", "more": "Three doors, one car; you pick one, the host, who knows where the car is, always opens a goat door you did not pick and always offers a switch. Switching wins with probability 2/3, because your first pick was wrong two times in three. Selvin published it in The American Statistician in 1975; vos Savant’s 1990 Parade column drew about 10,000 letters by the count reported at the time, most insisting she was wrong, including nearly 1,000 from readers with PhDs. A rational number.", "cites": [ "r-13.8" ], "links_to_cards": [] }, { "id": "diemean", "url": "https://www.teclamairme.com/#g-diemean", "name": "the expected value of a die roll", "symbol": "E[fair die]", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "7/2: the average of 1 through 6, a value the die never shows.", "more": "Expectation is the probability-weighted average of outcomes. For a fair die it is (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5, even though no face shows 3.5: an expected value need not be an attainable outcome. A rational number.", "cites": [ "r-13.9" ], "links_to_cards": [] }, { "id": "entropy0", "url": "https://www.teclamairme.com/#g-entropy0", "name": "the entropy of a certain event", "symbol": "H(certain)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "Shannon entropy measures surprise, and an event with probability 1 carries none: H = −1·log 1 = 0.", "more": "Claude Shannon defined entropy in A Mathematical Theory of Communication, 1948, as the average of −log p over the outcomes of a source; an event with probability 1 contributes −1·log 1 = 0, so a certain event carries no information. Shannon noted that the logarithmic form is recognizable as entropy in statistical mechanics, linking the information measure to the older physical one. The value is exactly 0, an integer.", "cites": [ "r-13.10" ], "links_to_cards": [] }, { "id": "onebit", "url": "https://www.teclamairme.com/#g-onebit", "name": "one bit", "symbol": "1 bit", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "The entropy of a fair coin: the unit of information, Shannon 1948.", "more": "The bit, a contraction of binary digit suggested by John Tukey and adopted by Shannon in 1948, is the entropy of a fair coin: two outcomes, each with probability ½, give H = −2·(½ log₂ ½) = 1. It is the unit of binary information. The value is the integer 1, measured in bits.", "cites": [ "r-13.11", "r-13.28" ], "links_to_cards": [] }, { "id": "mutinfo0", "url": "https://www.teclamairme.com/#g-mutinfo0", "name": "the mutual information of independent variables", "symbol": "I(X; Y), X ⟂ Y", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "0 bits: knowing one tells you nothing about the other, Shannon 1948.", "more": "Mutual information I(X;Y) = H(X) + H(Y) − H(X, Y) measures in bits how much one variable tells you about another; in Shannon’s framework it is 0 exactly when X and Y are independent, unlike covariance, whose zero can hide dependence. It is the Kullback–Leibler divergence of the joint distribution from the product of the marginals. An integer, 0.", "cites": [ "r-13.12" ], "links_to_cards": [] }, { "id": "kl0", "url": "https://www.teclamairme.com/#g-kl0", "name": "the divergence of a distribution from itself", "symbol": "DKL(P ‖ P)", "symbol_markup": "D_{KL}(P ‖ P)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "0: the Kullback–Leibler divergence, the information lost by using one distribution in place of another, vanishes only when they are the same.", "more": "Solomon Kullback and Richard Leibler defined relative entropy D(P‖Q) = Σ p log(p/q) in 1951; it is non-negative by Gibbs’ inequality and 0 if and only if P = Q, though it is not a distance, being asymmetric. For P = Q the defining sum vanishes term by term. An integer, 0.", "cites": [ "r-13.12" ], "links_to_cards": [] }, { "id": "xbarunbiased", "url": "https://www.teclamairme.com/#g-xbarunbiased", "name": "the sample mean is unbiased", "symbol": "E[X̄] − μ", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "The average of a sample drawn from one distribution with a finite mean has the same expectation as one draw, so this is exactly 0.", "more": "For an i.i.d. sample X₁, …, Xₙ with finite mean μ, linearity of expectation gives E[X̄] = E[(X₁+⋯+Xₙ)/n] = μ, so the sample mean is an unbiased estimator of μ and E[X̄] − μ = 0 for every n. The result needs only integrability and an identically distributed sample, not normality or even independence; the displayed identity follows directly. Exactly 0, an integer.", "cites": [ "r-13.13" ], "links_to_cards": [] }, { "id": "olsresid", "url": "https://www.teclamairme.com/#g-olsresid", "name": "OLS residuals sum to zero", "symbol": "Σᵢ(yᵢ − ŷᵢ)", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "With an intercept term, ordinary least squares chooses its line so that the residuals add up to exactly 0.", "more": "Ordinary least squares with an intercept minimizes Σ(yᵢ − a − bxᵢ)²; setting the derivative with respect to the intercept a to 0 gives Σ(yᵢ − ŷᵢ) = 0 directly, so the fitted residuals always sum to zero, whatever the data and whatever b turns out to be. The regression line through the data always passes through (x̄, ȳ) for the same reason. It fails only if the model omits the intercept. Exactly 0, an integer.", "cites": [ "r-13.14", "r-13.25" ], "links_to_cards": [ "leastsq" ] }, { "id": "z196", "url": "https://www.teclamairme.com/#g-z196", "name": "the two-sided 95% critical value", "symbol": "1.96", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "The familiar rounded standard-normal value used to build a 95% confidence interval; approximately, not exactly, the 97.5th percentile.", "more": "1.96 approximates the 97.5th percentile of the standard normal distribution, the value z with Φ(z) = 0.975, used because a two-sided 95% interval excludes 2.5% in each tail. A confidence interval’s half-width is this critical value times a standard error, not the bare number 1.96 by itself, and 1.96 is only the conventional two-decimal approximation. The displayed decimal is itself a rational number.", "cites": [ "r-13.16" ], "links_to_cards": [] }, { "id": "fisher005", "url": "https://www.teclamairme.com/#g-fisher005", "name": "Fisher’s significance level", "symbol": "0.05", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "One in twenty: Fisher’s convenient 1925 line for judging significance, a convention rather than a law of nature.", "more": "In Statistical Methods for Research Workers (1925) R. A. Fisher wrote that it is convenient to take a deviation of about twice the standard deviation, a probability of one in twenty, as the limit for judging a result significant. His wording presents one in twenty as a practical convention, not a universal theorem. A rational number.", "cites": [ "r-13.15" ], "links_to_cards": [] }, { "id": "markovstationary", "url": "https://www.teclamairme.com/#g-markovstationary", "name": "a stationary Markov distribution", "symbol": "‖πP−π‖1", "symbol_markup": "‖πP−π‖_{1}", "chapter": 13, "type": "int", "badge": "Integer or rational", "quick": "The distance between a Markov chain’s distribution and its next step; exactly 0 for P=[[1/2, 1/2], [1/4, 3/4]] and π=(1/3, 2/3), because π is stationary.", "more": "For the transition matrix P = [[1/2, 1/2], [1/4, 3/4]] and row distribution π = (1/3, 2/3), direct multiplication gives πP = π. Hence the ℓ¹ norm of πP−π is exactly 0. Markov introduced dependent chains in 1906; stationary distributions now organize random walks, mixing, and Markov-chain Monte Carlo. The concrete values make this a computation rather than a bare definition.", "cites": [ "r-13.17" ], "links_to_cards": [ "markov" ] }, { "id": "polya3", "url": "https://www.teclamairme.com/#g-polya3", "name": "Pólya’s random-walk constant", "symbol": "p₃", "chapter": 13, "type": "open", "badge": "Status open", "quick": "About 0.3405: the chance that a drunk on a three-dimensional grid ever returns to where they started. On a line or a plane the return has probability 1.", "more": "George Pólya proved in 1921 that a simple random walk on the integer lattice returns to its origin with probability 1 in one and two dimensions, but not in three, where the return probability is p₃ = 1 − 1/u with u = (√6/(32π³))·Γ(1/24)Γ(5/24)Γ(7/24)Γ(11/24) ≈ 1.5164, so p₃ ≈ 0.340537 (Watson 1939, closed form Glasser and Zucker 1977). The current reference literature surveyed here gives exact forms for p₃ but no proof that it is irrational.", "cites": [ "r-13.18", "r-13.24", "r-13.28" ], "links_to_cards": [] }, { "id": "bayes", "url": "https://www.teclamairme.com/#g-bayes", "name": "Bayes’ theorem", "symbol": "P(A|B) = P(B|A)P(A)/P(B)", "chapter": 13, "type": "fn", "badge": "Function or operator", "quick": "How evidence updates belief; Bayes’s posthumous essay was presented by Richard Price in 1763, and Laplace developed an independent version in 1774.", "more": "Bayes’ theorem follows in one line from the definition of conditional probability P(A|B) = P(A ∩ B)/P(B); Thomas Bayes formulated a special case in an essay that Richard Price presented in 1763, after Bayes’s death, and Laplace independently developed and generalized the method in 1774. It is the engine of Bayesian inference and makes prior odds explicit when evidence updates a probability. A theorem, not a number.", "cites": [ "r-13.20" ], "links_to_cards": [] }, { "id": "clt", "url": "https://www.teclamairme.com/#g-clt", "name": "the central limit theorem", "symbol": "(X̄ − μ)√n/σ → N(0, 1)", "chapter": 13, "type": "fn", "badge": "Function or operator", "quick": "Standardized averages of i.i.d. variables with finite variance tend to a bell curve: de Moivre’s early case, Laplace’s extension, and Pólya’s 1920 name.", "more": "In its familiar i.i.d. form, the central limit theorem says that standardized sums of variables with finite nonzero variance converge in distribution to the normal law. De Moivre found an early binomial case, Laplace extended it, later work supplied rigorous general versions, and George Pólya used the name central limit theorem in 1920. It is why √(2π) sits in the denominator: the bell curve is the limit all such standardized sums share. A theorem, not a number.", "cites": [ "r-13.21" ], "links_to_cards": [ "sqrt2pi" ] }, { "id": "lln", "url": "https://www.teclamairme.com/#g-lln", "name": "the law of large numbers", "symbol": "X̄n → μ", "symbol_markup": "X̄_{n} → μ", "chapter": 13, "type": "fn", "badge": "Function or operator", "quick": "Averages settle down: Jacob Bernoulli’s theorem, published in Ars Conjectandi in 1713, is an early law of large numbers.", "more": "Jacob Bernoulli proved in Ars Conjectandi (1713) that an observed frequency concentrates near its probability as the number of trials grows. Later strong-law forms make the convergence almost sure. A theorem, not a number.", "cites": [ "r-13.22" ], "links_to_cards": [] }, { "id": "normal", "url": "https://www.teclamairme.com/#g-normal", "name": "the standard normal density", "symbol": "φ(x) = e−x²/2/√(2π)", "symbol_markup": "φ(x) = e^{−x²/2}/√(2π)", "chapter": 13, "type": "fn", "badge": "Function or operator", "quick": "The bell curve, de Moivre 1733 and Gauss 1809: mean 0, variance 1, and area exactly 1 because of the √(2π) beneath it.", "more": "The normal distribution appeared as de Moivre’s limit of coin tosses in 1733, as Gauss’s law of errors in 1809, and as Laplace’s central limit in 1810; its density integrates to 1 precisely because ∫e^{−x²/2}dx = √(2π). About 68% of its mass lies within one standard deviation, 95% within two. A function, not a number.", "cites": [ "r-13.21", "r-13.23" ], "links_to_cards": [ "clt", "sqrt2pi" ] }, { "id": "markov", "url": "https://www.teclamairme.com/#g-markov", "name": "a Markov chain", "symbol": "P(Xₙ₊₁ | X₀…Xₙ) = P(Xₙ₊₁ | Xₙ)", "chapter": 13, "type": "fn", "badge": "Function or operator", "quick": "A process whose next step depends only on where it is now: Markov introduced these dependent chains in 1906.", "more": "A Markov chain is a sequence of random states in which the distribution of the next state depends on the present state alone, and not on the path that reached it. Andrei Markov’s 1906 paper introduced such dependent chains, extending the law of large numbers to variables that are not independent. Its transition matrix P collects the one-step probabilities, and a distribution π with πP = π is stationary, as the chapter’s two-state weather is. When the chain is finite, irreducible, and aperiodic—as that weather is, since every entry of its matrix is positive—the stationary distribution is unique and the chain walks toward it from any start at all, which is why the forecast forgets where it began. A process, not a number.", "cites": [ "r-13.17", "r-13.26" ], "links_to_cards": [ "lln" ] }, { "id": "jacobi", "url": "https://www.teclamairme.com/#g-jacobi", "name": "the Jacobi identity", "symbol": "[x, [y, z]] + [y, [z, x]] + [z, [x, y]]", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "In any Lie algebra the three nested brackets sum to 0; this is the Jacobi identity, the consistency law for the bracket.", "more": "A Lie algebra has a bilinear alternating bracket satisfying the Jacobi identity. In the equivalent derivation form, [x, [y, z]] = [[x, y], z] + [y, [x, z]]; moving every term to one side gives the cyclic zero above. The identity for Poisson brackets is in Jacobi’s Nova methodus, which Clebsch published in 1862, eleven years after Jacobi died, so the year is the edition’s and not his. Hermann Weyl introduced the name Lie algebra in 1934. Its value is the zero vector of the algebra, handed home as the integer 0.", "cites": [ "r-14.1", "r-14.18" ], "links_to_cards": [ "liealg" ] }, { "id": "e8", "url": "https://www.teclamairme.com/#g-e8", "name": "the dimension of E₈", "symbol": "dim E₈", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "248, the dimension of E₈, the largest of the five exceptional simple Lie algebras.", "more": "E₈ has dimension 248 and rank 8, making it the largest of the five exceptional simple Lie algebras. Its root system has 240 roots; the associated E₈ lattice gives the optimal sphere packing in eight dimensions, proved by Maryna Viazovska. An integer, 248.", "cites": [ "r-14.16", "r-14.2" ], "links_to_cards": [ "cayleyint" ] }, { "id": "liealt", "url": "https://www.teclamairme.com/#g-liealt", "name": "antisymmetry of the Lie bracket", "symbol": "[x, x]", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "A Lie bracket of anything with itself is 0; together with bilinearity and Jacobi, that is the definition of a Lie algebra.", "more": "The definition uses bilinearity, the alternating law [x, x] = 0, and the Jacobi identity. Expanding [x + y, x + y] = 0 forces [x, y] = −[y, x]. For matrices, the commutator [x, y] = xy − yx is the standard example. The value is the zero vector, handed home as the integer 0.", "cites": [ "r-14.1" ], "links_to_cards": [ "jacobi", "liealg" ] }, { "id": "su2", "url": "https://www.teclamairme.com/#g-su2", "name": "the dimension of su(2)", "symbol": "dim 𝔰𝔲(2)", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "3: the Lie algebra of spin and of rotations, spanned by the three Pauli matrices times i.", "more": "The real Lie algebra su(2) consists of traceless anti-Hermitian 2 × 2 matrices. It is three-dimensional, with a real basis given by i times the three Pauli matrices; SU(2) double-covers SO(3). An integer, 3.", "cites": [ "r-14.3" ], "links_to_cards": [ "liealg", "pauli" ] }, { "id": "g2", "url": "https://www.teclamairme.com/#g-g2", "name": "the dimension of G₂", "symbol": "dim G₂", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "14, the smallest exceptional Lie algebra: the symmetries of the octonions, which is why Chapters ᚌ(G) and ᚖ(W) share it.", "more": "G₂ is the smallest exceptional simple Lie algebra: it has dimension 14 and rank 2, and it is the derivation algebra of the octonions. That is the precise bridge to Chapter ᚖ(W). An integer, 14.", "cites": [ "r-14.2" ], "links_to_cards": [ "liealg", "octonions" ] }, { "id": "f4", "url": "https://www.teclamairme.com/#g-f4", "name": "the dimension of F₄", "symbol": "dim F₄", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "52, the exceptional Lie algebra of derivations of the exceptional Jordan algebra.", "more": "F₄ has dimension 52 and rank 4. It is the derivation algebra of the exceptional Jordan algebra formed from 3 × 3 Hermitian octonionic matrices. An integer, 52.", "cites": [ "r-14.2" ], "links_to_cards": [ "hermitian", "liealg" ] }, { "id": "e6", "url": "https://www.teclamairme.com/#g-e6", "name": "the dimension of E₆", "symbol": "dim E₆", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "78, the dimension of E₆, the rank-six exceptional simple Lie algebra.", "more": "E₆ has dimension 78 and rank 6, with a 27-dimensional fundamental representation. That representation is used in E₆ grand-unified models dating to 1976. An integer, 78.", "cites": [ "r-14.17", "r-14.2" ], "links_to_cards": [ "liealg" ] }, { "id": "e7", "url": "https://www.teclamairme.com/#g-e7", "name": "the dimension of E₇", "symbol": "dim E₇", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "133, the dimension of E₇, the rank-seven exceptional simple Lie algebra.", "more": "E₇ has dimension 133 and rank 7; its smallest nontrivial fundamental representation has dimension 56. An integer, 133.", "cites": [ "r-14.2" ], "links_to_cards": [ "liealg" ] }, { "id": "liealg", "url": "https://www.teclamairme.com/#g-liealg", "name": "a Lie algebra", "symbol": "𝔤", "chapter": 14, "type": "obj", "badge": "Non-real object", "quick": "A vector space with a bracket that is bilinear and alternating and obeys the Jacobi identity: what a continuous group of symmetries looks like near the identity, with matrix commutators as the standard example.", "more": "A Lie algebra is a vector space carrying a bracket [x, y] that is bilinear and alternating, so [x, x] = 0, and that satisfies the Jacobi identity; for matrices the bracket is the commutator xy − yx. It is the infinitesimal shadow of a continuous group of symmetries—the group’s moves near the identity, with composition traded for a bracket—which is why this chapter’s first brace is written with brackets and not with the group. Over an algebraically closed field of characteristic 0 the finite-dimensional simple ones fall into the four classical series of types A–D and the five exceptional types G₂, F₄, E₆, E₇, E₈. A structure rather than a number: the chapter’s first two zeros are its zero vector.", "cites": [ "r-14.1", "r-14.20" ], "links_to_cards": [ "e8", "g2", "infinitesimal", "jacobi", "liealt" ] }, { "id": "wedge", "url": "https://www.teclamairme.com/#g-wedge", "name": "a vector wedged with itself", "symbol": "e₁ ∧ e₁", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "In geometric algebra the outer product of a vector with itself is 0, because a vector spans no area with itself.", "more": "Grassmann introduced the exterior product in his 1844 Ausdehnungslehre; Clifford joined exterior and metric structure in his ~1876 algebra; the Encyclopedia of Mathematics dates the work to 1876 and other accounts to the 1878 paper, so the earliest is shown. The rule v ∧ v = 0 says that one vector spans no area with itself. Expanding (u + v) ∧ (u + v) = 0 gives u ∧ v = −v ∧ u. The value is the zero bivector, handed home as 0.", "cites": [ "r-14.4" ], "links_to_cards": [] }, { "id": "cliffanti", "url": "https://www.teclamairme.com/#g-cliffanti", "name": "orthogonal vectors anticommute", "symbol": "e₁e₂ + e₂e₁", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "In Clifford’s geometric algebra two perpendicular vectors anticommute, so this sum is 0: the rule that makes the geometric product work.", "more": "In a Clifford algebra, vectors satisfy uv + vu = 2B(u, v), where B(u, v) is zero when u and v are perpendicular, so perpendicular vectors anticommute. Thus e₁e₂ + e₂e₁ = 0. The value is the zero of the algebra, handed home as the integer 0.", "cites": [ "r-14.4" ], "links_to_cards": [] }, { "id": "cl3dim", "url": "https://www.teclamairme.com/#g-cl3dim", "name": "the dimension of the geometric algebra of space", "symbol": "dim Cl(3)", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "8 = 2³: one scalar, three vectors, three bivectors, and one trivector, every product of the three basis vectors of space.", "more": "A Clifford algebra on an n-dimensional space has a basis indexed by subsets of n basis vectors and therefore dimension 2^{n}. For n = 3 the grade dimensions are 1, 3, 3, 1, totaling 8; the even subalgebra is isomorphic to the quaternions. An integer, 8.", "cites": [ "r-14.5" ], "links_to_cards": [ "quaternions" ] }, { "id": "rootsum", "url": "https://www.teclamairme.com/#g-rootsum", "name": "the sum of the fifth roots of unity", "symbol": "1 + ζ₅ + ζ₅² + ζ₅³ + ζ₅⁴", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "The five points of a pentagon on the unit circle add to 0; the cyclic group cancels itself.", "more": "The n-th roots of unity are the vertices of a regular n-gon, and their sum is zero for n ≥ 2. In particular the five vertices shown here add to 0. Gauss’s construction of the regular 17-gon, recorded in his diary in March 1796, rested on sums of roots of unity like this one. The value is the complex number 0, handed home as an integer.", "cites": [ "r-14.19", "r-14.6" ], "links_to_cards": [] }, { "id": "s5", "url": "https://www.teclamairme.com/#g-s5", "name": "the order of the symmetric group on five letters", "symbol": "|S₅|", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "120, the first symmetric group that is not solvable, which is why the general quintic has no formula in radicals.", "more": "S₅ has 5! = 120 elements. Its normal subgroup A₅ is nonabelian simple, so S₅ is not solvable; that obstruction is why the general quintic is not solvable by radicals. An integer, 120.", "cites": [ "r-14.15", "r-14.8" ], "links_to_cards": [] }, { "id": "moonshine", "url": "https://www.teclamairme.com/#g-moonshine", "name": "the Monster’s smallest representation", "symbol": "196883", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "One less than the 196884 in the j-invariant; the coincidence that became monstrous moonshine.", "more": "196883 is the smallest nontrivial complex character degree of the Monster. McKay observed that 196883 + 1 = 196884, the coefficient of q in the j-function; Conway and Norton developed this into monstrous moonshine, proved by Borcherds in 1992. An integer, 196883.", "cites": [ "r-14.10", "r-14.11" ], "links_to_cards": [ "jinv" ] }, { "id": "charsum", "url": "https://www.teclamairme.com/#g-charsum", "name": "a nontrivial character sums to zero", "symbol": "Σg∈G χ(g), χ ≠ 1", "symbol_markup": "Σ_{g∈G} χ(g), χ ≠ 1", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "Add up any nontrivial irreducible character of a finite group over all its elements and you get 0: orthogonality of the characters Frobenius introduced in 1896.", "more": "For irreducible characters of a finite group, orthogonality with the trivial character gives Σ_{g∈G} χ(g) = 0 whenever χ is nontrivial. Frobenius introduced group characters in 1896. The value is the complex number 0, handed home as an integer.", "cites": [ "r-14.7" ], "links_to_cards": [] }, { "id": "a5", "url": "https://www.teclamairme.com/#g-a5", "name": "the order of the alternating group on five letters", "symbol": "|A₅|", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "60: the smallest nonabelian simple group, the rotations of an icosahedron, and the reason the general quintic has no formula in radicals.", "more": "A₅ has 5!/2 = 60 elements. It is the rotation group of the icosahedron and the smallest nonabelian simple group; its simplicity makes S₅ nonsolvable and supplies the group-theoretic obstruction for the general quintic. An integer, 60.", "cites": [ "r-14.8" ], "links_to_cards": [ "s5" ] }, { "id": "sporadic26", "url": "https://www.teclamairme.com/#g-sporadic26", "name": "the number of sporadic groups", "symbol": "26", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "Twenty-six finite simple groups belong to no infinite family; the classification theorem says there are no more.", "more": "The classification of finite simple groups has four parts: cyclic groups of prime order, alternating groups, groups of Lie type, and exactly 26 sporadic groups. The 2004 quasithin work closed the last recognized gap in the proof. An integer, 26.", "cites": [ "r-14.9" ], "links_to_cards": [] }, { "id": "monster", "url": "https://www.teclamairme.com/#g-monster", "name": "the order of the Monster group", "symbol": "|𝕄|", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "808017424794512875886459904961710757005754368000000000, about 8 × 10⁵³ elements, the largest sporadic group.", "more": "The Monster has order 2⁴⁶·3²⁰·5⁹·7⁶·11²·13³·17·19·23·29·31·41·47·59·71 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, about 8 × 10⁵³. It is the largest sporadic simple group, and its smallest nontrivial complex character degree is 196883. An integer.", "cites": [ "r-14.10" ], "links_to_cards": [] }, { "id": "titsgroup", "url": "https://www.teclamairme.com/#g-titsgroup", "name": "the Tits group", "symbol": "²F₄(2)′", "chapter": 14, "type": "obj", "badge": "Non-real object", "quick": "A finite simple group of order 17,971,200 that sits inside the Ree group ²F₄(2), classified by convention among the Lie-type families rather than counted as a twenty-seventh sporadic group.", "more": "The Tits group, named for Jacques Tits, is an index-2 subgroup of the Ree group ²F₄(2) and is simple in its own right, with order 17,971,200. It belongs to the infinite family ²F₄(2²ⁿ⁺¹)′ of derived subgroups of the Ree groups, though it is not strictly a group of Lie type itself. The classification of finite simple groups usually files it with the infinite Lie-type families rather than counting it as a sporadic group, which is why the tally of sporadic groups stays at twenty-six instead of becoming twenty-seven. Some sources call it a borderline or honorary sporadic group all the same.", "cites": [ "r-14.23" ], "links_to_cards": [] }, { "id": "jinv", "url": "https://www.teclamairme.com/#g-jinv", "name": "the j-invariant", "symbol": "j(τ)", "chapter": 14, "type": "fn", "badge": "Function or operator", "quick": "The modular j-function, with q-expansion 1/q + 744 + 196884q + …, is the basic modular function on the upper half-plane invariant under the full modular group.", "more": "The j-invariant is invariant under SL(2, ℤ) acting on the upper half-plane and generates the field of level-one modular functions; every elliptic curve over the complex numbers corresponds to a value of j, up to isomorphism. Its q-expansion, with q = e^(2πiτ), begins 1/q + 744 + 196884q + 21493760q² + …; the coefficient 196884 is one more than 196883, the dimension of the smallest nontrivial irreducible representation of the Monster group, the coincidence McKay noticed that grew into monstrous moonshine. j is also why Ramanujan’s constant e^(π√163) sits so close to an integer: at τ = (1+√−163)/2 the j-function takes the integer value −640320³, and because every term after 1/q + 744 is tiny there, e^(π√163) lies within about 10⁻¹² of 640320³ + 744.", "cites": [ "r-14.25" ], "links_to_cards": [ "ram" ] }, { "id": "moonshineconj", "url": "https://www.teclamairme.com/#g-moonshineconj", "name": "the monstrous moonshine conjecture", "symbol": "", "chapter": 14, "type": "fn", "badge": "Function or operator", "quick": "The Conway-Norton monstrous moonshine conjectures of 1979, proved by Borcherds in 1992, connect the coefficients of the j-function to the representations of the Monster group.", "more": "After McKay noticed that 196884, the first nontrivial coefficient of the j-invariant, is one more than 196883, the dimension of the smallest nontrivial representation of the Monster group, Conway and Norton turned the coincidence into a full conjectural correspondence in 1979: every coefficient of the modular functions attached to each element of the Monster matches a combination of the group’s representations. Richard Borcherds proved the conjectures in 1992 for the moonshine module, a vertex operator algebra with Monster symmetry, using the no-ghost theorem from string theory, work for which, in part, he received the Fields Medal in 1998.", "cites": [ "r-14.26" ], "links_to_cards": [ "jinv" ] }, { "id": "disc", "url": "https://www.teclamairme.com/#g-disc", "name": "the discriminant of a repeated root", "symbol": "disc((x − 1)²)", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "b² − 4ac for x² − 2x + 1 is 4 − 4 = 0; a zero discriminant means a root repeats.", "more": "The discriminant is, up to the leading-coefficient factor, the product of the squared differences of a polynomial’s roots. For ax² + bx + c it is b² − 4ac, and it vanishes exactly when a root repeats. For (x − 1)² it is 4 − 4 = 0, an integer.", "cites": [ "r-14.12" ], "links_to_cards": [] }, { "id": "degsqrt2", "url": "https://www.teclamairme.com/#g-degsqrt2", "name": "the degree of a field extension", "symbol": "[ℚ(√2) : ℚ]", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "2: adjoining √2 to the rationals doubles the dimension.", "more": "The degree [L : K] is the dimension of L as a vector space over K. Since ℚ(√2) has basis {1, √2} over ℚ, its degree is 2; extension degrees multiply in towers. An integer.", "cites": [ "r-14.13" ], "links_to_cards": [ "sqrt2" ] }, { "id": "galzeta5", "url": "https://www.teclamairme.com/#g-galzeta5", "name": "the order of a Galois group", "symbol": "|Gal(ℚ(ζ₅)/ℚ)|", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "4: the symmetries of the fifth roots of unity form (ℤ/5ℤ)^×, a central example in Galois theory.", "more": "For a cyclotomic field, Gal(ℚ(ζ_{n})/ℚ) is isomorphic to (ℤ/nℤ)^{×}, acting by ζ_{n} ↦ ζ_{n}^{k}. For n = 5 this group has order φ(5) = 4. An integer.", "cites": [ "r-14.14" ], "links_to_cards": [] }, { "id": "abelbracket", "url": "https://www.teclamairme.com/#g-abelbracket", "name": "Abel’s bracket", "symbol": "[the general quintic is solvable in radicals]", "chapter": 14, "type": "int", "badge": "Integer or rational", "quick": "0: Abel proved in 1824 that no formula using only arithmetic and root extraction solves every fifth-degree equation, and Galois explained why: S₅ is not solvable.", "more": "Ruffini’s 1799 proof had one gap in it and Abel proved in 1824 that no formula by radicals solves every quintic. Galois theory explains the obstruction: the general quintic has group S₅, which is not solvable because it contains the nonabelian simple group A₅. The Iverson bracket of the false proposition is 0, an integer.", "cites": [ "r-14.15", "r-14.21" ], "links_to_cards": [ "iverson", "s5" ] }, { "id": "braket", "url": "https://www.teclamairme.com/#g-braket", "name": "inner product of orthogonal states", "symbol": "⟨0|1⟩", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "0 in Dirac notation; orthogonal pure states can be perfectly distinguished.", "more": "Dirac’s 1939 paper introduced bra-ket notation, in which ⟨a|b⟩ is an inner product. The computational basis states |0⟩ and |1⟩ are orthonormal, so ⟨0|1⟩ = 0. Orthogonal pure states can be distinguished perfectly, whereas non-orthogonal states cannot; that distinction is part of the no-cloning limit. The value is the complex number 0, handed home as an integer.", "cites": [ "r-15.20", "r-15.3", "r-15.4" ], "links_to_cards": [] }, { "id": "norm00", "url": "https://www.teclamairme.com/#g-norm00", "name": "norm of a basis state", "symbol": "⟨0|0⟩", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "1: a normalized state’s overlap with itself, the rule every state obeys.", "more": "A normalized state has inner product 1 with itself. The Born rule turns squared amplitudes into measurement probabilities, so normalization makes the probabilities of all outcomes add to one. Thus ⟨0|0⟩ = 1, an integer.", "cites": [ "r-15.3", "r-15.5" ], "links_to_cards": [ "born" ] }, { "id": "plus0", "url": "https://www.teclamairme.com/#g-plus0", "name": "overlap of the plus state with zero", "symbol": "⟨+|0⟩", "chapter": 15, "type": "alg", "badge": "Algebraic irrational", "quick": "1/√2: the equal superposition |+⟩ has amplitude 1/√2 along |0⟩, so the chance of reading 0 is ½.", "more": "The Hadamard gate sends |0⟩ to the equal superposition |+⟩ = (|0⟩ + |1⟩)/√2. Its overlap with |0⟩ is 1/√2, whose square gives probability 1/2 for the outcome 0. It is algebraic, a root of 2x² − 1.", "cites": [ "r-15.6" ], "links_to_cards": [ "sqrt2" ] }, { "id": "sigx", "url": "https://www.teclamairme.com/#g-sigx", "name": "a Pauli-X matrix element", "symbol": "⟨1|σx|0⟩", "symbol_markup": "⟨1|σ_{x}|0⟩", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "1, because σx flips 0 to 1.", "more": "The Pauli X matrix sends |0⟩ to |1⟩ and is the single-qubit NOT gate, so ⟨1|σ_{x}|0⟩ = 1. Pauli introduced the three matrices in 1927 in his work on electron spin. An integer.", "cites": [ "r-15.7" ], "links_to_cards": [] }, { "id": "grass", "url": "https://www.teclamairme.com/#g-grass", "name": "the Grassmann anticommutator", "symbol": "θ₁θ₂ + θ₂θ₁", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "Anticommuting variables obey θ₁θ₂ = −θ₂θ₁, so the sum is 0; this is the sign rule used for fermion variables.", "more": "Grassmann variables anticommute: θ₁θ₂ = −θ₂θ₁, so θ₁θ₂ + θ₂θ₁ = 0 and each Grassmann variable squares to zero. They are the variables used in fermionic path integrals. Their anticommutation is that of Pauli’s 1925 exclusion principle, which says that two identical fermions cannot occupy the same quantum state. The value is the zero of the Grassmann algebra, handed home as the integer 0.", "cites": [ "r-15.20", "r-15.8" ], "links_to_cards": [ "exclusion" ] }, { "id": "commxpy", "url": "https://www.teclamairme.com/#g-commxpy", "name": "compatible observables commute", "symbol": "[x̂, p̂y]", "symbol_markup": "[x̂, p̂_{y}]", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "0: position along x and momentum along y can be known together, because their operators commute.", "more": "The canonical relations are [x̂_{i}, p̂_{j}] = iℏδ_{ij}I. Different Cartesian components therefore commute: [x̂, p̂_{y}] = 0. Born and Jordan’s 1925 work supplied the canonical matrix-mechanics relation. The value is the zero operator, handed home as the integer 0.", "cites": [ "r-15.20", "r-15.23", "r-15.9" ], "links_to_cards": [] }, { "id": "sigzplus", "url": "https://www.teclamairme.com/#g-sigzplus", "name": "the expected spin of a superposition", "symbol": "⟨+|σz|+⟩", "symbol_markup": "⟨+|σ_{z}|+⟩", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "0: measure σz on the equal superposition |+⟩ and you get +1 or −1 with equal chance, averaging to nothing.", "more": "The Pauli matrix σ_{z} sends |+⟩ = (|0⟩ + |1⟩)/√2 to |−⟩ = (|0⟩ − |1⟩)/√2. Since ⟨+|−⟩ = 0, the expectation ⟨+|σ_{z}|+⟩ vanishes; measuring σ_{z} gives +1 and −1 with equal probability. Exactly 0, an integer.", "cites": [ "r-15.10", "r-15.20" ], "links_to_cards": [ "sqrt2" ] }, { "id": "commxp", "url": "https://www.teclamairme.com/#g-commxp", "name": "the canonical commutator", "symbol": "[x̂, p̂]/(iℏ)", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "The identity: [x̂, p̂] = iℏ times the identity operator, and dividing by iℏ leaves that operator, which comes home as the number 1.", "more": "The canonical relation is [x̂, p̂] = iℏI, where I is the identity operator. Dividing by iℏ leaves I. No finite matrices can obey this relation, because the trace of every commutator is zero while tr(iℏI) is not. Under the chapter’s homecoming rule I contributes the integer 1.", "cites": [ "r-15.21", "r-15.9" ], "links_to_cards": [] }, { "id": "trsx2", "url": "https://www.teclamairme.com/#g-trsx2", "name": "the trace of a Pauli matrix squared", "symbol": "tr(σx²)", "symbol_markup": "tr(σ_{x}²)", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "2: every Pauli matrix squares to the identity, whose trace in two dimensions is 2.", "more": "Each Pauli matrix squares to the 2 × 2 identity, so tr(σ_{x}²) = tr(I) = 2. An integer.", "cites": [ "r-15.20", "r-15.7" ], "links_to_cards": [] }, { "id": "kennard", "url": "https://www.teclamairme.com/#g-kennard", "name": "the uncertainty bound", "symbol": "min ΔxΔp / ℏ", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "½: Kennard’s 1927 form of Heisenberg’s principle, ΔxΔp ≥ ℏ/2, with equality for a Gaussian wavepacket.", "more": "Heisenberg gave his qualitative uncertainty argument in 1927; later that year Kennard proved ΔxΔp ≥ ℏ/2, and Robertson generalized the inequality to arbitrary observables in 1929. Gaussian wavepackets attain equality. Dividing the minimum by ℏ gives 1/2, a rational.", "cites": [ "r-15.12" ], "links_to_cards": [ "hermitian" ] }, { "id": "tsirelson", "url": "https://www.teclamairme.com/#g-tsirelson", "name": "Tsirelson’s bound", "symbol": "2√2", "chapter": 15, "type": "alg", "badge": "Algebraic irrational", "quick": "For CHSH correlations, local hidden-variable models are bounded by 2 and quantum theory by 2√2.", "more": "The CHSH inequality bounds local hidden-variable correlations by 2. Quantum theory can reach 2√2 and cannot exceed that value; experiments have approached the quantum bound. Bell-test experiments, including loophole-free tests reported in 2015, underlie the 2022 Nobel Prize to Clauser, Aspect, and Zeilinger. The number 2√2 is algebraic, a root of x² − 8.", "cites": [ "r-15.13", "r-15.14" ], "links_to_cards": [ "chshineq" ] }, { "id": "diracnot", "url": "https://www.teclamairme.com/#g-diracnot", "name": "Dirac’s bra-ket notation", "symbol": "⟨a|b⟩", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "Dirac’s 1939 notation for an inner product: the bra ⟨a| and the ket |b⟩ close into ⟨a|b⟩.", "more": "Dirac’s 1939 paper A new notation for quantum mechanics introduced the notation this chapter is written in: ⟨a|b⟩ denotes an inner product, split into the bra ⟨a| and the ket |b⟩. Orthogonal vectors have inner product zero, and the computational basis states of a qubit are orthonormal, so ⟨0|1⟩ = 0 and ⟨0|0⟩ = 1. Every overlap term in this chapter—the quantum numerator’s ⟨0|1⟩ and ⟨+|σz|+⟩, the quantum denominator’s ⟨0|0⟩, ⟨+|0⟩, and ⟨1|σx|0⟩, and the Fourier numerator’s ⟨sin|cos⟩ — is written in it. A notation, not a number.", "cites": [ "r-15.3" ], "links_to_cards": [ "braket", "fourier", "norm00", "plus0", "sigx", "sigzplus" ] }, { "id": "exclusion", "url": "https://www.teclamairme.com/#g-exclusion", "name": "Pauli’s exclusion principle", "symbol": "", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "No two identical fermions can occupy the same quantum state: Wolfgang Pauli, 1925.", "more": "Pauli formulated the exclusion principle in 1925: two identical fermions cannot occupy the same quantum state. It is a statement about the half-integer-spin particles, the fermions, and not about the integer-spin bosons. The chapter’s Grassmann variables carry the same antisymmetric pattern, the minus sign that appears when two of them are swapped. A principle, not a number.", "cites": [ "r-15.8" ], "links_to_cards": [ "schrod" ] }, { "id": "chshineq", "url": "https://www.teclamairme.com/#g-chshineq", "name": "the CHSH inequality", "symbol": "S ≤ 2", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "The CHSH inequality bounds a combination of four correlation measurements by 2 for any local hidden-variable theory; quantum mechanics can reach 2√2.", "more": "Named for Clauser, Horne, Shimony, and Holt, who published it in 1969, the CHSH inequality combines four correlations measured between two separated observers, each choosing between two possible settings, into a single number S. Any theory of local hidden variables forces |S| ≤ 2; entangled quantum states can reach |S| = 2√2, the value fixed by Tsirelson’s bound, and experiments have approached that quantum limit. Violating the inequality is the standard modern form of a Bell test.", "cites": [ "r-15.25" ], "links_to_cards": [ "tsirelson" ] }, { "id": "fourier", "url": "https://www.teclamairme.com/#g-fourier", "name": "sine and cosine are orthogonal", "symbol": "⟨sin|cos⟩", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "∫₀^(2π) sin x cos x dx = 0: the inner product that makes Fourier series work.", "more": "Fourier presented his heat memoir in 1807 and published the Théorie analytique de la chaleur in 1822. Fourier coefficients rest on the orthogonality of sines and cosines over a full period; in particular ∫₀^{2π} sin x cos x dx = 0. Read as an L² inner product, this is ⟨sin|cos⟩ = 0. Exactly 0, an integer.", "cites": [ "r-15.15" ], "links_to_cards": [] }, { "id": "plancherel", "url": "https://www.teclamairme.com/#g-plancherel", "name": "Plancherel’s identity", "symbol": "‖f̂‖22−‖f‖22", "symbol_markup": "‖f̂‖_{2}^{2}−‖f‖_{2}^{2}", "chapter": 15, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the Fourier transform preserves L² energy when normalized with 2π in the exponent.", "more": "With f̂(ξ) = ∫f(x)e^{−2πixξ}dx, Plancherel’s theorem makes the Fourier transform unitary on L², so ‖f̂‖_{2} = ‖f‖_{2} and the displayed difference of squared norms is 0. Plancherel established the theorem in 1910. The result is an integer zero after evaluation.", "cites": [ "r-15.16" ], "links_to_cards": [ "fourierT" ] }, { "id": "gibbs", "url": "https://www.teclamairme.com/#g-gibbs", "name": "the Gibbs overshoot", "symbol": "Si(π)/π − ½", "chapter": 15, "type": "open", "badge": "Status open", "quick": "About 0.0895: the limiting Fourier overshoot at a jump is about nine percent of the jump; the spike narrows but does not shrink in height.", "more": "At a jump, Fourier partial sums have a limiting overshoot of Si(π)/π − 1/2 ≈ 0.089489872; increasing the number of terms narrows the spike without reducing its limiting relative height. Wilbraham described the phenomenon in 1848 and Gibbs reported it in 1899. No rational or irrational classification appears in the standard references checked for this chapter, so the piece leaves its arithmetic status open.", "cites": [ "r-15.17" ], "links_to_cards": [] }, { "id": "hilbert", "url": "https://www.teclamairme.com/#g-hilbert", "name": "Hilbert space", "symbol": "ℋ", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "A vector space with an inner product, complete in the distance it defines: the arena of quantum mechanics and of Fourier analysis.", "more": "A Hilbert space is a complete inner-product space; l² and L² are standard examples. When von Neumann set down his axioms he required infinitely many dimensions on purpose and said so by name, because that choice made his space unique, and uniqueness is what let him show that matrix mechanics and wave mechanics were one theory wearing two faces. His reason is not my reason, so this book takes the wider road and lets a finite space count: ℂ², the whole world of one qubit, is a Hilbert space. Hilbert’s integral-equation work around 1909 established the basis for the infinite-dimensional space that bears his name, and von Neumann made abstract Hilbert space the rigorous setting for quantum mechanics. A structure, not a number.", "cites": [ "r-15.1", "r-15.2", "r-15.24" ], "links_to_cards": [] }, { "id": "schrod", "url": "https://www.teclamairme.com/#g-schrod", "name": "the Schrödinger equation", "symbol": "iℏ ∂|ψ⟩/∂t = Ĥ|ψ⟩", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "How a quantum state changes in time, Erwin Schrödinger, 1926: the wave equation that the states of this chapter obey.", "more": "Schrödinger published his wave equation in 1926, initially in the time-independent eigenvalue form Ĥψ = Eψ, and recognized the equivalence of wave and matrix mechanics. The Hamiltonian is the self-adjoint operator representing energy. An equation, not a number.", "cites": [ "r-15.11" ], "links_to_cards": [] }, { "id": "hermitian", "url": "https://www.teclamairme.com/#g-hermitian", "name": "a Hermitian operator", "symbol": " = †", "symbol_markup": " = Â^{†}", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "An operator equal to its own conjugate transpose; its eigenvalues are real, which is why observables are Hermitian and measurements give real numbers.", "more": "A self-adjoint operator has real spectrum, and eigenvectors belonging to distinct eigenvalues are orthogonal. Quantum observables are represented by self-adjoint operators, with spectral values as possible measurement outcomes. In finite dimensions this becomes unitary diagonalization; in infinite dimensions the spectral theorem uses a resolution of the identity. A property, not a number.", "cites": [ "r-15.11" ], "links_to_cards": [ "reals", "spectralthm" ] }, { "id": "fourierT", "url": "https://www.teclamairme.com/#g-fourierT", "name": "the Fourier transform", "symbol": "f̂(ξ) = ∫ f(x) e−2πixξ dx", "symbol_markup": "f̂(ξ) = ∫ f(x) e^{−2πixξ} dx", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "A signal as a sum of waves, named for Fourier (his book of 1822), and the reason position and momentum are a conjugate pair.", "more": "The Fourier transform writes a function in terms of complex exponentials. It extends to a unitary map on L² by Plancherel’s theorem, carries position-space wavefunctions to momentum space, and sends Gaussians to Gaussians. Fourier’s 1822 book grew from his work on trigonometric series. An operator, not a number.", "cites": [ "r-15.15", "r-15.16", "r-15.19" ], "links_to_cards": [ "plancherel" ] }, { "id": "l2space", "url": "https://www.teclamairme.com/#g-l2space", "name": "the space of square-integrable functions", "symbol": "L²(ℝ)", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "Functions with a finite total ∫|f|²: the Hilbert space where wavefunctions live; its one-period version, from 0 to 2π, is where sine and cosine are orthogonal.", "more": "L² consists of measurable functions with finite ∫|f|², identified when they differ only on a set of measure zero; it is a complete inner-product space. Riesz and Fischer’s independent 1907 results connect L² Fourier coefficients with square-summable sequences. The Fourier numerator’s ⟨sin|cos⟩ is an inner product in the same kind of space, taken over one period from 0 to 2π. A space, not a number.", "cites": [ "r-15.1", "r-15.18" ], "links_to_cards": [ "fourier", "hilbert" ] }, { "id": "born", "url": "https://www.teclamairme.com/#g-born", "name": "the Born rule", "symbol": "P = |⟨φ|ψ⟩|²", "chapter": 15, "type": "fn", "badge": "Function or operator", "quick": "The probability of finding state ψ in state φ is the squared overlap; Max Born, 1926, and the reason ⟨0|1⟩ = 0 means perfect distinguishability.", "more": "The Born rule assigns an outcome probability from the squared modulus of its amplitude. Born proposed the statistical interpretation in 1926 and received half of the 1954 Nobel Prize for that work. Thus ⟨0|1⟩ = 0 gives probability zero, while |⟨+|0⟩|² = 1/2 gives a fair quantum coin. A rule, not a number.", "cites": [ "r-15.5" ], "links_to_cards": [ "braket", "plus0" ] }, { "id": "contour", "url": "https://www.teclamairme.com/#g-contour", "name": "contour integral around the unit circle", "symbol": "∮|z|=1", "symbol_markup": "∮_{|z|=1}", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "Integrate once counterclockwise around the circle of radius one.", "more": "The circle symbol on the integral sign means a path integral around a closed curve, here the circle of radius one centered at the origin, traversed once counterclockwise. Cauchy defined integrals along paths in the complex plane in his memoir of 1825. An operator, so uncolored.", "cites": [ "r-16.1" ], "links_to_cards": [ "cplane" ] }, { "id": "cauchy", "url": "https://www.teclamairme.com/#g-cauchy", "name": "Cauchy’s integral theorem", "symbol": "∮ ez dz", "symbol_markup": "∮ e^{z} dz", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "The integral of any entire function around any closed loop is 0 (Cauchy: path independence 1825, arbitrary closed loops 1846).", "more": "Cauchy’s integral theorem says that the integral of a holomorphic function around a closed loop is 0 when the loop and its interior lie in the function’s domain. Cauchy proved path-independence in his 1825 memoir Sur les intégrales définies, prises entre des limites imaginaires, but did not write the theorem over an arbitrary closed curve until the 1846 note Sur les intégrales qui s’étendent à tous les points d’une courbe fermée; the closed-loop form follows from the 1825 result as mathematics, though not as history: Cauchy proved it afresh by what is now called Green’s formula, and did not recall the 1825 memoir until 1851. Goursat’s 1884 and 1900 papers removed Cauchy’s extra continuity assumption on the derivative. Since e^z is entire, its integral around the unit circle is exactly 0, an integer.", "cites": [ "r-16.1", "r-16.2" ], "links_to_cards": [ "greenthm", "holomorphic" ] }, { "id": "dzz", "url": "https://www.teclamairme.com/#g-dzz", "name": "the integral that refuses to vanish", "symbol": "∮ dz/z", "chapter": 16, "type": "obj", "badge": "Non-real object", "quick": "Equal to 2πi once counterclockwise around the origin, the source of every residue.", "more": "∮ dz/z around the counterclockwise unit circle equals 2πi, because 1/z has a pole at the origin inside the loop and log z gains 2πi on each trip around it. It is the one integral every residue traces back to, and its value is a non-real complex number, which is why it carries the color of a non-real object. With the residue, the winding number, and the root count beside it, this chapter’s denominator is 2πi + 4, whose real part is 4, so it is certainly not zero.", "cites": [ "r-16.3" ], "links_to_cards": [ "argprin", "res", "winding" ] }, { "id": "res", "url": "https://www.teclamairme.com/#g-res", "name": "residue at an essential singularity", "symbol": "Res0 e1/z", "symbol_markup": "Res_{0} e^{1/z}", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "Equal to 1, read off the Laurent series of e^(1/z).", "more": "The residue of e^(1/z) at 0 is the coefficient of 1/z in its Laurent expansion 1 + 1/z + 1/(2!z²) + …, which is 1. The point is an essential singularity; Great Picard’s theorem says that, with at most one exception, every complex value occurs infinitely often in every neighborhood of such a point. Here the omitted value is 0, yet the residue is a plain integer.", "cites": [ "r-16.4" ], "links_to_cards": [ "laurent", "picard" ] }, { "id": "wirtinger", "url": "https://www.teclamairme.com/#g-wirtinger", "name": "the Cauchy–Riemann equations in one line", "symbol": "∂f/∂z̄", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0 for every holomorphic f: this one equation is the Cauchy–Riemann test for complex differentiability.", "more": "The Cauchy–Riemann condition for complex differentiability is the pair of real equations u_{x} = v_{y} and u_{y} = −v_{x}, where f = u + iv. Wirtinger’s operators, introduced in a ~1926 paper, fold them into ∂f/∂z̄ = ½(∂_{x} + i∂_{y})f = 0. The tilde is there because Range’s history calls it the 1926 paper while Springer’s own record dates the printed volume, Math. Ann. 97, 357–375, to 1927; both are cited at 16.5. It is the condition behind the chapter’s holomorphic calculations. Exactly 0, an integer.", "cites": [ "r-16.5" ], "links_to_cards": [ "holomorphic" ] }, { "id": "resentire", "url": "https://www.teclamairme.com/#g-resentire", "name": "the residue of an entire function", "symbol": "Res0 ez", "symbol_markup": "Res_{0} e^{z}", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: a function with no singularity has no 1/z term in its expansion, so its residue anywhere is 0.", "more": "The residue at a point is the coefficient of 1/(z − a) in the Laurent series there; for a function holomorphic at a the series has no negative powers, so the residue is 0. That is the residue theorem’s way of restating Cauchy’s theorem: the integral around a loop is 2πi times the residues inside, and an entire function has none. Exactly 0, an integer.", "cites": [ "r-16.4" ], "links_to_cards": [ "cauchy", "holomorphic", "laurent", "residuethm" ] }, { "id": "winding", "url": "https://www.teclamairme.com/#g-winding", "name": "the winding number of the unit circle about the origin", "symbol": "n(γ, 0)", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "Equal to 1: the counterclockwise circle goes round the origin once, which is exactly why ∮ dz/z is 2πi and not 0.", "more": "The winding number n(γ, a) = (1/2πi)∮_γ dz/(z − a) is an integer counting the signed turns of a closed curve γ around a; for the counterclockwise unit circle about the origin it is 1. The residue theorem weights each enclosed residue by this count, so deformations that do not cross a singularity leave the integral unchanged. An integer.", "cites": [ "r-16.3", "r-16.6" ], "links_to_cards": [ "dzz", "residuethm" ] }, { "id": "argprin", "url": "https://www.teclamairme.com/#g-argprin", "name": "the argument principle, counted", "symbol": "#{z : z2 + 1 = 0, |z| < 2}", "symbol_markup": "#{z : z^{2} + 1 = 0, |z| < 2}", "chapter": 16, "type": "int", "badge": "Integer or rational", "quick": "Equal to 2: (1/2πi)∮ f′/f dz counterclockwise around |z| = 2 counts the zeros of f = z² + 1 inside, which are i and −i.", "more": "For f meromorphic inside a positively oriented loop and nonzero on it, the argument principle says that (1/2πi)∮ f′(z)/f(z) dz equals the number of zeros minus the number of poles inside, counted with multiplicity. The polynomial z² + 1 has the two roots ±i and no poles, all inside |z| = 2, so the count is 2. Root-counting by winding number also gives a proof of the fundamental theorem of algebra. An integer.", "cites": [ "r-16.7" ], "links_to_cards": [ "fta" ] }, { "id": "holomorphic", "url": "https://www.teclamairme.com/#g-holomorphic", "name": "holomorphic and meromorphic", "symbol": "f ∈ 𝒪(U)", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "Holomorphic: complex-differentiable at every point of an open set; meromorphic: holomorphic except for poles. Holomorphic and complex analytic mean the same thing.", "more": "A holomorphic function is complex-differentiable on an open set U, which by Cauchy’s integral formula makes it infinitely differentiable and locally equal to its own Taylor series: holomorphic and complex analytic are equivalent. A meromorphic function is holomorphic except at isolated poles. A property, uncolored.", "cites": [ "r-16.8" ], "links_to_cards": [ "cauchyformula" ] }, { "id": "laurent", "url": "https://www.teclamairme.com/#g-laurent", "name": "the Laurent series", "symbol": "Σn∈ℤ an(z − a)n", "symbol_markup": "Σ_{n∈ℤ} a_{n}(z − a)^{n}", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "A power series allowed negative powers, valid on an annulus; the coefficient a₋₁ is the residue.", "more": "Pierre Alphonse Laurent submitted the expansion to the Academy of Sciences in 1843; Weierstrass had proved the result in 1841 but did not publish it then. On any annulus where f is holomorphic, f has a two-sided power series in z − a, and the coefficient of 1/(z − a) is its residue. A tool, uncolored.", "cites": [ "r-16.4", "r-16.9" ], "links_to_cards": [ "holomorphic" ] }, { "id": "residuethm", "url": "https://www.teclamairme.com/#g-residuethm", "name": "the residue theorem", "symbol": "∮γ f dz = 2πi Σ Res f", "symbol_markup": "∮_{γ} f dz = 2πi Σ Res f", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "Around a positively oriented simple closed curve, with the function holomorphic inside except at finitely many isolated singularities and none on the curve, the integral is 2πi times the sum of the residues inside: Cauchy’s theorem with the singularities put back.", "more": "Cauchy began the calculus of residues in 1826, in Sur un nouveau genre de calcul analogue au calcul infinitésimal, which is where the word residue enters mathematics; his Turin memoirs of 1831–32 were summarized in 1837 and republished in 1840–41. The theorem turns contour integrals into algebraic sums and evaluates real integrals such as ∫ dx/(1 + x²) = π over the whole real line. Every residue traces back to ∮ dz/z = 2πi in this chapter’s denominator. A theorem, uncolored.", "cites": [ "r-16.10", "r-16.3" ], "links_to_cards": [ "cauchy", "dzz", "holomorphic" ] }, { "id": "liouville", "url": "https://www.teclamairme.com/#g-liouville", "name": "Liouville’s theorem", "symbol": "bounded and entire ⇒ constant", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "A bounded holomorphic function on the whole plane is constant; Cauchy apparently first published it in 1844, Liouville got the name.", "more": "Cauchy apparently first published the bounded-entire theorem in 1844; Liouville presented it in lectures in 1847, which is how his name became attached to it. It gives a short proof of the fundamental theorem of algebra: if a nonconstant polynomial p had no root, 1/p would be bounded and entire. Chapter ᚔ(I) counts those roots. A theorem, uncolored.", "cites": [ "r-16.11" ], "links_to_cards": [ "fta", "holomorphic" ] }, { "id": "riemannsurf", "url": "https://www.teclamairme.com/#g-riemannsurf", "name": "a Riemann surface", "symbol": "Riemann surface", "chapter": 16, "type": "obj", "badge": "Non-real object", "quick": "The natural home of a multivalued function such as √z or log z: sheets glued along cuts so that the function becomes single-valued.", "more": "Riemann introduced them in his 1851 thesis to make sense of functions like √z, which needs two sheets, and log z, which needs infinitely many; they are one-dimensional complex manifolds, and the compact ones are classified topologically by genus, which ties this chapter to the topology of Chapter ᚊ(Q) and to the elliptic curves of Chapter ᚒ(U), which are Riemann surfaces of genus one. An object, so colored as a non-real object.", "cites": [ "r-16.12" ], "links_to_cards": [ "weierstrass" ] }, { "id": "mobiustrans", "url": "https://www.teclamairme.com/#g-mobiustrans", "name": "a Möbius transformation", "symbol": "z ↦ (az + b)/(cz + d)", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "The invertible holomorphic maps of the Riemann sphere that send circles to circles; the group of them is PSL(2, ℂ).", "more": "These fractional linear maps are exactly the holomorphic automorphisms of the Riemann sphere. They send generalized circles—ordinary circles or lines—to generalized circles and form PSL(2, ℂ), which is isomorphic to the proper orthochronous Lorentz group of special relativity. A map, uncolored.", "cites": [ "r-16.13" ], "links_to_cards": [ "holomorphic", "riemannsphere" ] }, { "id": "conformal", "url": "https://www.teclamairme.com/#g-conformal", "name": "conformal maps", "symbol": "f′(z) ≠ 0 ⇒ angles preserved", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "Wherever a holomorphic function has nonzero derivative it preserves angles; Riemann’s mapping theorem says any simply connected region but the plane can be mapped conformally and one-to-one onto the disc.", "more": "A holomorphic map with f′(z) ≠ 0 multiplies infinitesimal figures by the complex number f′(z), a rotation and a scaling, so angles survive. Riemann formulated his mapping theorem in 1851; later work supplied rigorous proofs. Mercator’s 1569 world map is a historic conformal projection. A property, uncolored.", "cites": [ "r-16.14" ], "links_to_cards": [ "holomorphic", "infinitesimal" ] }, { "id": "cauchyformula", "url": "https://www.teclamairme.com/#g-cauchyformula", "name": "Cauchy’s integral formula", "symbol": "f(z) = (1/2πi)∮ f(ζ)/(ζ − z) dζ", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "A holomorphic function’s values inside a loop are fixed by its values on the loop, and the fixing is an integral.", "more": "If f is holomorphic on a domain whose boundary is finitely many non-intersecting rectifiable curves, traversed in the positive sense, then f(z) = (1/2πi)∮ f(ζ)/(ζ − z) dζ at every interior point z. It is the reason holomorphic functions are as rigid as this chapter says: differentiate under the integral sign and one complex derivative becomes infinitely many, and expanding 1/(ζ − z) as a geometric series turns the same integral into the function’s own Taylor series. Cauchy’s theorem, from which the formula is derived, says that with nothing singular inside, a holomorphic function’s integral around the loop is 0, which is this chapter’s numerator. A theorem, uncolored.", "cites": [ "r-16.16" ], "links_to_cards": [ "cauchy", "holomorphic" ] }, { "id": "picard", "url": "https://www.teclamairme.com/#g-picard", "name": "Great Picard’s theorem", "symbol": "essential singularity ⇒ every value but one", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "In every neighborhood of an essential singularity a function takes every finite complex value, with at most one possible exception, infinitely often.", "more": "If f is holomorphic in a punctured neighborhood of an essential singular point, then in every neighborhood of that point, however small, each finite complex value with at most one possible exception is taken infinitely often. For e^(1/z) at the origin the one exception is 0, because the exponential is never 0; every other complex number is hit infinitely often inside any disc round the origin. The chapter puts it beside a residue that is nevertheless a plain 1. A theorem, uncolored.", "cites": [ "r-16.4" ], "links_to_cards": [ "holomorphic", "res" ] }, { "id": "greenthm", "url": "https://www.teclamairme.com/#g-greenthm", "name": "Green’s theorem", "symbol": "∮(Ldx+Mdy)=∬(∂M/∂x−∂L/∂y)dA", "chapter": 16, "type": "fn", "badge": "Function or operator", "quick": "Green’s theorem converts a line integral around a closed curve in the plane into a double integral over the region the curve encloses, relating circulation on the boundary to a derivative inside.", "more": "Named for George Green, who stated a version of it in an 1828 essay on electricity and magnetism, though a proof came only with Riemann in 1851, it says ∮(L dx + M dy) = ∬(∂M/∂x − ∂L/∂y) dA around a positively oriented simple closed curve. It is the two-dimensional case of the more general Stokes’ theorem, and in complex analysis it underlies Cauchy’s integral theorem, the statement that a holomorphic function’s integral around a closed loop vanishes. Applied to L = −y/2, M = x/2 it gives a way to compute a region’s area purely from its boundary.", "cites": [ "r-16.18" ], "links_to_cards": [ "cauchy", "holomorphic", "stokes" ] }, { "id": "li", "url": "https://www.teclamairme.com/#g-li", "name": "logarithmic integral", "symbol": "li", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The principal-value integral ∫ dt/ln t, the function in the prime number theorem, π(x) ~ li(x).", "more": "The logarithmic integral is the principal-value integral li(x) = PV∫₀ˣ dt/ln t across its singularity at t = 1. The prime number theorem can be written π(x) ~ li(x)—the form DLMF sharpens at §27.12 into a bound on |π(x) − li(x)|. A function, not a number, so uncolored.", "cites": [ "r-17.4" ], "links_to_cards": [ "pnt", "primecount" ] }, { "id": "mu", "url": "https://www.teclamairme.com/#g-mu", "name": "Ramanujan–Soldner constant", "symbol": "μ", "chapter": 17, "type": "open", "badge": "Status open", "quick": "The unique positive zero of li, about 1.45137; no cited source settles its irrationality.", "more": "The Ramanujan–Soldner constant, μ ≈ 1.4513692349, is the unique positive zero of li. Soldner tabulated the logarithmic integral in 1809, and Ramanujan later computed the constant. No source cited here settles whether μ is irrational, so it remains status open; li(μ) = 0 is its definition, which is why it can be a numerator.", "cites": [ "r-17.5" ], "links_to_cards": [ "li" ] }, { "id": "gammahalf", "url": "https://www.teclamairme.com/#g-gammahalf", "name": "gamma of one half", "symbol": "Γ(½)", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "Equal to √π.", "more": "Γ(½) = √π ≈ 1.7725, and the recurrence Γ(z + 1) = zΓ(z) makes Γ(½) the value (−½)! while (½)! = Γ(3/2) = √π/2. It is transcendental because π is transcendental.", "cites": [ "r-17.12", "r-17.3" ], "links_to_cards": [ "pi" ] }, { "id": "W1", "url": "https://www.teclamairme.com/#g-W1", "name": "the omega constant", "symbol": "W(1)", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "The solution of x·e^x = 1, about 0.5671, from Lambert’s W function; transcendental.", "more": "The omega constant Ω = W(1) ≈ 0.5671433 is the positive solution of Ωe^Ω = 1, where W is the multivalued inverse of w ↦ we^w. Corless, Gonnet, Hare, Jeffrey, and Knuth standardized its branches and the name W in 1996. Lindemann–Weierstrass proves Ω transcendental: if Ω were nonzero algebraic, e^Ω = 1/Ω could not also be algebraic. This Ω is Lambert’s, not Chaitin’s halting probability of Chapter ᚃ(F).", "cites": [ "r-17.11", "r-17.12" ], "links_to_cards": [ "lindweier", "omega" ] }, { "id": "j01", "url": "https://www.teclamairme.com/#g-j01", "name": "first zero of the Bessel function J₀", "symbol": "j0,1", "symbol_markup": "j_{0,1}", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "About 2.4048, the note a drumhead cannot help playing; transcendental by Siegel’s theorem of 1929.", "more": "The first positive zero of J₀ is j₀,₁ ≈ 2.4048256. Zeros of J₀ set the frequencies of a circular drumhead’s axisymmetric modes, and this first zero gives the fundamental one. Siegel’s 1929 theorem implies that every nonzero zero of J₀ is transcendental.", "cites": [ "r-17.13" ], "links_to_cards": [] }, { "id": "erf1", "url": "https://www.teclamairme.com/#g-erf1", "name": "error function at one", "symbol": "erf(1)", "chapter": 17, "type": "open", "badge": "Status open", "quick": "About 0.8427, the Gaussian probability within √2 standard deviations; no cited source settles its irrationality.", "more": "erf(1) ≈ 0.8427008 equals the probability that a standard normal variable lies within √2 standard deviations of its mean, because erf(x) = (2/√π)∫₀ˣ e^(−t²)dt. A 2023 Siegel–Shidlovsky argument proves the unnormalized integral ∫₀¹e^(−t²)dt transcendental, but does not determine the arithmetic nature of the ratio erf(1); no cited source settles its irrationality, so it remains status open.", "cites": [ "r-17.14", "r-17.23" ], "links_to_cards": [ "erffn", "normal", "sqrt2" ] }, { "id": "ai0", "url": "https://www.teclamairme.com/#g-ai0", "name": "Airy function at zero", "symbol": "Ai(0)", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "About 0.3550, equal to 1/(3^(2/3) Γ(2/3)), transcendental via Chudnovsky’s result on Γ(1/3).", "more": "Ai(0) = 1/(3^(2/3)·Γ(⅔)) ≈ 0.3550280. Airy functions solve the Airy differential equation and occur in linear-potential quantum problems. Chudnovsky’s algebraic-independence work in the 1970s makes π and Γ(⅓) algebraically independent; together with Γ(⅓)Γ(⅔) = 2π/√3, this proves Ai(0) transcendental.", "cites": [ "r-17.15", "r-17.24", "r-17.27" ], "links_to_cards": [ "airyfn", "sqrt3" ] }, { "id": "recipgamma", "url": "https://www.teclamairme.com/#g-recipgamma", "name": "the reciprocal gamma function at a pole", "symbol": "1/Γ(0)", "chapter": 17, "type": "int", "badge": "Integer or rational", "quick": "Γ has a pole at 0, so its reciprocal, which extends smoothly across the poles, is zero there.", "more": "The gamma function has simple poles at 0, −1, −2, …, while its reciprocal 1/Γ is entire and has a simple zero at each of those points. Thus (1/Γ)(0) = 0 exactly, hence an integer; the displayed 1/Γ(0) is standard shorthand for that entire reciprocal-gamma value.", "cites": [ "r-17.6" ], "links_to_cards": [ "gammafn" ] }, { "id": "sincpi", "url": "https://www.teclamairme.com/#g-sincpi", "name": "the sinc function at pi", "symbol": "sinc(π)", "chapter": 17, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: unnormalized sinc x = sin x / x, and the sine vanishes at π while the denominator does not.", "more": "For the unnormalized convention sinc(x) = sin x/x, continuity gives sinc(0) = 1 and the nonzero integer multiples of π are its zeros. The sinc kernel also reconstructs a band-limited signal from sufficiently dense samples. Thus sinc(π) = 0 exactly, an integer.", "cites": [ "r-17.7" ], "links_to_cards": [] }, { "id": "j1zero", "url": "https://www.teclamairme.com/#g-j1zero", "name": "the Bessel function of order one at zero", "symbol": "J1(0)", "symbol_markup": "J_{1}(0)", "chapter": 17, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: every Bessel function Jₙ with n ≥ 1 starts from nothing, while J₀(0) = 1.", "more": "For integer n ≥ 0, the defining series of Jₙ begins with (x/2)ⁿ/n!. Consequently J₀(0) = 1 and Jₙ(0) = 0 for every n ≥ 1, including J₁(0). Exactly 0, an integer.", "cites": [ "r-17.8" ], "links_to_cards": [ "besselfn" ] }, { "id": "betahalf", "url": "https://www.teclamairme.com/#g-betahalf", "name": "the Beta function at one half, one half", "symbol": "B(½, ½)", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "Equal to π: the Beta function is Γ(a)Γ(b)/Γ(a + b), and here the two Γ(½) give √π·√π.", "more": "The beta integral satisfies B(a, b) = Γ(a)Γ(b)/Γ(a+b). Therefore B(½, ½) = Γ(½)²/Γ(1) = π, equivalently the integral of 1/√(t(1−t)) over [0, 1]. It is transcendental because π is.", "cites": [ "r-17.12", "r-17.3", "r-17.9" ], "links_to_cards": [ "betafn", "gammahalf" ] }, { "id": "digamma1", "url": "https://www.teclamairme.com/#g-digamma1", "name": "the digamma function at one", "symbol": "|ψ(1)|", "chapter": 17, "type": "open", "badge": "Status open", "quick": "ψ(1) = −γ: the logarithmic derivative of Γ at 1 is minus the Euler–Mascheroni constant, whose irrationality is unknown.", "more": "The digamma function is ψ(x) = Γ′(x)/Γ(x), and its special values satisfy ψ(1) = −γ and ψ(n+1) = Hₙ − γ. Current literature still does not establish whether the Euler–Mascheroni constant γ is irrational, so |ψ(1)| remains status open. It is the same γ as Chapter ᚃ(F)’s.", "cites": [ "r-17.10", "r-17.23" ], "links_to_cards": [ "digammafn", "gamma" ] }, { "id": "wm1", "url": "https://www.teclamairme.com/#g-wm1", "name": "the other real branch of Lambert W", "symbol": "|W−1(−1/e)|", "symbol_markup": "|W_{−1}(−1/e)|", "chapter": 17, "type": "int", "badge": "Integer or rational", "quick": "Equal to 1: the two real branches of W meet at the branch point (−1/e, −1), so W₋₁(−1/e) = −1.", "more": "Lambert W is multivalued because w ↦ we^w is not one-to-one. On the real interval −1/e < x < 0 it has the branches W₀ and W₋₁, and both meet at W(−1/e) = −1. Therefore |W₋₁(−1/e)| = 1, an integer.", "cites": [ "r-17.11" ], "links_to_cards": [] }, { "id": "K0", "url": "https://www.teclamairme.com/#g-K0", "name": "the complete elliptic integral of the first kind at zero", "symbol": "K(0)", "chapter": 17, "type": "tr", "badge": "Proven transcendental", "quick": "Equal to π/2: with modulus zero the integral ∫₀^(π/2) dθ/√(1 − k² sin²θ) is just a quarter turn.", "more": "The complete elliptic integral of the first kind is K(k) = ∫₀^(π/2) dθ/√(1−k²sin²θ); it gives the finite-amplitude pendulum period. At k = 0 the integrand is 1, so K(0) = π/2, transcendental because π is. The arc length of an ellipse uses a different one, the elliptic integral of the second kind E.", "cites": [ "r-17.12", "r-17.16" ], "links_to_cards": [] }, { "id": "fresnelS", "url": "https://www.teclamairme.com/#g-fresnelS", "name": "the Fresnel sine integral at infinity", "symbol": "S(∞)", "chapter": 17, "type": "int", "badge": "Integer or rational", "quick": "Equal to ½: ∫₀^∞ sin(πt²/2) dt converges to ½, the point the Cornu spiral winds toward.", "more": "The Fresnel integrals are S(x) = ∫₀ˣ sin(πt²/2)dt and C(x) = ∫₀ˣ cos(πt²/2)dt. They parametrically draw the Cornu spiral used in diffraction, and both tend to ½ as x → ∞. Thus S(∞) is exactly the rational number ½.", "cites": [ "r-17.17" ], "links_to_cards": [ "cornu", "fresnelfn" ] }, { "id": "hypergeom", "url": "https://www.teclamairme.com/#g-hypergeom", "name": "the Gauss hypergeometric function", "symbol": "2F1(a, b; c; z)", "symbol_markup": "_{2}F_{1}(a, b; c; z)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The series Σ (a)ₙ(b)ₙ/(c)ₙ · zⁿ/n!, with logarithm, arcsine, and polynomial special cases.", "more": "The Gauss hypergeometric function is defined for |z| < 1 by ₂F₁(a, b;c;z) = Σₙ₌₀∞ (a)ₙ(b)ₙzⁿ/((c)ₙn!), with analytic continuation elsewhere. Parameter choices give elementary functions including logarithms and inverse trigonometric functions, while negative integer parameters produce polynomial cases. A function, uncolored.", "cites": [ "r-17.18" ], "links_to_cards": [ "analcont" ] }, { "id": "polylog", "url": "https://www.teclamairme.com/#g-polylog", "name": "the polylogarithm", "symbol": "Lis(z)", "symbol_markup": "Li_{s}(z)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "Σ zⁿ/nˢ: Li₁(z) = −ln(1 − z), Liₛ(1) = ζ(s) where the series converges, and Li₂ is the dilogarithm.", "more": "The polylogarithm Liₛ(z) = Σₙ₌₁∞ zⁿ/nˢ generalizes the logarithm through Li₁(z) = −ln(1−z) and specializes to the zeta function through Liₛ(1) = ζ(s), where the defining series converges. Its dilogarithm value Li₂(½) = π²/12 − (ln 2)²/2 is a standard special case. A function, uncolored.", "cites": [ "r-17.19" ], "links_to_cards": [ "zeta" ] }, { "id": "thetafn", "url": "https://www.teclamairme.com/#g-thetafn", "name": "the Jacobi theta function", "symbol": "ϑ(z; τ)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "Σ exp(πin²τ + 2πinz): a building block of elliptic functions, and the function whose transformation law gives the functional equation of ζ.", "more": "Jacobi’s theta function ϑ₃(z;τ) = Σₙ∈ℤ exp(πin²τ + 2πinz) is related to elliptic functions and to the Riemann zeta function. At τ = i, ϑ₃(0;i) = Γ(¼)/(√2·π^(3/4)); by Γ(¼)Γ(¾) = π√2 this is exactly the same as π^(1/4)/Γ(¾), not a competing value. Its modular transformation supplies the theta identity used in the zeta functional equation. A function, uncolored.", "cites": [ "r-17.20" ], "links_to_cards": [ "funceq", "gamma14", "sqrt2", "zeta" ] }, { "id": "orthopoly", "url": "https://www.teclamairme.com/#g-orthopoly", "name": "the classical orthogonal polynomials", "symbol": "Pn(x), Hn(x), Ln(x)", "symbol_markup": "P_{n}(x), H_{n}(x), L_{n}(x)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "Legendre, Hermite, and Laguerre: polynomial families orthogonal under weights, used for spherical harmonics, the quantum oscillator, and the hydrogen atom.", "more": "Legendre, Hermite, and Laguerre polynomials are each orthogonal under their own weight. DLMF’s physical applications show Hermite polynomials in harmonic-oscillator eigenfunctions and generalized Laguerre polynomials in hydrogenic radial eigenfunctions; Legendre polynomials are tied to spherical harmonics. Functions, uncolored.", "cites": [ "r-17.21" ], "links_to_cards": [] }, { "id": "incgamma", "url": "https://www.teclamairme.com/#g-incgamma", "name": "the incomplete gamma function", "symbol": "Γ(s, x)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The gamma integral run from x to infinity instead of from 0, its lower companion γ(s, x) stopping at x; the error function and the chi-squared distribution are both cases of it.", "more": "The upper incomplete gamma function is Γ(s, x) = ∫ₓ∞ t^(s−1)e^(−t)dt; its lower companion γ(s, x) runs from 0 to x, and their sum is Γ(s). The identity γ(½, x²) = √π erf(x) connects it to the error function, while regularized incomplete-gamma functions give chi-square distribution functions. A function, uncolored.", "cites": [ "r-17.22" ], "links_to_cards": [ "erffn", "gammafn" ] }, { "id": "gammafn", "url": "https://www.teclamairme.com/#g-gammafn", "name": "the gamma function", "symbol": "Γ", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "Extends the factorial, Γ(n + 1) = n!: Euler’s integral ∫₀^∞ e^(−t) t^(z−1) dt where z has positive real part, continued to the whole plane, never zero, with poles at 0, −1, −2, ….", "more": "Euler’s integral, Γ(z) = ∫₀^∞ e^(−t) t^(z−1) dt for Re z > 0, extends by analytic continuation to a meromorphic function on the whole plane with no zeros and a simple pole at each nonpositive integer, the pole at −n having residue (−1)ⁿ/n!. It extends the factorial: Γ(1) = 1 and n! = Γ(n + 1). Euler introduced it in 1729 correspondence with Goldbach, and the symbol Γ is Legendre’s, from his volume of ~1811. Infinitely many smooth functions interpolate the factorial, so why this one? Bohr and Mollerup answered that in 1922: Γ is the only positive function on x > 0 with f(1) = 1 and f(x + 1) = x f(x) whose logarithm is convex. It is the family this chapter’s first brace is drawn from. A function, not a number, so uncolored.", "cites": [ "r-17.2", "r-17.26" ], "links_to_cards": [ "analcont", "permfact" ] }, { "id": "besselfn", "url": "https://www.teclamairme.com/#g-besselfn", "name": "the Bessel functions", "symbol": "Jₙ", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The solutions of z²w″ + zw′ + (z² − ν²)w = 0: the shapes a circular drumhead can hold.", "more": "Bessel’s equation arises when Laplace’s equation or the Helmholtz equation is separated in cylindrical coordinates, so its solutions are the radial shapes of round things that vibrate or conduct heat: a drumhead, a waveguide, a cylinder. Jₙ, the Bessel function of the first kind, is the solution that stays finite at the center, given by an ascending series in which the order ν sets the power of z it starts from. That one fact decides the two values this chapter uses: J₀(0) = 1, because the series for order zero begins with a constant term, while every Jₙ with n ≥ 1 begins with a power of z and so starts from nothing. The zeros of J₀ are the axisymmetric notes a circular drum can play, and the first of them, about 2.4048, is the fundamental. A family of functions, uncolored.", "cites": [ "r-17.28", "r-17.8" ], "links_to_cards": [ "j01" ] }, { "id": "airyfn", "url": "https://www.teclamairme.com/#g-airyfn", "name": "the Airy functions", "symbol": "Ai, Bi", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The two standard solutions of w″ = zw, the simplest equation whose behavior changes character at a point.", "more": "The Airy equation, w″ = zw, is about as simple as a differential equation can be while still being interesting, and what makes it interesting is that the sign of z flips the answer’s character: to the left of the origin the solutions oscillate, to the right one of them decays and the other grows. That is the mathematics of a turning point—a light ray at a caustic, a rainbow’s edge, a quantum particle meeting a linear potential it does not have the energy to climb. Ai and Bi are the two standard solutions, both entire, and Ai is the one that decays. Its value at the origin, 1/(3^(2/3)Γ(2/3)), is one of this chapter’s numbers. A family of functions, uncolored.", "cites": [ "r-17.15", "r-17.28" ], "links_to_cards": [ "ai0" ] }, { "id": "erffn", "url": "https://www.teclamairme.com/#g-erffn", "name": "the error function", "symbol": "erf", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "erf z = (2/√π) ∫₀^z e^(−t²) dt: the bell curve, integrated, and normalized so that it runs to 1.", "more": "The Gaussian e^(−t²) has no antiderivative in elementary terms, so the integral of it is given a name and becomes a function in its own right. The 2/√π in front is chosen so that erf runs from −1 to 1, which makes it the cumulative normal distribution in disguise. Its complement erfc z = 1 − erf z is the tail, the quantity a physicist actually wants when asking how much is left beyond a threshold. The chapter uses erf(1), the probability inside √2 standard deviations. A family of functions, uncolored.", "cites": [ "r-17.14", "r-17.28" ], "links_to_cards": [ "erf1", "sqrt2" ] }, { "id": "fresnelfn", "url": "https://www.teclamairme.com/#g-fresnelfn", "name": "the Fresnel integrals", "symbol": "C, S", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "C(z) = ∫₀^z cos(πt²/2) dt and S(z) = ∫₀^z sin(πt²/2) dt: the two coordinates of the spiral light draws at an edge.", "more": "Integrate a cosine and a sine whose argument grows as the square of the variable, and plot one against the other: the curve is the Cornu spiral, which winds tighter and tighter toward a point without ever arriving. That point is (½, ½), which is why the chapter’s Fresnel value is a half. The spiral is not a decoration: Cornu drew it in 1874 so that the diffraction of light past a straight edge could be read off a graph. A family of functions, uncolored.", "cites": [ "r-17.17", "r-17.28" ], "links_to_cards": [ "cornu" ] }, { "id": "pvalue", "url": "https://www.teclamairme.com/#g-pvalue", "name": "the Cauchy principal value", "symbol": "p.v. ∫", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "Cut a symmetric notch out around the bad point and shrink it. Two infinities that would each be undefined cancel, and the integral has a value after all.", "more": "An integral whose integrand blows up inside the interval may have no ordinary value: the piece to the left and the piece to the right can each be infinite. The Cauchy principal value removes a symmetric interval around the singularity, integrates what is left, and lets the interval shrink to nothing; because the two halves are cut away at exactly the same rate, their infinities cancel and a finite limit can survive. The symmetry is the whole content—cut asymmetrically and any answer at all can be arranged. It is what makes the logarithmic integral li(x) defined past t = 1, where ln t vanishes, and it is how the distribution p.v.(1/x) is defined. A convention of the framework, uncolored.", "cites": [ "r-17.29" ], "links_to_cards": [ "li" ] }, { "id": "lambertw", "url": "https://www.teclamairme.com/#g-lambertw", "name": "the Lambert W function", "symbol": "W(x)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The Lambert W function is the multivalued inverse of w ↦ w·e^w, used to solve equations where the unknown appears both inside and outside an exponential.", "more": "W(x) answers the question: for which w does w·e^w = x? Because w·e^w is not one-to-one on the reals, W is multivalued; its principal branch W₀ is real for x ≥ -1/e, and a second real branch W₋₁ exists for -1/e ≤ x < 0, meeting W₀ at the branch point (-1/e, -1). The function solves equations from self-referential exponentials, such as x^x = z, to problems in combinatorics and delay differential equations. Lambert studied the related equation in 1758 and Euler solved a special case in 1783; Corless and collaborators standardized the modern notation and theory in 1996.", "cites": [ "r-17.30" ], "links_to_cards": [ "reals" ] }, { "id": "betafn", "url": "https://www.teclamairme.com/#g-betafn", "name": "the beta function", "symbol": "B(a, b)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The beta function B(a, b) = Γ(a)Γ(b)/Γ(a+b) is the two-variable relative of the gamma function, equal to the integral of t^(a−1)(1−t)^(b−1) from 0 to 1.", "more": "Euler introduced the beta integral, ∫₀¹ t^(a−1)(1−t)^(b−1) dt, which equals Γ(a)Γ(b)/Γ(a+b), tying it directly to the gamma function. It appears throughout probability, where it normalizes the beta distribution, and in physics, in the Veneziano amplitude that helped launch string theory. The special value B(½, ½) = π, which has a card of its own, follows from Γ(½)² = π.", "cites": [ "r-17.31" ], "links_to_cards": [ "gammafn" ] }, { "id": "digammafn", "url": "https://www.teclamairme.com/#g-digammafn", "name": "the digamma function", "symbol": "ψ(x)", "chapter": 17, "type": "fn", "badge": "Function or operator", "quick": "The digamma function ψ(x) = Γ′(x)/Γ(x) is the logarithmic derivative of the gamma function.", "more": "ψ is defined as Γ′/Γ, and it obeys its own recurrence ψ(x+1) = ψ(x) + 1/x, mirroring the gamma function’s own functional equation. Its special values include ψ(1) = −γ, the negative of the Euler–Mascheroni constant, and ψ(n+1) = Hₙ − γ for positive integers n, tying it to the harmonic numbers. It is the first of the polygamma functions, the higher derivatives of log Γ, and it has an asymptotic expansion of its own, ψ(z) ∼ ln z − 1/(2z) − …, set beside Stirling’s series for the logarithm of the gamma function.", "cites": [ "r-17.33" ], "links_to_cards": [ "digamma1", "gammafn", "harmonicnum" ] }, { "id": "cornu", "url": "https://www.teclamairme.com/#g-cornu", "name": "the Cornu spiral", "symbol": "", "chapter": 17, "type": "obj", "badge": "Non-real object", "quick": "The Cornu spiral is the curve traced by plotting the Fresnel integrals C(t) against S(t) as t runs from −∞ to ∞, winding into a tight double spiral around the two points (½, ½) and (−½, −½).", "more": "Its curvature is proportional to its arc length, a property Leonhard Euler studied in 1744, so it is also called the Euler spiral; Alfred Cornu later used it to analyze diffraction, and the diffraction of light passing a straight edge can be read off its graph. The two points the curve winds toward, at (½, ½) and (−½, −½), are the limiting values of the Fresnel integrals themselves. Outside optics the same curve, sometimes called a clothoid, is used to design smooth, jerk-free transitions in roads and railway track.", "cites": [ "r-17.35" ], "links_to_cards": [ "fresnelfn" ] }, { "id": "zetam2", "url": "https://www.teclamairme.com/#g-zetam2", "name": "the first trivial zero", "symbol": "ζ(−2)", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the functional equation forces zeros at every negative even integer, and −2 is the first.", "more": "Riemann’s functional equation relates ζ(s) to ζ(1 − s) through a factor sin(πs/2), which is zero at every negative even integer, so ζ(−2) = ζ(−4) = ζ(−6) = … = 0 exactly. These are called the trivial zeros because they are the ones the formula gives away for free. Exactly 0, hence an integer.", "cites": [ "r-18.5" ], "links_to_cards": [ "funceq", "zeta" ] }, { "id": "zeta2", "url": "https://www.teclamairme.com/#g-zeta2", "name": "the Basel value", "symbol": "ζ(2)", "chapter": 18, "type": "tr", "badge": "Proven transcendental", "quick": "π²/6, found by Euler in 1734.", "more": "The Basel problem, posed by Pietro Mengoli in 1650 and solved by Euler in 1734: 1 + 1/4 + 1/9 + … = π²/6 ≈ 1.6449. Transcendental because π is transcendental (Lindemann, 1882), so π² is too, and a nonzero rational multiple of a transcendental number is transcendental.", "cites": [ "r-11.4", "r-18.2", "r-18.3" ], "links_to_cards": [ "pi" ] }, { "id": "zeta4", "url": "https://www.teclamairme.com/#g-zeta4", "name": "zeta of four", "symbol": "ζ(4)", "chapter": 18, "type": "tr", "badge": "Proven transcendental", "quick": "π⁴/90.", "more": "Euler found ζ(4) = π⁴/90 ≈ 1.0823 by the same method as the Basel value, and in fact found ζ(2n) for every positive integer n as a rational multiple of π^(2n) involving the Bernoulli numbers. Transcendental for the same reason as ζ(2).", "cites": [ "r-11.4", "r-18.2" ], "links_to_cards": [ "bern2", "zeta2" ] }, { "id": "zeta0", "url": "https://www.teclamairme.com/#g-zeta0", "name": "zeta of zero", "symbol": "ζ(0)", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly −1/2 by analytic continuation.", "more": "The continued function takes the value −1/2 at s = 0, which can be read off the functional equation. It is a rational number and therefore an integer-or-rational in this piece’s coloring; it enters the denominator in absolute value so that the product stays positive.", "cites": [ "r-18.1", "r-18.2" ], "links_to_cards": [ "analcont", "funceq" ] }, { "id": "zetam1", "url": "https://www.teclamairme.com/#g-zetam1", "name": "zeta of minus one", "symbol": "ζ(−1)", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly −1/12, the infamous value attached to 1 + 2 + 3 + ….", "more": "Euler worked with such divergent sums formally in the 1740s, though it is unclear whether he ever summed this one to −1/12; Ramanujan states it in his letter to Hardy of 27 February 1913; Riemann’s continuation makes it the honest value ζ(−1). The divergent series itself has no sum; the unique analytic function agreeing with the series where it converges takes the value −1/12 at −1. Rational, so uncolored, and used here in absolute value.", "cites": [ "r-18.1", "r-18.18", "r-18.24", "r-18.28" ], "links_to_cards": [] }, { "id": "t1", "url": "https://www.teclamairme.com/#g-t1", "name": "height of the first nontrivial zero", "symbol": "t1", "symbol_markup": "t_{1}", "chapter": 18, "type": "open", "badge": "Status open", "quick": "About 14.1347, the first place ζ vanishes on the critical line.", "more": "The imaginary part of the lowest nontrivial zero, 14.134725…, first published by Gram in 1903. Hardy proved in 1914 that infinitely many zeros lie on the critical line; Gourdon checked the first 10¹³ zeros in 2004, and Platt and Trudgian proved rigorously in 2021 that every zero up to height 3×10¹², the lowest 12,363,153,437,138 of them, lies on it. No source cited here settles whether t₁ is irrational, let alone transcendental, so it is status open.", "cites": [ "r-18.22", "r-18.7", "r-18.8" ], "links_to_cards": [ "critline", "zeta" ] }, { "id": "zetarho", "url": "https://www.teclamairme.com/#g-zetarho", "name": "the first nontrivial zero", "symbol": "ζ(½ + i·t1)", "symbol_markup": "ζ(½ + i·t_{1})", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0 by the definition of t₁, the height of the first zero on the critical line.", "more": "Riemann showed that every nontrivial zero lies in the closed strip 0 ≤ Re(s) ≤ 1 and conjectured in 1859 that all of them lie on the line Re(s) = ½. The lowest one has imaginary part t₁ ≈ 14.1347, computed by Gram in 1903 (Siegel later found that Riemann had computed it himself, unpublished). The value ζ(½ + i·t₁) is exactly 0 because t₁ is defined as the height at which ζ first vanishes on the line, so this is an integer.", "cites": [ "r-18.17", "r-18.28", "r-18.7", "r-18.8" ], "links_to_cards": [ "critline", "critstrip", "t1" ] }, { "id": "recipzeta1", "url": "https://www.teclamairme.com/#g-recipzeta1", "name": "the reciprocal of zeta at its pole", "symbol": "1/ζ(1)", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: ζ has a simple pole at s = 1, because the harmonic series diverges, so its reciprocal vanishes there.", "more": "The harmonic series 1 + ½ + ⅓ + … diverges, as Oresme showed around 1350, and Riemann’s continuation turns that into a simple pole at s = 1 with residue 1, the only pole ζ has. Its reciprocal is therefore analytic at 1 and equal to 0 there, the same move as 1/Γ(0) in Chapter ᚐ(A). Exactly 0, an integer.", "cites": [ "r-18.6" ], "links_to_cards": [ "recipgamma", "zeta" ] }, { "id": "zeta6", "url": "https://www.teclamairme.com/#g-zeta6", "name": "zeta of six", "symbol": "ζ(6)", "chapter": 18, "type": "tr", "badge": "Proven transcendental", "quick": "π⁶/945, the third of Euler’s even values, about 1.0173.", "more": "Euler’s formula, in the Bernoulli-number form of his 1755 book, ζ(2n) = (−1)^(n+1) B₂ₙ (2π)^(2n) / (2·(2n)!), gives ζ(6) = π⁶/945 ≈ 1.0173, with the Bernoulli number B₆ = 1/42 doing the work. Every even value is a rational multiple of a power of π and so transcendental by Lindemann (1882); no odd positive value is known to have any such formula.", "cites": [ "r-11.4", "r-18.2" ], "links_to_cards": [] }, { "id": "zetahalf", "url": "https://www.teclamairme.com/#g-zetahalf", "name": "zeta at the center of the critical strip", "symbol": "|ζ(½)|", "chapter": 18, "type": "open", "badge": "Status open", "quick": "About 1.4603545 in absolute value: ζ(½) = −1.4603545…, the value at the one real point of the critical line; no source cited here settles whether it is irrational.", "more": "ζ(½) = −1.4603545088…, computed from the alternating series (1 − 2^(1−s))^(−1) Σ (−1)^(n+1)/n^s, which converges for Re(s) > 0. It is the value at the one real point of the critical line and the center of the functional equation’s symmetry. No source cited here settles whether it is irrational, so it is status open; it enters the denominator in absolute value.", "cites": [ "r-18.10" ], "links_to_cards": [ "critline", "funceq" ] }, { "id": "t2", "url": "https://www.teclamairme.com/#g-t2", "name": "height of the second nontrivial zero", "symbol": "t2", "symbol_markup": "t_{2}", "chapter": 18, "type": "open", "badge": "Status open", "quick": "About 21.0220, the second zero on the critical line; the third is at about 25.0109.", "more": "The second nontrivial zero sits at ½ + 21.022040i and the third at ½ + 25.010858i, both found by Gram in 1903. The average gap between zeros at height t shrinks like 2π/ln t, and Montgomery’s pair-correlation conjecture of 1973, which Dyson recognized as the statistics of random Hermitian matrices, describes how the gaps are distributed. No source cited here settles whether t₂ is irrational: status open.", "cites": [ "r-18.7" ], "links_to_cards": [ "critline", "hermitian" ] }, { "id": "eulerproduct2", "url": "https://www.teclamairme.com/#g-eulerproduct2", "name": "Euler’s product at two", "symbol": "ζ(2)∏p prime(1−p−2)−1", "symbol_markup": "ζ(2)∏_{p prime}(1−p^{−2})−1", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: multiplying ζ(2) by the reciprocal of its prime product gives one, and the trailing −1 subtracts it.", "more": "Euler showed in 1737 that for s>1, ζ(s)=∏_{p}(1−p^{−s})^{−1}, a consequence of unique prime factorization. At s=2 this gives ζ(2)∏_{p}(1−p^{−2})=1, so the displayed residual is 0. The identity is the central bridge between the zeta function and the primes.", "cites": [ "r-18.16", "r-18.23", "r-18.4" ], "links_to_cards": [ "eulerprod", "ufd", "zeta", "zeta2" ] }, { "id": "eulerfactor2", "url": "https://www.teclamairme.com/#g-eulerfactor2", "name": "the first Euler factor at two", "symbol": "(1 − 2−2)−1", "symbol_markup": "(1 − 2^{−2})^{−1}", "chapter": 18, "type": "int", "badge": "Integer or rational", "quick": "4/3: the prime 2’s share of ζ(2). Multiply the factors for every prime and the product is π²/6.", "more": "Euler’s product writes ζ(s) as a product over the primes of 1/(1 − p^{−s}), because expanding each factor as a geometric series and multiplying out lists every integer exactly once, by unique factorization. At s = 2 the factor for p = 2 is 1/(1 − ¼) = 4/3; for p = 3 it is 9/8; the infinite product is π²/6. A rational number.", "cites": [ "r-18.16", "r-18.4" ], "links_to_cards": [ "eulerprod", "ufd", "zeta", "zeta2" ] }, { "id": "zeta", "url": "https://www.teclamairme.com/#g-zeta", "name": "Riemann zeta function", "symbol": "ζ", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Σ n^(−s), which converges only for Re(s) > 1 and is extended to the whole plane except s = 1 by analytic continuation.", "more": "Euler studied the sum of 1/n^s for real s from the 1730s and found the product over primes in 1737. Riemann, in his only paper on number theory (1859), made s complex and extended the function to the whole plane except s = 1, then connected its zeros to the distribution of primes. It is a function, not a number, so it carries no color.", "cites": [ "r-18.1", "r-18.23", "r-18.28", "r-18.3" ], "links_to_cards": [ "analcont", "eulerprod" ] }, { "id": "eulerprod", "url": "https://www.teclamairme.com/#g-eulerprod", "name": "the Euler product", "symbol": "Πp (1 − p−s)−1", "symbol_markup": "Π_{p} (1 − p^{−s})^{−1}", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "For Re(s) > 1, ζ(s) is also a product over every prime, which is why a function about a sum knows anything about primes.", "more": "Euler proved in 1737 that Σ n^(−s) = Π_p (1 − p^(−s))^(−1) for real s > 1: expand each factor as a geometric series and multiply, and unique factorization makes every n appear exactly once. Taking s = 1 gives a new proof that there are infinitely many primes, and the product is why the zeros of ζ control how the primes are spread. A function identity, so uncolored.", "cites": [ "r-18.23", "r-18.3", "r-18.4" ], "links_to_cards": [ "euclidprimes", "ufd", "zeta" ] }, { "id": "funceq", "url": "https://www.teclamairme.com/#g-funceq", "name": "the functional equation", "symbol": "ζ(s) = 2s πs−1 sin(πs/2) Γ(1 − s) ζ(1 − s)", "symbol_markup": "ζ(s) = 2^{s} π^{s−1} sin(πs/2) Γ(1 − s) ζ(1 − s)", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Riemann’s 1859 symmetry between s and 1 − s; the sine factor is where the trivial zeros come from.", "more": "Riemann proved in 1859 that the continued function satisfies this reflection formula, relating ζ(s) to ζ(1 − s). The factor sin(πs/2) vanishes at every negative even integer, which produces the trivial zeros, and the symmetry about the line Re(s) = ½ is why that line is the natural home of the nontrivial ones. Euler had guessed the real version in 1749. A function identity, uncolored.", "cites": [ "r-18.27", "r-18.5" ], "links_to_cards": [ "thetafn", "xifn", "zeta" ] }, { "id": "xifn", "url": "https://www.teclamairme.com/#g-xifn", "name": "the Riemann xi function", "symbol": "ξ(s)", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "½ s(s − 1) π^(−s/2) Γ(s/2) ζ(s): entire, symmetric under s ↔ 1 − s, and zero exactly at the nontrivial zeros.", "more": "Riemann’s completed zeta function, ξ(s) = ½ s(s − 1) π^(−s/2) Γ(s/2) ζ(s), removes the pole and the trivial zeros, leaving an entire function with ξ(s) = ξ(1 − s) whose zeros are exactly the nontrivial zeros of ζ. The Riemann Hypothesis says all of them have real part ½. It is this function that de Bruijn deformed in 1950 and Newman turned in 1976 into the constant Λ_dBN that closes the piece. A function, uncolored.", "cites": [ "r-18.28", "r-18.5" ], "links_to_cards": [ "dbn", "rh", "zeta" ] }, { "id": "critstrip", "url": "https://www.teclamairme.com/#g-critstrip", "name": "the critical strip", "symbol": "0 < Re(s) < 1", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "The vertical band where every nontrivial zero lives; the critical line runs down its middle.", "more": "Riemann showed in 1859 that all nontrivial zeros lie in the closed strip 0 ≤ Re(s) ≤ 1; Hadamard and de la Vallée Poussin proved in 1896 that none lie on its edges Re(s) = 1 or 0, which is exactly the prime number theorem. The hypothesis is that they all lie on the middle line Re(s) = ½. A region rather than a number, so uncolored.", "cites": [ "r-18.12", "r-18.17", "r-18.8" ], "links_to_cards": [ "critline", "pnt" ] }, { "id": "primecount", "url": "https://www.teclamairme.com/#g-primecount", "name": "the prime-counting function", "symbol": "π(x)", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "The number of primes up to x: π(10) = 4, π(100) = 25, π(10⁶) = 78498.", "more": "π(x) counts the primes not exceeding x; this π is not the circle constant, and every chapter has its own key. Gauss, at fifteen or sixteen, in 1792 or 1793, guessed from tables that π(x) grows like x/ln x, and Riemann’s 1859 explicit formula writes π(x) exactly as a sum over the zeros of ζ. Exact values have been computed as far as π(10²⁹). A function, uncolored.", "cites": [ "r-18.11", "r-18.28" ], "links_to_cards": [ "zeta" ] }, { "id": "pnt", "url": "https://www.teclamairme.com/#g-pnt", "name": "the prime number theorem", "symbol": "π(x) ~ x/ln x", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "The primes thin out like 1/ln x: conjectured by Gauss and Legendre, proved by Hadamard and de la Vallée Poussin in 1896.", "more": "Gauss conjectured it around 1792 and Legendre published a version in 1798; Chebyshev proved in 1850 that π(x) is within constant factors of x/ln x; Hadamard and de la Vallée Poussin proved the theorem independently in 1896 by showing ζ has no zeros on Re(s) = 1; Erdős and Selberg each published an elementary proof in 1949. The logarithmic integral li(x) of Chapter ᚐ(A) is the better approximation. A theorem, uncolored.", "cites": [ "r-18.12", "r-18.27", "r-18.28" ], "links_to_cards": [ "li", "primecount", "zeta" ] }, { "id": "mertensM", "url": "https://www.teclamairme.com/#g-mertensM", "name": "the Mertens function", "symbol": "M(x)", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "The running sum of the Möbius function; the Riemann Hypothesis is equivalent to M(x) growing no faster than x^(½ + ε) for every ε > 0.", "more": "M(x) = Σ_(n ≤ x) μ(n), with μ the Möbius function of Chapter ᚄ(S). Stieltjes conjectured in a letter of 1885, and Mertens in print in 1897, that |M(x)| < √x, which would have implied the Riemann Hypothesis; Odlyzko and te Riele disproved it in 1985 without finding a counterexample. Kotnik and te Riele showed in 2006 that one exists below about e^(1.59×10⁴⁰); the bound was later lowered to about e^(1.017×10²⁹) and, by Kim and Nguyen in 2024, to about e^(1.96×10¹⁹). RH itself is equivalent to M(x) = O(x^(½ + ε)) for every ε > 0. A function, uncolored.", "cites": [ "r-18.13", "r-18.27" ], "links_to_cards": [ "rh" ] }, { "id": "dirichletL", "url": "https://www.teclamairme.com/#g-dirichletL", "name": "a Dirichlet L-function", "symbol": "L(s, χ)", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Zeta twisted by a character: Dirichlet’s 1837 tool for primes in arithmetic progressions, and the ancestor of the L-function in the epilogue.", "more": "L(s, χ) = Σ χ(n) n^(−s) for a Dirichlet character χ modulo q. Dirichlet introduced them in 1837 to prove that every arithmetic progression a, a + q, a + 2q, … with gcd(a, q) = 1 contains infinitely many primes, using L(1, χ) ≠ 0. The generalized Riemann Hypothesis says their nontrivial zeros also lie on Re(s) = ½; the L-function of an elliptic curve in Chapter ᚒ(U) and the epilogue is a further cousin. A function, uncolored.", "cites": [ "r-18.14" ], "links_to_cards": [ "LfuncE", "rh", "zeta" ] }, { "id": "grampts", "url": "https://www.teclamairme.com/#g-grampts", "name": "the first Gram point", "symbol": "g0", "symbol_markup": "g_{0}", "chapter": 18, "type": "open", "badge": "Status open", "quick": "About 17.8456, the Gram point g₀, where the phase of ζ on the critical line is a multiple of π.", "more": "Gram points gₙ are the heights at which the Riemann–Siegel theta function θ(t) equals nπ; g₀ ≈ 17.8456, and Gram observed in 1903 that zeros tend to fall between consecutive Gram points, a rule of thumb (Gram’s law) that fails infinitely often, though Gram points still underlie the usual method of verifying zeros. No source cited here settles whether g₀ is irrational, so status open.", "cites": [ "r-18.15", "r-18.27" ], "links_to_cards": [ "critline", "zeta" ] }, { "id": "analcont", "url": "https://www.teclamairme.com/#g-analcont", "name": "analytic continuation", "symbol": "f₁ = f₂ on an arc ⇒ f₁ = f₂ on D₁ ∩ D₂", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Two analytic functions that agree on an arc, or on any infinite set with a limit point, agree on the whole overlap—so an extension, where one exists, is the only one.", "more": "If f₁ is analytic in a domain D₁ and f₂ is analytic in D₂, and the two agree on an arc in D = D₁ ∩ D₂—or on just an infinite number of points with a limit point in D—then they are equal throughout D, and f₂ is called an analytic continuation of f₁. That rigidity is what lets the series Σ n^(−s), which converges only for Re(s) > 1, name a single function on the rest of the plane: any analytic function agreeing with the series where it converges has no freedom left anywhere else. It does not make a divergent series converge and it does not manufacture a value out of nothing; and it cannot remove a genuine singularity, which is why ζ’s continuation is meromorphic and keeps its pole at s = 1. A construction, uncolored.", "cites": [ "r-18.1", "r-18.21" ], "links_to_cards": [ "zeta" ] }, { "id": "rh", "url": "https://www.teclamairme.com/#g-rh", "name": "the Riemann Hypothesis", "symbol": "every nontrivial zero of ζ has Re(s) = ½", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Riemann’s 1859 remark that every nontrivial zero of ζ very probably lies on the line Re(s) = ½; still open, and one of the seven Clay Millennium Prize Problems.", "more": "In his memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, published in the Monatsberichte der Berliner Akademie in November 1859, Riemann confined the nontrivial zeros to the strip with Re(s) between 0 and 1 and wrote that it is very probable that all of them lie on the middle line, adding that he had put the search for a proof aside as unnecessary for his immediate purpose. It is still open. Platt and Trudgian verified in 2021 that every zero up to height 3×10¹² is on the line—the lowest 12,363,153,437,138 of them—and the Clay Mathematics Institute lists it among its seven Millennium Prize Problems, with a million dollars for a proof. A conjecture, uncolored.", "cites": [ "r-18.17", "r-18.19", "r-18.8" ], "links_to_cards": [ "clay", "critstrip", "zeta" ] }, { "id": "casimir", "url": "https://www.teclamairme.com/#g-casimir", "name": "the Casimir force", "symbol": "⟨E⟩/A = −ħcπ²/720a³", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "The vacuum force between two parallel plates: the three-dimensional calculation regularizes with ζ(−3) = 1/120, while ζ(−1) = −1/12 is the one-dimensional scalar case.", "more": "Zeta regularization assigns a finite value to the divergent sum of vacuum-mode energies. The one-dimensional scalar field is the case that runs through ζ(−1) = −1/12, the value popular accounts attach to 1 + 2 + 3 + …; the familiar two-parallel-plate problem in three dimensions runs the same argument through ζ(−3) = 1/120 and gives an energy per unit area of −ħcπ²/720a³. The difference is why ζ needs values at every negative integer and not just at −1. A physical effect, uncolored.", "cites": [ "r-18.18" ], "links_to_cards": [ "zetam1" ] }, { "id": "ufd", "url": "https://www.teclamairme.com/#g-ufd", "name": "unique factorization", "symbol": "n = ∏ pᵉ", "chapter": 18, "type": "fn", "badge": "Function or operator", "quick": "Every integer above 1 is a product of primes in exactly one way. It is why the Euler product works, and why 1 is not a prime.", "more": "The fundamental theorem of arithmetic says every integer greater than 1 factors into primes in exactly one way up to the order of the factors; it can be read out of Euclid’s Elements, Book VII propositions 30 to 32 with Book IX proposition 14. It is the hinge of this chapter: multiply out ∏ₚ (1 + p⁻ˢ + p⁻²ˢ + ⋯) and each integer appears once and only once because each has one factorization, which is exactly what turns a product over the primes into a sum over the integers. It is also the reason 1 is not a prime—admit it and every factorization could be padded with copies of it—and the reason unique factorization domains have a name at all, since rings of algebraic integers need not have the property. A theorem, uncolored.", "cites": [ "r-18.26" ], "links_to_cards": [ "eulerprod", "ints" ] }, { "id": "ptinf", "url": "https://www.teclamairme.com/#g-ptinf", "name": "the point at infinity, projective X-coordinate", "symbol": "X(𝒪)", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "The identity of an elliptic curve’s group is the point at infinity, which in projective coordinates is (0 : 1 : 0), so its projective X-coordinate is 0.", "more": "In projective coordinates a point of the plane is (X : Y : Z) up to scaling; the affine plane is Z = 1 and the points with Z = 0 are at infinity. Every elliptic curve in Weierstrass form passes through exactly one point at infinity, 𝒪 = (0 : 1 : 0), and the group law takes it as the identity. Its projective X-coordinate is 0, unchanged by rescaling, so the numerator is exactly 0, an integer.", "cites": [ "r-19.1" ], "links_to_cards": [ "grouplaw", "projcoord", "weierstrass" ] }, { "id": "discE", "url": "https://www.teclamairme.com/#g-discE", "name": "the discriminant of the curve y² = x³ − x", "symbol": "Δ(y² = x³ − x)", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "64, for the curve y² = x³ − x; a nonzero discriminant is what makes a cubic an elliptic curve at all.", "more": "The discriminant of y² = x³ + ax + b is Δ = −16(4a³ + 27b²); it is nonzero exactly when the cubic has three distinct roots, which is the definition of an elliptic curve. For y² = x³ − x, a = −1 and b = 0 give Δ = 64, confirmed by LMFDB, where the curve is 32.a3 (Cremona 32a2), with conductor 32, rank 0, and complex multiplication by i. An integer. The curve of conductor 37 in the epilogue is a different curve; this one is chosen for its clean numbers.", "cites": [ "r-19.5" ], "links_to_cards": [ "E37", "cmult", "rank32", "weierstrass" ] }, { "id": "j1728", "url": "https://www.teclamairme.com/#g-j1728", "name": "the j-invariant of the same curve", "symbol": "j(y² = x³ − x)", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "1728, the cube of 12: this curve’s value of the invariant that tells two complex curves apart.", "more": "The j-invariant, 1728·4a³/(4a³ + 27b²), classifies elliptic curves over the complex numbers up to isomorphism: two curves are the same shape if and only if they share a j. For y² = x³ − x it is 1728 = 12³, the value taken by every curve with complex multiplication by i. An integer.", "cites": [ "r-19.5" ], "links_to_cards": [ "cmult", "jinv", "weierstrass" ] }, { "id": "cond37", "url": "https://www.teclamairme.com/#g-cond37", "name": "the conductor of the curve of conductor 37", "symbol": "N(E₃₇)", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "37, the smallest conductor of any elliptic curve of rank one.", "more": "The conductor of an elliptic curve over the rationals is an integer recording the primes of bad reduction and how bad they are; the modularity theorem, proved for semistable curves such as this one by Wiles and Taylor and in full by Breuil, Conrad, Diamond, and Taylor in 2001, says the curve corresponds to a modular form of that level. The curve 37a1, y² + y = x³ − x, has conductor 37, the smallest conductor of any curve of rank one, found in the tables of Birch, Swinnerton-Dyer, and later Cremona. An integer, and the same curve whose L-value vanishes in the epilogue.", "cites": [ "r-19.13", "r-19.8" ], "links_to_cards": [ "E37", "modularity" ] }, { "id": "rank32", "url": "https://www.teclamairme.com/#g-rank32", "name": "the rank of the curve y² = x³ − x", "symbol": "rank(y² = x³ − x)", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the curve has only its four torsion points over ℚ, which is Fermat’s theorem that 1 is not a congruent number.", "more": "The Mordell–Weil group of y² = x³ − x is ℤ/2 × ℤ/2 with no points of infinite order, so its rank is 0 (LMFDB 32.a3). A rational right triangle of area 1 would give a rational point with y ≠ 0 on this curve, so the rank being 0 is exactly Fermat’s theorem, proved by infinite descent, that 1 is not a congruent number, and the same descent is the one proof of his last theorem he ever wrote down, for n = 4. Exactly 0, an integer.", "cites": [ "r-19.18", "r-19.4" ], "links_to_cards": [ "descent", "fermat", "mordellweil", "rankE", "weierstrass" ] }, { "id": "tors4", "url": "https://www.teclamairme.com/#g-tors4", "name": "the torsion group of y² = x³ − x", "symbol": "|E(ℚ)tors|", "symbol_markup": "|E(ℚ)_{tors}|", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "4: the identity, of order 1, and the three points where y = 0, namely (0, 0), (1, 0), and (−1, 0), each of order 2.", "more": "The points of finite order on y² = x³ − x are 𝒪 and the three points with y = 0, since doubling a point with y = 0 gives 𝒪; the group is ℤ/2 × ℤ/2. Nagell (1935) and Lutz (1937) proved that torsion points on a curve with integer coefficients have integer coordinates and y = 0 or y² dividing the discriminant, which is how such groups are computed by hand. An integer.", "cites": [ "r-19.21", "r-19.6" ], "links_to_cards": [ "weierstrass" ] }, { "id": "pts5", "url": "https://www.teclamairme.com/#g-pts5", "name": "the points of y² = x³ − x over the field with five elements", "symbol": "#E(𝔽5)", "symbol_markup": "#E(𝔽_{5})", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "8, counted by hand: seven affine points plus the point at infinity, inside Hasse’s bound |8 − 6| ≤ 2√5.", "more": "Over 𝔽₅ the curve y² = x³ − x has the points (0, 0), (1, 0), (4, 0), (2,±1), (3,±2) and 𝒪, eight in all. Hasse proved in 1933 that #E(𝔽_p) is within 2√p of p + 1, here |8 − 6| = 2 ≤ 4.47; the differences a_p = p + 1 − #E(𝔽_p), here a₅ = −2, are the numbers the curve’s L-function is built from, and by the modularity theorem that L-function is the L-function of a modular form. Counting points over 𝔽_p is also the arithmetic under elliptic-curve cryptography. An integer.", "cites": [ "r-19.24", "r-19.7" ], "links_to_cards": [ "LfuncE", "ecc", "modularity", "weierstrass" ] }, { "id": "mazur16", "url": "https://www.teclamairme.com/#g-mazur16", "name": "Mazur’s bound on torsion", "symbol": "16", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "No elliptic curve over ℚ has more than 16 rational points of finite order, and the fifteen possible torsion groups are known exactly.", "more": "Barry Mazur proved in 1977 and 1978 that the torsion subgroup of an elliptic curve over ℚ is one of exactly fifteen groups: cyclic of order 1 to 10 or 12, or ℤ/2 × ℤ/2n for n = 1 to 4. The largest, ℤ/2 × ℤ/8, has order 16, and the largest cyclic order is 12. Merel proved in 1996 that over every number field the torsion is bounded by a number depending only on the field’s degree. An integer.", "cites": [ "r-19.20", "r-19.9" ], "links_to_cards": [] }, { "id": "congruent5", "url": "https://www.teclamairme.com/#g-congruent5", "name": "the smallest congruent number", "symbol": "5", "chapter": 19, "type": "int", "badge": "Integer or rational", "quick": "5 is the smallest whole number that is the area of a right triangle with rational sides (the triangle 3/2, 20/3, 41/6); 1, 2, 3, and 4 are not.", "more": "A congruent number is the area of a right triangle with rational sides; 5 is the smallest, then 6 (the 3-4-5 triangle) and 7. Fermat proved that 1 is not congruent, and n is congruent exactly when the curve y² = x³ − n²x has positive rank, which is why this chapter’s rank-0 numerator and this denominator entry are two sides of the same question. Tunnell’s criterion of 1983 decides the question for every n, assuming the Birch and Swinnerton-Dyer conjecture. An integer.", "cites": [ "r-19.10", "r-19.24" ], "links_to_cards": [ "bsdconj", "pythag345" ] }, { "id": "weierstrass", "url": "https://www.teclamairme.com/#g-weierstrass", "name": "the Weierstrass form", "symbol": "y² = x³ + ax + b", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Every elliptic curve over a field of characteristic not 2 or 3 can be written this way; a and b determine Δ and j.", "more": "Weierstrass’s ℘-function of the 1860s satisfies (℘′)² = 4℘³ − g₂℘ − g₃, which is why cubics in this shape are called Weierstrass form and why every complex elliptic curve is a torus ℂ/Λ. The general form y² + a₁xy + a₃y = x³ + a₂x² + a₄x + a₆, needed over fields of characteristic 2 and 3 and for minimal models such as y² + y = x³ − x, reduces to it by completing the square and cube whenever the characteristic is not 2 or 3. A form, uncolored.", "cites": [ "r-19.21", "r-19.5" ], "links_to_cards": [ "E37" ] }, { "id": "grouplaw", "url": "https://www.teclamairme.com/#g-grouplaw", "name": "the group law", "symbol": "P + Q + R = 𝒪", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Three points of the curve on one line add to the identity; that rule, with 𝒪 at infinity, makes the rational points an abelian group.", "more": "Draw the line through P and Q; it meets the cubic in a third point R, and P + Q is defined as the reflection of R in the x-axis, so that any three collinear points sum to 𝒪. The chord-and-tangent construction goes back to Diophantus and Newton; Mordell answered in 1922 a question apparently posed by Poincaré in his memoir of 1901: whether finitely many rational points generate all the others by this construction. Historians differ on how explicitly Poincaré had the group in hand, so the book claims no more than that. The associativity of this addition is the hard part; it can be proved through the ℘-function or by direct computation. A rule, uncolored.", "cites": [ "r-19.16", "r-19.19", "r-19.2" ], "links_to_cards": [] }, { "id": "mordellweil", "url": "https://www.teclamairme.com/#g-mordellweil", "name": "the Mordell–Weil theorem", "symbol": "E(ℚ) ≅ ℤr ⊕ E(ℚ)tors", "symbol_markup": "E(ℚ) ≅ ℤ^{r} ⊕ E(ℚ)_{tors}", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "The rational points of an elliptic curve form a finitely generated group: a finite torsion part and r independent points of infinite order, r the rank.", "more": "Mordell proved in 1922 that E(ℚ) is finitely generated, answering Poincaré’s 1901 question, and Weil extended it to Jacobians of higher-genus curves over arbitrary number fields in 1928. The rank r is the number the Birch and Swinnerton-Dyer conjecture ties to the L-function; no algorithm is proven to compute it, curves of rank at least 31 are known (Alpöge and Howell, 2026) while the largest rank ever pinned down exactly is 20 (Elkies and Klagsbrun, 2020), and whether ranks are bounded is open. A theorem, uncolored.", "cites": [ "r-19.17", "r-19.19", "r-19.3" ], "links_to_cards": [ "LfuncE", "bsdconj", "rankE" ] }, { "id": "LfuncE", "url": "https://www.teclamairme.com/#g-LfuncE", "name": "the L-function of an elliptic curve", "symbol": "L(E, s)", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "An Euler product with one factor for every prime, built from the numbers a_p that come from counting its points modulo each prime; continued past where the product converges, its behavior at s = 1 is what the Birch and Swinnerton-Dyer conjecture is about.", "more": "L(E, s) = Π_p (1 − a_p p^(−s) + p^(1−2s))^(−1) for primes of good reduction, times a simpler factor (1 − a_p p^(−s))^(−1) at each of the finitely many bad primes, with a_p = p + 1 − #E(𝔽_p); it converges for Re(s) > 3/2 and, by the modularity theorem, continues to the whole plane with a functional equation of its own about s = 1. The conjecture says the order of vanishing at s = 1 equals the rank; for the curve of conductor 37 the value L(E₃₇, 1) = 0 closes the piece. A function, uncolored.", "cites": [ "r-19.11", "r-19.21", "r-28.5" ], "links_to_cards": [ "E37", "bsdconj", "lE37", "modularity", "rankE" ] }, { "id": "modularity", "url": "https://www.teclamairme.com/#g-modularity", "name": "the modularity theorem", "symbol": "E ↔ f ∈ S2(Γ0(N))", "symbol_markup": "E ↔ f ∈ S_{2}(Γ_{0}(N))", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Every elliptic curve over ℚ comes from a modular form of weight two and level equal to its conductor: its semistable case, proved by Wiles with Taylor, is what settled Fermat’s last theorem.", "more": "Conjectured by Taniyama in 1955, sharpened with Shimura by 1957 and given its first serious evidence by Weil in 1967, proved for semistable curves by Wiles with Taylor in 1995 and for all curves by Breuil, Conrad, Diamond, and Taylor in 2001. Ribet showed in the summer of 1986 that a counterexample to Fermat’s last theorem would give a non-modular curve, publishing the proof in 1990, so modularity settled Fermat, the bit in Chapter ᚆ(H). It is also what proved that the L-function has an analytic continuation to the whole plane. A theorem, uncolored.", "cites": [ "r-19.13", "r-19.24" ], "links_to_cards": [ "LfuncE", "analcont", "fermat" ] }, { "id": "cmult", "url": "https://www.teclamairme.com/#g-cmult", "name": "complex multiplication", "symbol": "End(E) ⊋ ℤ", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "A curve with extra endomorphisms beyond multiplication by integers; y² = x³ − x has the map (x, y) ↦ (−x, iy), so its endomorphisms over ℂ are ℤ[i].", "more": "Most elliptic curves have only the integer multiplication maps; a curve with complex multiplication has an extra symmetry by an imaginary quadratic order, as y² = x³ − x has by the Gaussian integers, which is why its j-invariant is 1728. Kronecker’s Jugendtraum, realized by the theory of complex multiplication, builds abelian extensions of imaginary quadratic fields from such curves, and CM curves are where the Birch and Swinnerton-Dyer conjecture was first proved in part (Coates and Wiles, 1977). A property, uncolored.", "cites": [ "r-19.14", "r-19.24" ], "links_to_cards": [ "bsdconj", "j1728", "weierstrass" ] }, { "id": "ecc", "url": "https://www.teclamairme.com/#g-ecc", "name": "elliptic-curve cryptography", "symbol": "kP", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Multiplying a point by a large integer k is easy; recovering k from kP is the elliptic-curve discrete logarithm problem, on which much of today’s key exchange rests.", "more": "Proposed independently by Koblitz and Miller in 1985: on a curve over a large finite field, computing kP by repeated doubling is fast, but no known classical algorithm recovers k from P and kP in fewer than about √n steps, where n is the order of the point. The curve secp256k1, y² = x³ + 7 over a 256-bit prime field, underlies Bitcoin; Shor’s algorithm would break all of it on a sufficiently large fault-tolerant quantum computer. A construction, uncolored.", "cites": [ "r-19.15", "r-19.24" ], "links_to_cards": [] }, { "id": "descent", "url": "https://www.teclamairme.com/#g-descent", "name": "Fermat’s method of infinite descent", "symbol": "solution ⇒ smaller solution", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Assume a solution in positive integers, build a strictly smaller one from it, and repeat: positive integers cannot descend forever, so there was no solution.", "more": "Fermat coined the name and used the method on Diophantine equations. His one complete surviving proof runs this way: from any Pythagorean triangle whose area is a square he derives a smaller one, so no such triangle exists. That single argument is three statements at once—1 is not a congruent number, the curve y² = x³ − x has rank 0, and Fermat’s last theorem holds for fourth powers. The fourth-power case is the one case of his last theorem he ever wrote down, which is why Chapter ᚆ(H) says the proof he left covers fourth powers only and this chapter uses the same descent to show that no right triangle with rational sides has area 1: they are the same argument. A method, uncolored.", "cites": [ "r-19.18" ], "links_to_cards": [ "fermat", "rank32", "weierstrass" ] }, { "id": "projcoord", "url": "https://www.teclamairme.com/#g-projcoord", "name": "projective coordinates", "symbol": "(X : Y : Z)", "chapter": 19, "type": "fn", "badge": "Function or operator", "quick": "Three numbers instead of two, with scaling ignored. The ordinary plane is the part with Z ≠ 0, which scaling can always bring to Z = 1, and the points at infinity finally get coordinates of their own.", "more": "A point of the projective plane is a triple (X : Y : Z), not all zero, and two triples name the same point when one is a nonzero multiple of the other. The ordinary plane sits inside as the part with Z ≠ 0, read off as (X/Z, Y/Z); the triples with Z = 0 are the points at infinity, one for each direction. Möbius introduced the coordinates in Der barycentrische Calcul in 1827, and the payoff is exactly the one this chapter needs: the identity of an elliptic curve’s group is the point at infinity, and in projective coordinates it stops being a figure of speech and becomes the perfectly ordinary triple (0 : 1 : 0), whose X-coordinate is 0. A notation of the framework, uncolored.", "cites": [ "r-19.23" ], "links_to_cards": [ "ptinf" ] }, { "id": "lamC", "url": "https://www.teclamairme.com/#g-lamC", "name": "measure of the Cantor set", "symbol": "λ(𝒞)", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "Zero in Lebesgue measure, even though the set has as many points as the whole line.", "more": "The Lebesgue measure of the Cantor set is 0: after n steps the set lies inside 2^n intervals of total length (2/3)^n, which tends to 0. Exactly 0, an integer, despite the set having the cardinality of the continuum. It is the first of the three zeros in the numerator.", "cites": [ "r-20.1", "r-20.2" ], "links_to_cards": [ "cantor", "cont", "lebesgue" ] }, { "id": "dimH", "url": "https://www.teclamairme.com/#g-dimH", "name": "Hausdorff dimension of the Cantor set", "symbol": "dimH 𝒞", "symbol_markup": "dim_{H} 𝒞", "chapter": 20, "type": "tr", "badge": "Proven transcendental", "quick": "ln 2 / ln 3, about 0.6309, a transcendental dimension.", "more": "Hausdorff introduced fractional dimension in 1918; for a set made of m copies of itself scaled by 1/r the dimension is ln m / ln r, so the Cantor set has dimension ln 2 / ln 3 = log₃ 2 ≈ 0.6309. It is irrational because no power of 2 is a power of 3, and then transcendental by the Gelfond–Schneider theorem (1934), since 3 raised to an algebraic irrational cannot equal 2.", "cites": [ "r-20.7" ], "links_to_cards": [ "cantor", "gelfschn" ] }, { "id": "removed", "url": "https://www.teclamairme.com/#g-removed", "name": "the removed length", "symbol": "Σ 2n−1/3n", "symbol_markup": "Σ 2^{n−1}/3^{n}", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "The middle thirds sum to exactly 1, which is why nothing is left to weigh.", "more": "The lengths removed at each step, 1/3 + 2/9 + 4/27 + … = Σ 2^(n−1)/3^n, form a geometric series with sum exactly 1. The whole length of the interval is gone and uncountably many points remain. Exactly 1, an integer.", "cites": [ "r-20.2" ], "links_to_cards": [] }, { "id": "cantorfn", "url": "https://www.teclamairme.com/#g-cantorfn", "name": "the Cantor function at one half", "symbol": "c(½)", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "The devil’s staircase, flat almost everywhere in Lebesgue measure, reaches ½ at ½.", "more": "The Cantor function, introduced by Cantor in 1884 and nicknamed the devil’s staircase: continuous, non-decreasing from 0 to 1, constant on every removed interval, so its derivative is zero almost everywhere. By symmetry c(½) = ½ exactly. Exactly ½, an integer-or-rational.", "cites": [ "r-20.20", "r-20.6" ], "links_to_cards": [ "aeverywhere", "lebesgue" ] }, { "id": "lamQ", "url": "https://www.teclamairme.com/#g-lamQ", "name": "the measure of the rationals", "symbol": "λ(ℚ ∩ [0, 1])", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: a countable set can be covered by intervals of total length as small as you like, so the rationals weigh nothing.", "more": "Cover the n-th rational by an interval of length ε/2ⁿ; the total is ε, and ε was arbitrary, so the Lebesgue measure of the rationals in [0, 1] is 0 even though they are dense. Countable sets are null: the reason almost every real number is irrational, transcendental and, in the sense of Chapter ᚏ(R), undefinable. Exactly 0, an integer.", "cites": [ "r-20.22", "r-20.5" ], "links_to_cards": [ "dense", "lebesgue" ] }, { "id": "cantorderiv", "url": "https://www.teclamairme.com/#g-cantorderiv", "name": "the derivative of the Cantor function at one half", "symbol": "c′(½)", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the devil’s staircase is flat across the whole middle third (⅓, ⅔), where it sits at height ½, so its slope at ½ is 0; it is flat almost everywhere in Lebesgue measure, yet it climbs from 0 to 1.", "more": "The Cantor function is constant on each removed middle third, so its derivative is 0 on a set of Lebesgue measure 1; it rises only on the Cantor set, which has Lebesgue measure 0. One half lies inside the very first removed interval, (⅓, ⅔), where the staircase is flat at height ½, so c′(½) is exactly 0 and not merely 0 almost everywhere. The function is not absolutely continuous, which is the hypothesis the fundamental theorem of calculus needs if a function’s whole rise is to be recovered by integrating its derivative. Cantor introduced the function in 1884 and reported Scheeffer’s point that it is a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack; Scheeffer (1884), Lebesgue (1904), and Vitali (1905) discussed and popularized it. Exactly 0, an integer.", "cites": [ "r-20.19", "r-20.20", "r-20.6" ], "links_to_cards": [ "abscont", "aeverywhere", "cantor", "cantorfn", "ftc", "lebesgue" ] }, { "id": "fatcantor", "url": "https://www.teclamairme.com/#g-fatcantor", "name": "the fat Cantor set", "symbol": "λ(SVC)", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "Exactly ½: remove middle quarters, then sixteenths, and so on, and what is left is still nowhere dense but weighs half the interval.", "more": "The Smith–Volterra–Cantor set (Smith 1874, Volterra 1881, Cantor) removes from [0, 1] the middle 1/4, then an interval of length 1/16 from the middle of each remaining piece, then 1/64, and so on, so the removed lengths sum to ½ and the remainder has Lebesgue measure ½ while containing no interval at all. It shows that measure and topological size are different things. Exactly ½, an integer-or-rational.", "cites": [ "r-20.24", "r-20.8" ], "links_to_cards": [ "cantor", "dense", "lebesgue" ] }, { "id": "koch", "url": "https://www.teclamairme.com/#g-koch", "name": "the dimension of the Koch curve", "symbol": "dimH Koch", "symbol_markup": "dim_{H} Koch", "chapter": 20, "type": "tr", "badge": "Proven transcendental", "quick": "ln 4 / ln 3, about 1.2619: four copies at one third the size, transcendental by Gelfond–Schneider.", "more": "Helge von Koch’s 1904 curve, a continuous curve with no tangent anywhere, replaces each segment by four of one third the length, so its Hausdorff dimension is ln 4 / ln 3 ≈ 1.2619 and its length is infinite; the snowflake it bounds has area 8/5 of the starting triangle. The dimension is irrational because no power of 4 is a power of 3, and then transcendental by Gelfond–Schneider (1934).", "cites": [ "r-20.24", "r-20.7", "r-20.9" ], "links_to_cards": [ "boxdim", "gelfschn", "third" ] }, { "id": "sierp", "url": "https://www.teclamairme.com/#g-sierp", "name": "the dimension of the Sierpiński triangle", "symbol": "dimH Sierpiński", "symbol_markup": "dim_{H} Sierpiński", "chapter": 20, "type": "tr", "badge": "Proven transcendental", "quick": "ln 3 / ln 2, about 1.585: three copies at half the size.", "more": "Wacław Sierpiński described the triangle in 1915, three copies of itself at half scale, so its Hausdorff dimension is ln 3 / ln 2 ≈ 1.5850; the same pattern appears in Pascal’s triangle taken modulo 2 and in the Tower of Hanoi’s state graph. Transcendental by Gelfond–Schneider, since 2^x = 3 has no algebraic irrational solution.", "cites": [ "r-20.10", "r-20.7" ], "links_to_cards": [ "dimH", "gelfschn" ] }, { "id": "menger", "url": "https://www.teclamairme.com/#g-menger", "name": "the dimension of the Menger sponge", "symbol": "dimH Menger", "symbol_markup": "dim_{H} Menger", "chapter": 20, "type": "tr", "badge": "Proven transcendental", "quick": "ln 20 / ln 3, about 2.727: twenty copies at one third the size, with zero volume and infinite surface area, yet topologically only a curve.", "more": "Karl Menger’s 1926 sponge keeps twenty of the twenty-seven sub-cubes at each step, so its Hausdorff dimension is ln 20 / ln 3 ≈ 2.7268, its volume is 0, and its surface area infinite; it is a universal curve: every curve, meaning every compact metric space of covering dimension one, is homeomorphic to a subset of it. Transcendental by Gelfond–Schneider.", "cites": [ "r-20.11", "r-20.24", "r-20.7" ], "links_to_cards": [ "dimH", "gelfschn", "homeo", "third" ] }, { "id": "banachtarski", "url": "https://www.teclamairme.com/#g-banachtarski", "name": "the pieces of the Banach–Tarski paradox", "symbol": "5", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "A solid ball can be cut into five pieces—built with the axiom of choice — that reassemble into two balls of the same size; some of them have no measure at all, which is why measure theory is not contradicted.", "more": "Banach and Tarski proved in 1924, using the axiom of choice, that a ball in ℝ³ can be decomposed into finitely many pieces and reassembled by rotations and translations into two balls each the size of the original; Raphael Robinson showed in 1947 that five pieces suffice and four do not. Some of the pieces are non-measurable, which is why measure theory is not contradicted: it simply refuses to weigh them. An integer.", "cites": [ "r-20.12" ], "links_to_cards": [ "ac", "vitali" ] }, { "id": "aeverywhere", "url": "https://www.teclamairme.com/#g-aeverywhere", "name": "almost everywhere", "symbol": "a.e.", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "True except on a set of Lebesgue measure zero—which is weaker than everywhere, because a null set can still be uncountable.", "more": "A property holds almost everywhere if the set on which it fails is measurable with Lebesgue measure 0. The gap between that and everywhere is exactly the subject of this chapter: the Cantor function’s derivative is 0 almost everywhere, and yet the function climbs from 0 to 1, because all of its rise happens on the Cantor set, which is uncountable and has measure 0. A qualifier on a statement, so uncolored.", "cites": [ "r-20.2", "r-20.23", "r-20.6" ], "links_to_cards": [ "cantor", "cantorderiv", "lebesgue" ] }, { "id": "abscont", "url": "https://www.teclamairme.com/#g-abscont", "name": "absolute continuity", "symbol": "Σ|Δf| < ε", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "The hypothesis the Cantor function lacks: for every ε there is a δ such that any finite set of disjoint intervals of total length under δ moves the function by less than ε in total.", "more": "A function on an interval is absolutely continuous when, for every ε > 0, there is a δ > 0 such that every finite collection of pairwise disjoint subintervals whose lengths add to less than δ has Σ|f(bᵢ) − f(aᵢ)| < ε. It is stronger than uniform continuity, and it is exactly the condition under which the fundamental theorem of calculus survives the move to the Lebesgue integral: f is absolutely continuous precisely when f(x) = f(a) + ∫ₐˣ f′, an equivalence known as the fundamental theorem of Lebesgue integral calculus. The Cantor function is the standard witness that differentiability almost everywhere is not enough—it is continuous, nondecreasing, and of bounded variation, its derivative is 0 almost everywhere, and it still climbs from 0 to 1. The Banach–Zaretsky theorem says the missing ingredient is the Luzin N property: absolute continuity is continuity plus bounded variation plus sending every null set to a null set. A hypothesis, uncolored.", "cites": [ "r-20.25" ], "links_to_cards": [ "aeverywhere", "cantorderiv", "ftc", "lebesgueint" ] }, { "id": "dense", "url": "https://www.teclamairme.com/#g-dense", "name": "dense and nowhere dense", "symbol": "cl A = X", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "Dense: every interval, however small, contains one of its points, as the rationals do in the reals. Nowhere dense: not even its closure contains an interval, as with the Cantor set. Neither says anything about length.", "more": "A subset is dense when its closure is the whole space—equivalently, when the interior of its complement is empty—and the rationals are the standard example in the reals. A subset is nowhere dense when the closure has empty interior, equivalently when the complement of the closure is dense. The two words describe where a set sits, not how much of the line it takes up, which is the whole point of this chapter: the rationals are dense and have measure zero, the Cantor set is nowhere dense and also has measure zero, and Volterra’s nowhere-dense set of 1881 is nowhere dense with positive measure. The Baire category theorem, which says that in a complete metric space the complement of a countable union of nowhere dense sets is never empty, is what makes nowhere dense the topological word for small. A property of sets, uncolored.", "cites": [ "r-20.26", "r-20.27" ], "links_to_cards": [ "cantor", "fatcantor", "reals", "union" ] }, { "id": "logisticfixed", "url": "https://www.teclamairme.com/#g-logisticfixed", "name": "a logistic-map fixed point", "symbol": "f2(1/2)−1/2, fr(x)=rx(1−x)", "symbol_markup": "f_{2}(1/2)−1/2, f_{r}(x)=rx(1−x)", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: one half is a fixed point of the logistic map at r=2.", "more": "For the logistic family f_{r}(x)=rx(1−x), f_{2}(1/2)=2·(1/2)·(1/2)=1/2. The residual is therefore 0. The family is a standard model of iteration, bifurcation, and deterministic chaos; its period-doubling route is governed by the Feigenbaum constants in Chapter ᚃ(F). A rational computation carrying dynamical systems.", "cites": [ "r-20.13", "r-20.14" ], "links_to_cards": [ "logisticmap", "logisticr" ] }, { "id": "lebesgue", "url": "https://www.teclamairme.com/#g-lebesgue", "name": "Lebesgue measure", "symbol": "λ", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "Length, made rigorous enough to measure very strange sets.", "more": "Lebesgue measure, from Henri Lebesgue’s 1902 thesis, extends length from intervals to a vast class of sets while keeping countable additivity: the measure of a countable disjoint union is the sum of the measures. It is the foundation of modern integration and of probability. A function on sets, so uncolored.", "cites": [ "r-20.24", "r-20.5" ], "links_to_cards": [ "lebesgueint" ] }, { "id": "cantor", "url": "https://www.teclamairme.com/#g-cantor", "name": "Cantor set", "symbol": "𝒞", "chapter": 20, "type": "obj", "badge": "Non-real object", "quick": "What remains of [0, 1] after removing middle thirds forever: uncountable, yet weightless.", "more": "The middle-thirds set, published by Cantor in 1883 (Henry John Stephen Smith had described such sets in 1874): the numbers in [0, 1] whose base-3 expansion uses only the digits 0 and 2. Closed, nowhere dense, uncountable, and self-similar. It is a set rather than a number, so it is colored as a non-real object.", "cites": [ "r-20.1", "r-20.2", "r-20.3" ], "links_to_cards": [ "dense" ] }, { "id": "lebesgueint", "url": "https://www.teclamairme.com/#g-lebesgueint", "name": "the Lebesgue integral", "symbol": "∫ f dλ", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "Integration by slicing the range instead of the domain, which is why it can integrate functions Riemann’s integral cannot and, under the convergence theorems, passes limits inside the integral.", "more": "Lebesgue’s 1902 thesis defines the integral by measuring the sets on which f takes each value, rather than by summing over intervals of the domain; the indicator function of the rationals, which has no Riemann integral, has Lebesgue integral 0. The dominated and monotone convergence theorems make it the integral of modern analysis and probability, and Chapter ᚋ(M)’s probabilities are Lebesgue integrals. A tool, uncolored.", "cites": [ "r-20.16", "r-20.24", "r-20.5" ], "links_to_cards": [] }, { "id": "sigmaalg", "url": "https://www.teclamairme.com/#g-sigmaalg", "name": "a σ-algebra", "symbol": "σ-algebra", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "The family of sets that a measure is allowed to weigh: closed under complements and countable unions.", "more": "A measure needs a domain of measurable sets, and a σ-algebra is one closed under complement and countable union; on the line the standard ones are the Borel σ-algebra, generated by the open sets and named for Borel (his measure appears in his 1898 Leçons), and its completion, the Lebesgue σ-algebra. Kolmogorov’s 1933 axioms make probability a measure on a σ-algebra of events. A structure, uncolored.", "cites": [ "r-20.17", "r-20.24" ], "links_to_cards": [] }, { "id": "vitali", "url": "https://www.teclamairme.com/#g-vitali", "name": "a non-measurable set", "symbol": "Vitali set", "chapter": 20, "type": "obj", "badge": "Non-real object", "quick": "A subset of [0, 1] that Lebesgue measure cannot weigh, built with the axiom of choice by Vitali in 1905.", "more": "Pick one representative from each class of reals differing by a rational; the result, Vitali’s set of 1905, cannot have a Lebesgue measure, because countably many rational shifts of it tile [0, 1] up to a null set, and countably many equal measures cannot add up to 1: zero gives 0 and anything positive gives infinity. It is the reason Lebesgue measure lives on a σ-algebra and not on all sets, and one of the precursors of the Banach–Tarski paradox. An object, colored as a non-real object.", "cites": [ "r-20.18" ], "links_to_cards": [ "ac", "banachtarski", "lebesgue", "sigmaalg" ] }, { "id": "boxdim", "url": "https://www.teclamairme.com/#g-boxdim", "name": "the box-counting dimension", "symbol": "dimbox", "symbol_markup": "dim_{box}", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "Count the boxes of side ε needed to cover a set; if the count grows like ε^(−d), the dimension is d. Easier to compute than Hausdorff’s, and equal to it for the self-similar sets here, though not for the rationals.", "more": "The Minkowski–Bouligand or box-counting dimension, named for Hermann Minkowski and Georges Bouligand, measures how the number N(ε) of ε-boxes covering a set scales: d = lim ln N(ε) / ln(1/ε). For self-similar sets like the Cantor set and the Koch curve it agrees with the Hausdorff dimension; for the rationals it gives 1 while Hausdorff gives 0, so the two can differ; Hausdorff’s is often very hard to compute, and this is the practical one. A quantity, uncolored.", "cites": [ "r-20.18" ], "links_to_cards": [ "cantor", "dimH", "koch" ] }, { "id": "logisticr", "url": "https://www.teclamairme.com/#g-logisticr", "name": "the logistic parameter", "symbol": "r", "chapter": 20, "type": "int", "badge": "Integer or rational", "quick": "2: the growth rate of the map f(x) = rx(1 − x) on the page. Between 1 and 3 every start inside the interval settles to a fixed point; above 3 the doubling cascade to chaos begins.", "more": "Robert May’s 1976 paper in Nature made the logistic map rx(1 − x) a standard example of simple rules with complicated behavior. For 1 < r < 3 there is one attracting fixed point, 1 − 1/r (here ½); at r = 3 it splits into a cycle of two, then four, then eight, at parameter values whose spacing shrinks by Feigenbaum’s δ ≈ 4.669 of Chapter ᚃ(F), until chaos at r ≈ 3.5699. An integer.", "cites": [ "r-20.13", "r-20.14", "r-20.15" ], "links_to_cards": [ "feig", "logisticmap" ] }, { "id": "logisticmap", "url": "https://www.teclamairme.com/#g-logisticmap", "name": "the logistic map", "symbol": "f(x) = rx(1 − x)", "chapter": 20, "type": "fn", "badge": "Function or operator", "quick": "One line of arithmetic, iterated: this year’s population as a fraction of its ceiling becomes next year’s.", "more": "Robert May’s 1976 paper in Nature made f(x) = rx(1 − x) the standard example of a simple deterministic rule with complicated behavior. For r between 1 and 3 every start strictly inside the interval settles to the fixed point 1 − 1/r; at r = 3 that point splits into a cycle of two, then four, then eight, and the period-doubling cascade reaches chaos near r ≈ 3.5699. This chapter takes it at r = 2, the tame end of the road, and leaves the cascade to Chapter ᚃ(F). A map, so uncolored.", "cites": [ "r-20.13", "r-20.14" ], "links_to_cards": [ "logisticr" ] }, { "id": "euler", "url": "https://www.teclamairme.com/#g-euler", "name": "Euler’s identity", "symbol": "eiπ + 1", "symbol_markup": "e^{iπ} + 1", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "e^(iπ) = −1, so the numerator is 0.", "more": "Euler published the formula e^{iθ} = cos θ + i sin θ in 1748 in his Introductio in analysin infinitorum; setting θ = π gives e^{iπ} = −1, so e^{iπ} + 1 = 0. Five fundamental constants appear once each—0, 1, i, π, and e—and the polls keep noticing: a 1990 poll by the Mathematical Intelligencer named it the most beautiful theorem in mathematics, and a 2004 Physics World poll tied it with Maxwell’s equations as the greatest equation ever. It is exactly zero, a rational number, so it carries the integer color: the arithmetic happens in ℂ but the answer is 0.", "cites": [ "r-21.1", "r-21.2" ], "links_to_cards": [ "eulerformula", "maxwell" ] }, { "id": "ibar", "url": "https://www.teclamairme.com/#g-ibar", "name": "i times its conjugate", "symbol": "i·ī", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "1: i times its conjugate −i is −i² = 1, the first of three ways this denominator makes i disappear.", "more": "The conjugate z̄ of z = a + bi is a − bi, the reflection across the real axis. For any z the product z·z̄ = a² + b² = |z|² is the square of the modulus, so i·ī = i·(−i) = −i² = 1, the first of three ways this denominator makes i vanish. Value 1, an integer.", "cites": [ "r-21.3" ], "links_to_cards": [] }, { "id": "gaussint", "url": "https://www.teclamairme.com/#g-gaussint", "name": "norm of the Gaussian integer 1 + i", "symbol": "(1+i)(1−i)", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "2, which is why 2 is not prime among the Gaussian integers.", "more": "Gauss introduced the Gaussian integers ℤ[i] = {a + bi : a, b ∈ ℤ} in his second memoir on biquadratic reciprocity (1832). The norm N(a + bi) = (a + bi)(a − bi) = a² + b² is multiplicative, and N(1 + i) = 2, so 2 = −i(1 + i)² is no longer prime in ℤ[i]. The Gaussian primes are 1 + i and its associates, the ordinary primes p ≡ 3 (mod 4) such as 3, 7, 11, and the factors a ± bi of the primes p ≡ 1 (mod 4), which split because they are sums of two squares (5 = (2 + i)(2 − i)). Value 2, an integer.", "cites": [ "r-21.4" ], "links_to_cards": [] }, { "id": "unitcirc", "url": "https://www.teclamairme.com/#g-unitcirc", "name": "modulus of a point on the unit circle", "symbol": "|eiθ|", "symbol_markup": "|e^{iθ}|", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "1 for every real θ.", "more": "For any real θ, |e^{iθ}| = √(cos²θ + sin²θ) = 1: the point e^{iθ} sits on the unit circle at angle θ. This is Euler’s formula of 1748 read as geometry, and it is why multiplying by e^{iθ} is a rotation with no stretching. Value 1 for every θ, an integer.", "cites": [ "r-21.1", "r-21.5" ], "links_to_cards": [ "eulerformula" ] }, { "id": "splitj", "url": "https://www.teclamairme.com/#g-splitj", "name": "the split-complex unit squared", "symbol": "j²", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "In the split-complex numbers j² = 1 with j ≠ ±1, the third way to build a two-component number from the reals; this j is not the quaternion one.", "more": "The split-complex numbers a + bj, with j² = +1 and j ≠ ±1, go back to James Cockle’s tessarines of 1848, of which they are the subalgebra spanned by 1 and j; Clifford used them in the 1870s as coefficients for his split-biquaternions. They are the third and last way to build a two-component number from the real line (the others being i² = −1 and ε² = 0), and their unit hyperbola plays the role the unit circle plays for ℂ, which is why, since the late twentieth century, their multiplication has been read as the Lorentz boost of a spacetime plane. This j is unrelated to the quaternion j of Chapter ᚕ(J). Value j² = 1, an integer.", "cites": [ "r-21.6" ], "links_to_cards": [ "dual", "reals" ] }, { "id": "dual", "url": "https://www.teclamairme.com/#g-dual", "name": "the dual-number zero", "symbol": "ε²", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "In the dual numbers ε² = 0 with ε ≠ 0, the second way to build a two-component number from the reals; this ε is not the surreal one.", "more": "The dual numbers a + bε with ε² = 0 and ε ≠ 0 were introduced by William Clifford in 1873. Because (a + bε)² = a² + 2abε, a function evaluated at x + ε returns f(x) + f′(x)ε: the derivative comes for free, which is the idea behind forward-mode automatic differentiation. ε² is exactly 0 by definition, so the rider carries the integer color. This ε is not the surreal infinitesimal of Chapter ᚙ(X).", "cites": [ "r-21.7" ], "links_to_cards": [ "epsS", "reals" ] }, { "id": "re_i", "url": "https://www.teclamairme.com/#g-re_i", "name": "the real part of i", "symbol": "Re(i)", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "0: a complex number z = x + iy has real part Re z = x and imaginary part Im z = y, and i sits entirely on the imaginary axis.", "more": "Re and Im are the two coordinates of the complex plane, the picture Caspar Wessel presented in 1797 (printed 1799), Argand drew in 1806, and Gauss made respectable in 1831. Re(i) = 0 and Im(1) = 0 say that 1 and i lie on different axes: a real number has no imaginary part and i has no real part. Exactly 0, an integer, riding beside Euler’s identity.", "cites": [ "r-21.8" ], "links_to_cards": [ "cplane", "euler" ] }, { "id": "demoivre", "url": "https://www.teclamairme.com/#g-demoivre", "name": "de Moivre’s formula, at a sixth root of unity", "symbol": "(cos π/3 + i sin π/3)6", "symbol_markup": "(cos π/3 + i sin π/3)^{6}", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "1: de Moivre’s formula (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, with θ = π/3 and n = 6, lands on cos 2π + i sin 2π = 1.", "more": "The formula (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ goes back to Abraham de Moivre’s work on roots of equations (1707) and on cos nθ (1722, 1730), and it follows in one line from Euler’s e^{iθ} = cos θ + i sin θ. Raising a point on the unit circle to the nth power multiplies its angle by n, so the sixth power of the point at 60° returns to 1; the six sixth roots of unity are the corners of a regular hexagon, and this is one of them. Value 1, an integer.", "cites": [ "r-21.1", "r-21.9" ], "links_to_cards": [ "eulerformula" ] }, { "id": "fta", "url": "https://www.teclamairme.com/#g-fta", "name": "the fundamental theorem of algebra, counted", "symbol": "#{z : z2 + 1 = 0}", "symbol_markup": "#{z : z^{2} + 1 = 0}", "chapter": 21, "type": "int", "badge": "Integer or rational", "quick": "2: every degree-n polynomial with complex coefficients has exactly n complex roots counted with multiplicity, and z² + 1 has the two roots i and −i.", "more": "The fundamental theorem of algebra says that every nonconstant polynomial with complex coefficients has a root in ℂ, so that a degree-n polynomial has n roots counted with multiplicity. d’Alembert attempted it in 1746; Gauss’s doctoral proof of 1799 had a topological gap that Ostrowski filled only in 1920; Argand gave the first rigorous proof in 1806, and was the first to state the theorem for complex rather than real coefficients. Here it counts the roots of z² + 1: exactly two, i and −i. Value 2, an integer.", "cites": [ "r-21.10" ], "links_to_cards": [ "liouville" ] }, { "id": "argi", "url": "https://www.teclamairme.com/#g-argi", "name": "the argument of i", "symbol": "arg i", "chapter": 21, "type": "tr", "badge": "Proven transcendental", "quick": "π/2 is the principal argument of i; an argument is an angle from the positive real axis and is defined modulo 2π.", "more": "The polar form z = |z|e^{i arg z} writes a complex number by its modulus and its angle; arg i = π/2 because i is one unit up the imaginary axis. The argument is defined only up to multiples of 2π, and the principal value is taken in (−π, π]. Since π is transcendental (Lindemann, 1882), so is π/2, hence the transcendental color.", "cites": [ "r-21.11", "r-21.18" ], "links_to_cards": [ "pi" ] }, { "id": "i", "url": "https://www.teclamairme.com/#g-i", "name": "imaginary unit", "symbol": "i", "chapter": 21, "type": "obj", "badge": "Non-real object", "quick": "A square root of −1: i² = −1.", "more": "Cardano computed with such roots in the Ars Magna of 1545 and called the result “as subtle as it is useless”; Rafael Bombelli gave the multiplication rules that made them work in L’algebra of 1572, and used them to reach the three real roots of particular irreducible cubics; Descartes coined the word imaginary for them in La Géométrie (1637), to stress their unreal nature, and Newton was using the word by 1670; Euler introduced the letter i (a paper of 1777, printed 1794); Wessel (1797), Argand (1806), and Gauss (1831) drew it as the point one unit above the origin, and Gauss gave the name complex number, and suggested lateral in place of imaginary. Its powers cycle with period four: i, −1, −i, 1. It is a non-real object, a tool of this chapter’s framework rather than a value the formula hands home, so it is tinted as an object rather than a number.", "cites": [ "r-21.12", "r-21.16", "r-21.18", "r-21.8" ], "links_to_cards": [] }, { "id": "riemannsphere", "url": "https://www.teclamairme.com/#g-riemannsphere", "name": "the Riemann sphere", "symbol": "ℂ ∪ {∞}", "chapter": 21, "type": "obj", "badge": "Non-real object", "quick": "The complex plane with one point at infinity added, which makes it a sphere; 1/0 = ∞ is legal there, and the formula does not go there.", "more": "Riemann introduced the extended complex plane in 1857: adjoin a single point ∞ and the plane closes up into a sphere, via stereographic projection. Möbius transformations z ↦ (az + b)/(cz + d) become bijections of the sphere, and 1/0 = ∞ becomes a legitimate value. It is a non-real object and a tool of this chapter only; the formula’s rule against dividing by zero is a rule of the plane, not the sphere, which is why the sphere is listed here as a tool but plays no part in the chapter’s fraction.", "cites": [ "r-21.13" ], "links_to_cards": [ "cplane", "mobiustrans" ] }, { "id": "eulerformula", "url": "https://www.teclamairme.com/#g-eulerformula", "name": "Euler’s formula", "symbol": "eiθ = cos θ + i sin θ", "symbol_markup": "e^{iθ} = cos θ + i sin θ", "chapter": 21, "type": "fn", "badge": "Function or operator", "quick": "The engine of the chapter, from Euler’s Introductio of 1748: the exponential of an imaginary angle is the point on the unit circle at that angle, and θ = π gives the identity in the numerator.", "more": "Euler published e^{iθ} = cos θ + i sin θ in the Introductio in analysin infinitorum of 1748; Roger Cotes had the logarithmic form ln(cos θ + i sin θ) = iθ in 1714. It puts e^{iθ} on the unit circle, so |e^{iθ}| = 1, and it turns multiplication of complex numbers into addition of angles, which is why de Moivre’s formula follows in one line, and why the polar form z = |z|e^{i arg z} exists. Euler’s identity is the case θ = π. A formula rather than a value, hence the tool tag.", "cites": [ "r-21.1", "r-21.5" ], "links_to_cards": [ "demoivre", "euler", "unitcirc" ] }, { "id": "cplane", "url": "https://www.teclamairme.com/#g-cplane", "name": "the complex plane", "symbol": "ℂ ≅ ℝ²", "chapter": 21, "type": "obj", "badge": "Non-real object", "quick": "Every complex number is a point: a + bi sits at (a, b). Wessel drew it in 1797, Argand in 1806, and Gauss’s announcement of 1831 made it respectable; it is also called the Argand diagram.", "more": "Identifying a + bi with the point (a, b) makes addition a translation and multiplication a rotation combined with a scaling, which is the geometric content of Euler’s formula. Caspar Wessel presented the picture to the Danish Academy in 1797 (printed 1799, and unnoticed), Jean-Robert Argand published it in 1806, and Gauss’s 1831 announcement of his second memoir on biquadratic residues, printed in 1832, fixed the modern view and the name. The plane cannot be ordered in a way compatible with its arithmetic, which is the property the first rung of the ladder gives up. A non-real object and the chapter’s picture.", "cites": [ "r-21.5", "r-21.8" ], "links_to_cards": [ "eulerformula" ] }, { "id": "twosq", "url": "https://www.teclamairme.com/#g-twosq", "name": "the Brahmagupta–Fibonacci two-square identity", "symbol": "(a² + b²)(c² + d²) = (ac − bd)² + (ad + bc)²", "chapter": 21, "type": "fn", "badge": "Function or operator", "quick": "The complex norm is multiplicative, |zw| = |z||w|, written out in integers: Diophantus stated it in the third century, Brahmagupta c. 628, Fibonacci in 1225.", "more": "The identity appears in Diophantus’s Arithmetica (III, 19), was rediscovered by Brahmagupta c. 628 and generalized to his identity for Pell’s equation, and reached Europe in Fibonacci’s Liber Quadratorum of 1225, which is why it often carries his name. It says the norm of a product of Gaussian integers is the product of the norms, and it is the n = 2 case of the ladder’s chain of norm identities: Euler’s four squares (1748) for the quaternions, Degen’s eight squares (c. 1818) for the octonions, and, by Hurwitz’s theorem of 1898, no more. A formula rather than a value, hence the tool tag.", "cites": [ "r-21.14" ], "links_to_cards": [ "hurwitz", "octonions", "quaternions" ] }, { "id": "fermat2sq", "url": "https://www.teclamairme.com/#g-fermat2sq", "name": "Fermat’s theorem on sums of two squares", "symbol": "p ≡ 1 (mod 4)", "chapter": 21, "type": "fn", "badge": "Function or operator", "quick": "Fermat’s theorem on sums of two squares says an odd prime is expressible as a sum of two squares exactly when it leaves remainder 1 on division by 4, and 13 = 2² + 3² is the smallest example after 5.", "more": "Pierre de Fermat stated the theorem around 1640 without a surviving proof; Leonhard Euler found the first proof after much effort, announcing it to Goldbach in letters of 1747 and 1749 and publishing it between 1752 and 1755. A prime p ≡ 1 (mod 4) is always expressible as a² + b², uniquely up to order and sign, while a prime p ≡ 3 (mod 4) never is; the theorem reflects the fact that such primes split, rather than stay prime, when factored inside the Gaussian integers, where a² + b² = (a+bi)(a−bi). It connects to two other cards of this chapter: the Brahmagupta–Fibonacci identity, which says a product of two sums of two squares is again a sum of two squares, and the norm of the Gaussian integer 1 + i.", "cites": [ "r-21.20" ], "links_to_cards": [ "gaussint" ] }, { "id": "ijk", "url": "https://www.teclamairme.com/#g-ijk", "name": "Hamilton’s identity", "symbol": "ijk + 1", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "i² = j² = k² = ijk = −1, carved into Brougham Bridge in 1843.", "more": "Hamilton’s fundamental formula, discovered on 16 October 1843 and cut into the stone of Brougham (now Broom) Bridge on the Royal Canal in Dublin: i² = j² = k² = ijk = −1. From it the whole quaternion multiplication table follows, including ij = k and ji = −k. Gauss had found the same multiplication in an unpublished note of 1819, printed only in 1900, and Olinde Rodrigues had parametrized rotations by four numbers in 1840; Hamilton was the one who saw a number system. The numerator adds 1 to ijk, giving exactly 0, a rational number computed in ℍ, hence the integer color.", "cites": [ "r-22.1", "r-22.12", "r-22.2", "r-22.8" ], "links_to_cards": [] }, { "id": "qnorm", "url": "https://www.teclamairme.com/#g-qnorm", "name": "norm of the all-ones quaternion", "symbol": "‖1+i+j+k‖", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "2: the norm √(1² + 1² + 1² + 1²) of the simplest quaternion with every component present.", "more": "The quaternion norm ‖a + bi + cj + dk‖ = √(a² + b² + c² + d²) is the four-dimensional Pythagoras, and it is multiplicative: the norm of a product is the product of the norms. That fact is Euler’s four-square identity, sent to Goldbach in a letter of 4 May 1748, a century before Hamilton found it again, as a statement about quaternions in 1843; Lagrange used it in 1770 to prove that every positive integer is a sum of four squares. For 1 + i + j + k the norm is √4 = 2. Value 2, an integer, one of the four nonzero factors of the denominator.", "cites": [ "r-22.12", "r-22.17", "r-22.3", "r-22.7" ], "links_to_cards": [ "quaternions" ] }, { "id": "comm", "url": "https://www.teclamairme.com/#g-comm", "name": "the quaternion commutator", "symbol": "ij − ji", "chapter": 22, "type": "obj", "badge": "Non-real object", "quick": "2k, nonzero: quaternions do not commute.", "more": "The commutator ij − ji measures how far two quaternions are from commuting; for the units i and j it is k − (−k) = 2k, nonzero. It is a quaternion, not a real number, so it is tinted as a non-real object. Multiplied by the norm 2, the conjugate product 4 and the order 8 of the quaternion group, it makes the denominator 128k, which has an inverse because every nonzero quaternion does (Hamilton, 1843), so dividing 0 by it is legal and gives 0, the real scalar 0 that ℍ shares with ℝ.", "cites": [ "r-22.12", "r-22.13", "r-22.17", "r-22.2", "r-22.5" ], "links_to_cards": [ "q8", "qconj", "qnorm", "quaternions" ] }, { "id": "qrot", "url": "https://www.teclamairme.com/#g-qrot", "name": "Hamilton’s rotation formula, turning i into j", "symbol": "q i q−1 − j, q = (1 + k)/√2", "symbol_markup": "q i q^{−1} − j, q = (1 + k)/√2", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "0: conjugating a vector by the unit quaternion q = cos(π/4) + k sin(π/4) rotates it a quarter turn about the k-axis, and a quarter turn carries i exactly onto j.", "more": "A unit quaternion q = cos(θ/2) + u sin(θ/2), with u a unit pure quaternion, rotates any vector v by the angle θ about the axis u through the map v ↦ q v q^{−1}. Olinde Rodrigues had the composition rule for rotations, in four parameters, in 1840, three years before quaternions existed; Cayley published the conjugation form in 1845. Here θ = 90° about k, so i becomes j, and the difference q i q^{−1} − j is exactly 0 (verified by computation). The unit quaternions form the 3-sphere S³ and the group SU(2); because q and −q give the same rotation, S³ double-covers the rotation group SO(3), which is why a spinor needs a 720° turn to come home. Games and spacecraft use this formula instead of Euler angles because it has no gimbal lock. Exactly 0, an integer, riding beside Hamilton’s identity.", "cites": [ "r-22.10", "r-22.4", "r-22.6", "r-22.8" ], "links_to_cards": [ "ijk", "quaternions", "s3su2", "versor" ] }, { "id": "qconj", "url": "https://www.teclamairme.com/#g-qconj", "name": "a quaternion times its conjugate", "symbol": "(1 + i + j + k)(1 − i − j − k)", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "4: the conjugate q̄ reverses the signs of i, j, and k, and q·q̄ = ‖q‖² always, so this product is 2² = 4.", "more": "The conjugate of q = a + bi + cj + dk is q̄ = a − bi − cj − dk, and q·q̄ = q̄·q = a² + b² + c² + d² = ‖q‖² is real; this is how quaternions are divided, since q^{−1} = q̄/‖q‖². For the all-ones quaternion the product is 4 (verified by computation). Hurwitz’s quaternion integers, the a + bi + cj + dk whose coordinates are all integers or all half-integers, use this norm to run a Euclidean algorithm that proves Lagrange’s four-square theorem: every positive integer is a sum of four squares. Hurwitz published them in the Göttingen Nachrichten of 1896 and set them out at length in his Vorlesungen of 1919. Value 4, an integer, a factor of the denominator.", "cites": [ "r-22.12", "r-22.7" ], "links_to_cards": [ "euclidalg", "quaternions" ] }, { "id": "q8", "url": "https://www.teclamairme.com/#g-q8", "name": "the order of the quaternion group", "symbol": "|Q₈|", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "8: the eight elements ±1, ±i, ±j, ±k close under multiplication and form the smallest group in which Hamilton’s minus sign lives.", "more": "The quaternion group Q₈ = {±1, ±i, ±j, ±k} is a non-abelian group of order eight, generated by Hamilton’s relations i² = j² = k² = ijk = −1. It and the symmetry group of the square are the only two non-abelian groups of order eight; every subgroup of Q₈ is normal, which makes it the model Hamiltonian group, a name Dedekind gave in 1897. Chapter ᚌ(G) holds the larger groups. Value 8, an integer, a factor of the denominator.", "cites": [ "r-22.16", "r-22.5" ], "links_to_cards": [ "ijk" ] }, { "id": "versor", "url": "https://www.teclamairme.com/#g-versor", "name": "a versor, a unit quaternion", "symbol": "‖(1 + i + j + k)/2‖", "chapter": 22, "type": "int", "badge": "Integer or rational", "quick": "1: Hamilton’s name for a quaternion of norm one; the versors form the 3-sphere S³, and every rotation of space is represented by exactly two of them, q and −q.", "more": "Hamilton coined versor in the 1840s, and fixed it in his Lectures on Quaternions of 1853, for a quaternion of norm one, a pure turning with no stretching, just as e^{iθ} is in the plane. (1 + i + j + k)/2 is a versor; as a rotation it turns space by 120° about the axis (1, 1, 1), cycling i → j → k → i (verified by computation). The versors form the unit 3-sphere S³ ⊂ ℝ⁴, which is the Lie group SU(2), the double cover of the rotation group SO(3). A key entry, not in the factor: it is the unit that Hamilton’s rotation formula rides on. Value 1, an integer.", "cites": [ "r-22.10", "r-22.11" ], "links_to_cards": [ "s3su2" ] }, { "id": "pauli", "url": "https://www.teclamairme.com/#g-pauli", "name": "the Pauli matrices", "symbol": "σ₁, σ₂, σ₃", "chapter": 22, "type": "obj", "badge": "Non-real object", "quick": "Three 2 × 2 complex matrices from Wolfgang Pauli’s 1927 theory of electron spin; sending i, j, k to −iσ₁, −iσ₂, −iσ₃, where the i in front is the complex √−1, turns Hamilton’s table into matrix multiplication.", "more": "σ₁ = [[0, 1], [1, 0]], σ₂ = [[0, −i], [i, 0]], σ₃ = [[1, 0], [0, −1]]: traceless, Hermitian, unitary, and σ₁σ₂ = iσ₃. Pauli introduced them in 1927 to describe the spin of the electron. The algebra generated by −iσ₁, −iσ₂, −iσ₃ is isomorphic to the quaternions, so ℍ sits inside the 2 × 2 complex matrices, and the spin of the electron can be described with quaternions; the same matrices return in Chapter ᚍ(K)’s Hilbert space. Non-real objects and a tool of the chapter, not in the factor.", "cites": [ "r-22.9" ], "links_to_cards": [ "hermitian", "hilbert", "quaternions" ] }, { "id": "s3su2", "url": "https://www.teclamairme.com/#g-s3su2", "name": "the unit quaternions as a group", "symbol": "S³ ≅ SU(2)", "chapter": 22, "type": "obj", "badge": "Non-real object", "quick": "The quaternions of norm 1 form the 3-sphere, which is the Lie group SU(2); it double-covers the rotation group SO(3), q and −q giving the same rotation.", "more": "The unit quaternions {q : ‖q‖ = 1} are the 3-sphere S³ in ℝ⁴, closed under multiplication, and as a group they are SU(2), the 2 × 2 unitary matrices of determinant 1. The map q ↦ (v ↦ q v q^{−1}) sends S³ onto the rotation group SO(3) two-to-one, since q and −q rotate identically; that double cover is why a spinor returns to itself only after 720°, and S³ = SU(2) is the Lie group whose Lie algebra su(2), of dimension 3, opens Chapter ᚌ(G)’s brace. A non-real object and a tool of the chapter, not in the factor.", "cites": [ "r-22.10" ], "links_to_cards": [ "liealg", "quaternions", "su2" ] }, { "id": "quaternions", "url": "https://www.teclamairme.com/#g-quaternions", "name": "the quaternions", "symbol": "ℍ = {a + bi + cj + dk : a,b,c,d ∈ ℝ}", "chapter": 22, "type": "obj", "badge": "Non-real object", "quick": "The four-dimensional real algebra generated by i² = j² = k² = ijk = −1: associative, with every nonzero element invertible, but multiplication is not commutative.", "more": "A quaternion has the form a + bi + cj + dk with real coefficients. Addition is coordinatewise, while Hamilton’s relations determine multiplication; in particular ij = k but ji = −k. Multiplication remains associative, and conjugation gives q·q̄ = ‖q‖² and q⁻¹ = q̄/‖q‖² for q ≠ 0. Thus ℍ is a four-dimensional real division algebra, the second rung of the ladder, after ℂ: it keeps division but gives up commutativity. Quaternions were influential in mathematical physics until vector analysis displaced them in many applications. A non-real object and the ambient number system of this chapter.", "cites": [ "r-22.15", "r-22.2", "r-22.3" ], "links_to_cards": [ "ijk" ] }, { "id": "assoc0", "url": "https://www.teclamairme.com/#g-assoc0", "name": "an associator with a repeated entry", "symbol": "(e₁e₁)e₂ − e₁(e₁e₂)", "chapter": 23, "type": "int", "badge": "Integer or rational", "quick": "0 by the alternative law, a surviving associativity promise the octonions keep.", "more": "The associator [a, b, c] = (ab)c − a(bc) measures the failure of associativity. An algebra is alternative when the associator vanishes whenever two of its three arguments coincide, so that (aa)b = a(ab) and (ba)a = b(aa); the octonions, found by John Graves in December 1843 and independently by Arthur Cayley, who printed them in 1845, are alternative, so (e₁e₁)e₂ − e₁(e₁e₂) is exactly 0. Artin’s theorem goes further: any two octonions generate an associative subalgebra, which is why octonion powers are unambiguous. A real number computed in 𝕆, hence the integer color.", "cites": [ "r-23.1", "r-23.2", "r-23.3" ], "links_to_cards": [ "octonions" ] }, { "id": "assoc", "url": "https://www.teclamairme.com/#g-assoc", "name": "an associator of three independent units", "symbol": "(e₁e₂)e₄ − e₁(e₂e₄)", "chapter": 23, "type": "obj", "badge": "Non-real object", "quick": "Nonzero, equal to 2e₇ under the Cayley–Dickson table: octonions do not associate.", "more": "With three independent imaginary units the associator does not vanish: under the Cayley–Dickson multiplication table, (e₁e₂)e₄ − e₁(e₂e₄) = 2e₇, verified by computation. It is a nonzero octonion, tinted as a non-real object, and it has an inverse because the octonions are a division algebra: the norm is multiplicative, so every nonzero octonion has an inverse (Graves, 1843), and 0 divided by it is 0. Hurwitz’s theorem of 1898 says only that there are no further algebras of the kind. Under the Fano-plane labeling in Baez’s 2002 survey the triple e₁, e₂, e₄ is a quaternion line and a nonzero associator comes instead from e₁, e₂, e₃: (e₁e₂)e₃ − e₁(e₂e₃) = −2e₆; the book uses the Cayley–Dickson table so that this chapter and Chapter ᚗ(V) agree.", "cites": [ "r-23.19", "r-23.4", "r-23.6", "r-23.7" ], "links_to_cards": [ "cdrule", "hurwitz", "octonions" ] }, { "id": "moufang", "url": "https://www.teclamairme.com/#g-moufang", "name": "a Moufang identity, evaluated", "symbol": "(e₄e₁)(e₂e₄) − e₄((e₁e₂)e₄)", "chapter": 23, "type": "int", "badge": "Integer or rational", "quick": "0: the octonions obey Ruth Moufang’s identities, of which (zx)(yz) = z((xy)z) is one, and this instance vanishes even though e₁, e₂, e₄ do not associate.", "more": "Ruth Moufang introduced in 1935 the structures now called Moufang loops; the nonzero octonions, though not a group, satisfy identities such as (zx)(yz) = z((xy)z), z(x(zy)) = ((zx)z)y, and ((xz)y)z = x(z(yz)), and the unit octonions are the standard example. With x = e₁, y = e₂, z = e₄, whose ordinary associator is 2e₇, the Moufang difference is exactly 0 under the Cayley–Dickson table (verified by computation). Exactly 0, an integer, riding beside the alternative law.", "cites": [ "r-23.15", "r-23.5" ], "links_to_cards": [ "cdrule", "octonions" ] }, { "id": "onorm", "url": "https://www.teclamairme.com/#g-onorm", "name": "the norm of the all-ones octonion, squared", "symbol": "‖1 + e₁ + … + e₇‖2", "symbol_markup": "‖1 + e₁ + … + e₇‖^{2}", "chapter": 23, "type": "int", "badge": "Integer or rational", "quick": "8: the sum of eight squares, and the octonion norm is multiplicative, which is the eight-square identity of Degen, Graves, and Cayley.", "more": "The octonion norm ‖x‖² = x x̄ = x₀² + … + x₇² is multiplicative, ‖xy‖ = ‖x‖‖y‖, and that statement is Degen’s eight-square identity, found by Carl Ferdinand Degen around 1818 and rediscovered by Graves in his letter to Hamilton of 26 December 1843 and by Cayley in 1845. Hurwitz proved in 1898 that such an identity exists only for 1, 2, 4, and 8 squares. For the all-ones octonion the norm squared is 8 (verified by computation). Value 8, an integer, a factor of the denominator.", "cites": [ "r-23.2", "r-23.6", "r-23.7" ], "links_to_cards": [] }, { "id": "fano", "url": "https://www.teclamairme.com/#g-fano", "name": "the Fano plane", "symbol": "PG(2, 2)", "chapter": 23, "type": "obj", "badge": "Non-real object", "quick": "The smallest projective plane, seven points and seven lines with three points on each line: the multiplication table of the seven imaginary octonion units.", "more": "Gino Fano described the seven-point plane in 1892. Label its points e₁ … e₇ and orient each of its seven lines; then each line is a copy of the quaternion units i, j, k, and the whole octonion multiplication table is read off the picture. The book uses the Cayley–Dickson labeling, in which the lines are {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6} (read off the computed products); Baez’s 2002 survey uses a different labeling, with the same geometry. Its symmetry group has order 168, which is also the number of signed zero-divisor pairs Chapter ᚗ(V) counts. A non-real object and a tool of the chapter.", "cites": [ "r-23.10", "r-23.19", "r-23.4", "r-23.9", "r-24.8" ], "links_to_cards": [ "cdrule" ] }, { "id": "cayleyint", "url": "https://www.teclamairme.com/#g-cayleyint", "name": "units of the integral octonions", "symbol": "240", "chapter": 23, "type": "int", "badge": "Integer or rational", "quick": "The integral octonions (Coxeter’s integral Cayley numbers) have exactly 240 elements of norm 1, and they sit at the 240 roots of E₈.", "more": "Dickson (1923) and Kirmse (1924) studied octonions with integer-like coordinates, Kirmse with an error that Coxeter corrected in 1946: there are seven maximal orders, not eight, all isometric to the E₈ lattice scaled by 1/√2, and each has exactly 240 elements of norm 1, forming a Moufang loop of order 240. Those 240 points are the root system of E₈, whose 248-dimensional Lie algebra is in Chapter ᚌ(G)’s denominator, and those same 240 vectors realize the kissing number in eight dimensions, which Levenshtein, and independently Odlyzko and Sloane, fixed at 240 in 1979. Viazovska proved the E₈ sphere packing itself optimal in 2016. A key entry, not in the factor. Value 240, an integer.", "cites": [ "r-23.11", "r-23.12" ], "links_to_cards": [ "e8", "liealg", "octonions", "sqrt2" ] }, { "id": "hurwitz", "url": "https://www.teclamairme.com/#g-hurwitz", "name": "Hurwitz’s theorem", "symbol": "1, 2, 4, 8", "chapter": 23, "type": "int", "badge": "Integer or rational", "quick": "The only dimensions, 1, 2, 4, and 8, in which a normed division algebra over the reals exists: ℝ, ℂ, ℍ, 𝕆, and no more.", "more": "Adolf Hurwitz proved in 1898 that a sum-of-squares identity (x₁² + … + xₙ²)(y₁² + … + yₙ²) = z₁² + … + zₙ² with bilinear z exists only for n = 1, 2, 4, 8, and hence that the normed division algebras over ℝ are exactly ℝ, ℂ, ℍ, and 𝕆; the general classification of composition algebras appeared posthumously in 1923. Frobenius had shown in 1877 that the associative ones are ℝ, ℂ, ℍ. The Cayley–Dickson ladder keeps climbing past 8, but from the sedenions on the norm stops multiplying and zero divisors appear, which is Chapter ᚗ(V). A key entry, four integers.", "cites": [ "r-23.14", "r-23.18", "r-23.7" ], "links_to_cards": [ "cdrule", "reals", "sedenions", "zd" ] }, { "id": "cross7", "url": "https://www.teclamairme.com/#g-cross7", "name": "the seven-dimensional cross product", "symbol": "e₁ × e₂ = e₃", "chapter": 23, "type": "obj", "badge": "Non-real object", "quick": "The imaginary octonions carry a bilinear cross product, orthogonal to both factors and of length ‖x‖‖y‖ sin θ, x × y = Im(xy) for pure imaginary x, y; such a product exists only in three and seven dimensions.", "more": "A bilinear product on ℝⁿ that is orthogonal to its two factors and has length ‖x‖‖y‖ sin θ exists only for n = 3 and n = 7 (Brown and Gray 1967, Massey 1983; the restriction is related to Hurwitz’s theorem); the three-dimensional one comes from the quaternions and the seven-dimensional one from the octonions, as x × y = Im(xy) = ½(xy − yx) for imaginary x, y. Under the Cayley–Dickson table e₁ × e₂ = e₃ and, as always, x × x = 0; unlike its three-dimensional cousin it fails the Jacobi identity and is preserved only by G₂, not by all of SO(7). A non-real object and a tool of the chapter.", "cites": [ "r-23.13" ], "links_to_cards": [ "cdrule", "g2", "hurwitz", "jacobi", "octonions", "quaternions" ] }, { "id": "octonions", "url": "https://www.teclamairme.com/#g-octonions", "name": "the octonions", "symbol": "𝕆 = {x₀ + x₁e₁ + ··· + x₇e₇ : xᵢ ∈ ℝ}", "chapter": 23, "type": "obj", "badge": "Non-real object", "quick": "The eight-dimensional real normed division algebra 𝕆: alternative but not associative, the last Cayley–Dickson algebra with a multiplicative norm and no zero divisors.", "more": "An octonion has eight real coordinates. The third Cayley–Dickson doubling gives the ladder ℝ → ℂ → ℍ → 𝕆: every nonzero octonion is invertible and the norm is multiplicative, but multiplication is not associative. The octonions remain alternative, so every subalgebra generated by two elements is associative. Hurwitz’s theorem makes 𝕆 the last real normed division algebra; the next doubling, the sedenions, has zero divisors and loses norm multiplicativity. A non-real object and the ambient number system of this chapter.", "cites": [ "r-23.1", "r-23.14", "r-23.3", "r-23.7" ], "links_to_cards": [ "cdrule", "hurwitz", "sedenions", "zd" ] }, { "id": "zd", "url": "https://www.teclamairme.com/#g-zd", "name": "sedenion zero divisors", "symbol": "(e₃+e₁₀)(e₆−e₁₅)", "chapter": 24, "type": "int", "badge": "Integer or rational", "quick": "Two nonzero sedenions whose product is 0, impossible in ℝ, ℂ, ℍ, or 𝕆.", "more": "The sedenions are the fourth Cayley–Dickson doubling, and they have zero divisors: nonzero elements whose product is zero. The chapter’s example, (e₃ + e₁₀)(e₆ − e₁₅) = 0 under the Cayley–Dickson convention, is verified by exact integer computation; there are 84 ways to split a set of four units into two pairs so that some product (e_{a} ± e_{b})(e_{c} ± e_{d}) vanishes, 42 four-unit sets with two splittings each, and each splitting vanishes for exactly two of its four sign patterns, giving 168 signed unordered pairs and 336 when factor order is counted. Guillermo Moreno showed in 1998 that the space of pairs of norm-one sedenions that multiply to zero is homeomorphic to the compact form of the exceptional Lie group G₂, the same group that is the octonions’ symmetry group in Chapter ᚖ(W). The product is exactly 0, a rational number computed in 𝕊, hence the integer color.", "cites": [ "r-24.2", "r-24.9" ], "links_to_cards": [ "cdrule", "g2", "homeo", "octonions", "sedenions" ] }, { "id": "zdnorm", "url": "https://www.teclamairme.com/#g-zdnorm", "name": "norms of the two zero divisors", "symbol": "‖e₃+e₁₀‖·‖e₆−e₁₅‖", "chapter": 24, "type": "int", "badge": "Integer or rational", "quick": "√2 · √2 = 2, while the product of the two factors has norm 0: the sedenion norm is not multiplicative.", "more": "The norm of a sedenion is the square root of the sum of the squares of its sixteen coordinates, so ‖e₃ + e₁₀‖ = ‖e₆ − e₁₅‖ = √2 and the product of the norms is 2. Hurwitz’s theorem of 1898 says the norm is multiplicative only in dimensions 1, 2, 4, and 8, and this pair is the demonstration: the product of the norms is 2 while the norm of the product is 0. Value 2, an integer, and a factor of the denominator, so it must be nonzero, which it is.", "cites": [ "r-24.4", "r-24.5" ], "links_to_cards": [ "hurwitz", "sqrt2" ] }, { "id": "powassoc", "url": "https://www.teclamairme.com/#g-powassoc", "name": "power associativity survives", "symbol": "(aa)a − a(aa), a = e₃ + e₁₀", "chapter": 24, "type": "int", "badge": "Integer or rational", "quick": "0: the sedenions are no longer alternative, but powers of a single element are still unambiguous, so (aa)a = a(aa); they are also flexible, (ab)a = a(ba).", "more": "Each Cayley–Dickson doubling loses something: ℂ loses order, ℍ commutativity, 𝕆 associativity, 𝕊 the alternative law and the absence of zero divisors. What every rung keeps is power associativity, a·a·a meaning one thing however it is bracketed, and flexibility, (ab)a = a(ba); Schafer proved in 1954 that all Cayley–Dickson algebras are flexible, and all of them are power-associative. For a = e₃ + e₁₀, one of the zero divisors, (aa)a − a(aa) = 0 (verified by exact integer computation). Exactly 0 in 𝕊, read as the scalar 0, an integer, riding beside the zero divisors.", "cites": [ "r-24.6", "r-24.7" ], "links_to_cards": [ "cdrule", "sedenions", "zd" ] }, { "id": "zd84", "url": "https://www.teclamairme.com/#g-zd84", "name": "the sedenion zero-divisor pairs", "symbol": "84", "chapter": 24, "type": "int", "badge": "Integer or rational", "quick": "The number of ways to choose the two unit pairs {a, b} and {c, d} so that (ea ± eb)(ec ± ed) can vanish, ignoring factor order and which ± signs do it: 42 four-unit sets, each split into two pairs in two different ways.", "quick_markup": "The number of ways to choose the two unit pairs {a, b} and {c, d} so that (e_{a} ± e_{b})(e_{c} ± e_{d}) can vanish, ignoring factor order and which ± signs do it: 42 four-unit sets, each split into two pairs in two different ways.", "more": "Wikipedia’s table lists 84 zero divisors of the shape (e_{a} ± e_{b})(e_{c} ± e_{d}) = 0 among the fifteen imaginary sedenion units. My own exact count agrees, and finds that the 84 come from only 42 distinct four-element sets, none of which contains the real unit: each set, like {e₃, e₁₀, e₆, e₁₅}, splits into a vanishing product in exactly two of its three possible ways, (e₃ + e₁₀)(e₆ − e₁₅) and (e₃ + e₁₅)(e₆ + e₁₀), and each of those splittings vanishes for exactly two of its four sign patterns, which is where the 168 signed pairs and the 336 ordered ones come from. Robert de Marrais’s 2000 catalog reaches the same two numbers by another route: 42 “assessor” planes, each spanned by two units, carrying 168 unit zero divisors, four to a plane. Moreno’s 1998 theorem describes the space of pairs of norm-one sedenions that multiply to zero as homeomorphic to the compact form of G₂. Value 84, an integer, a factor of the denominator.", "cites": [ "r-24.12", "r-24.8", "r-24.9" ], "links_to_cards": [ "g2", "homeo", "sedenions", "zd" ] }, { "id": "cdrule", "url": "https://www.teclamairme.com/#g-cdrule", "name": "the Cayley–Dickson doubling rule", "symbol": "(a, b)(c, d) = (ac − d̄b, da + bc̄)", "chapter": 24, "type": "fn", "badge": "Function or operator", "quick": "The one recipe that builds every rung of the ladder: a pair from the rung below multiplies by this rule, with the conjugate (a, b)̄ = (ā, −b).", "more": "Leonard Dickson gave the general doubling in 1919, showing how the octonions can be built as a two-dimensional algebra over the quaternions; the construction carries Arthur Cayley’s name as well, for the octonions he printed in 1845. Applied to ℝ it gives ℂ, then ℍ, 𝕆, 𝕊, the 32-dimensional trigintaduonions, and so on forever; the book uses this exact rule, with this sign convention, for every product it computes in Chapters ᚔ(I) to ᚗ(V), which is why its octonion table differs from Baez’s Fano labeling. A function, not a value.", "cites": [ "r-24.10", "r-24.15" ], "links_to_cards": [ "octonions", "quaternions", "trig" ] }, { "id": "trig", "url": "https://www.teclamairme.com/#g-trig", "name": "the trigintaduonions", "symbol": "𝕋", "chapter": 24, "type": "obj", "badge": "Non-real object", "quick": "The 32-dimensional rung above the sedenions, and the ladder keeps going: 64, 128, and every power of two above, each with zero divisors.", "more": "Doubling the sedenions gives a 32-dimensional algebra, the trigintaduonions, written 𝕋, and the construction never stops, giving an algebra of dimension 2ⁿ for every n. From the sedenions on, every rung has zero divisors (the same enumeration that gives the sedenions their 84 gives 1,260 here, which is the number Wikipedia reports, derived from 147 distinct triples), is neither alternative nor associative nor commutative, and keeps power associativity, flexibility, and distributivity. A non-real object, the first rung the book does not climb.", "cites": [ "r-24.1", "r-24.11" ], "links_to_cards": [ "powassoc", "sedenions", "zd" ] }, { "id": "sedenions", "url": "https://www.teclamairme.com/#g-sedenions", "name": "the sedenions", "symbol": "𝕊 = {(a, b) : a,b ∈ 𝕆}", "chapter": 24, "type": "obj", "badge": "Non-real object", "quick": "The sixteen-dimensional Cayley–Dickson algebra built from pairs of octonions: noncommutative, nonassociative, and nonalternative, with zero divisors, but still power-associative and flexible.", "more": "A sedenion is an ordered pair of octonions and therefore has sixteen real coordinates. It is the fourth Cayley–Dickson doubling of ℝ, after ℂ, ℍ, and 𝕆. Its multiplication is noncommutative, nonassociative, and nonalternative; zero divisors defeat cancellation and unique division, so 𝕊 is not a division algebra. It remains power-associative and flexible, and doubling continues to 32 dimensions and beyond. A non-real object and the ambient algebra of this chapter.", "cites": [ "r-24.1", "r-24.2", "r-24.6" ], "links_to_cards": [ "cdrule", "octonions", "zd" ] }, { "id": "adicid", "url": "https://www.teclamairme.com/#g-adicid", "name": "a carry that never ends", "symbol": "…2223 + 1", "symbol_markup": "…222_{3} + 1", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "Adding 1 sends a carry to infinity and leaves all zeros; arguably the cleanest zero in the piece.", "more": "The numerator of the chapter: …222₃ plus one. Adding 1 to the last digit gives 10 in base three, write 0 and carry 1; the carry then does the same to every digit in turn and never lands, so every digit is 0 and the result is exactly 0. Read digit by digit, no limit is needed, and the 3-adic distance agrees: a truncation with n twos is 3ⁿ − 1, plus one is 3ⁿ, and 3ⁿ shrinks to zero 3-adically. That is why I call it the cleanest zero in the piece. It is the integer 0 and is tinted as one.", "cites": [ "r-25.3" ], "links_to_cards": [ "threeadic" ] }, { "id": "tenadic", "url": "https://www.teclamairme.com/#g-tenadic", "name": "the 10-adic automorphic pair", "symbol": "…890625 + …109376", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "Each squares to itself, they sum to exactly 1, and they multiply to 0.", "more": "The 10-adic automorphic numbers, …890625 and …109376, each of which equals its own square; they arise because 10 = 2·5 is not prime, so ℤ₁₀ splits into a 2-adic and a 5-adic part and has idempotents other than 0 and 1. They are the limits of the automorphic truncations 5, 25, 625, 0625, 90625, 890625, … and 6, 76, 376, 9376, 09376, 109376, …. The one-digit cases 5 and 6 were already called circular numbers in later antiquity and the Middle Ages. Their sum is exactly 1 and their product exactly 0, both checkable digit by digit (890625 + 109376 = 1000001 and 890625 × 109376 ends in six zeros). Here the pair hands back the integer 1, which is why it is tinted as an integer although each member lives in ℤ₁₀.", "cites": [ "r-25.22", "r-25.6", "r-25.7" ], "links_to_cards": [ "nines10" ] }, { "id": "adicabs", "url": "https://www.teclamairme.com/#g-adicabs", "name": "3-adic absolute value of 27", "symbol": "|27|3", "symbol_markup": "|27|_{3}", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "1/27, because in the p-adics the divisible numbers are the small ones; its inverse is 27.", "more": "The 3-adic absolute value |n|₃ = 3^(−v₃(n)), where v₃(n) is the exponent of 3 in n. Hensel built the p-adic numbers from 1897; Josef Kürschák founded valuation theory, announcing it at the Cambridge congress of 1912 and putting his axioms for the absolute value in print the year after, and Alexander Ostrowski proved in 1916 that these p-adic absolute values and the ordinary one are, up to equivalence, the only nontrivial absolute values the rationals have. So |27|₃ = 1/27 and the reciprocal used in the formula is 27; since the ordinary absolute value of 27 is also 27, the product formula reads |27|₃ · |27|_∞ = (1/27) · 27 = 1. A rational number, tinted as one.", "cites": [ "r-25.1", "r-25.10", "r-25.18", "r-25.2", "r-25.5" ], "links_to_cards": [ "prodformula" ] }, { "id": "sqrt2_7", "url": "https://www.teclamairme.com/#g-sqrt2_7", "name": "a 7-adic square root of two, squared", "symbol": "α², α = √2 in ℤ₇", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "√2 exists in the 7-adics because 3² = 9 ≡ 2 mod 7, and its square is 2.", "more": "A square root of 2 in the 7-adic integers, squared. Since 3² = 9 ≡ 2 (mod 7), Hensel’s lemma, whose monic special case he stated in his Neue Grundlagen der Arithmetik of 1904, lifts the residue 3 to an exact 7-adic root of x² = 2, whose expansion begins 3 + 1·7 + 2·7² + … (checked digit by digit: 10² = 100 ≡ 2 mod 49, 108² = 11664 ≡ 2 mod 343); the rationals have no such root at all. Its square is exactly 2, which is what comes home, so the item is tinted as an integer although the root itself lives in ℤ₇.", "cites": [ "r-25.8", "r-25.9" ], "links_to_cards": [ "Zp", "hensel", "sqrt2" ] }, { "id": "val3_27", "url": "https://www.teclamairme.com/#g-val3_27", "name": "the 3-adic valuation of 27", "symbol": "v3(27)", "symbol_markup": "v_{3}(27)", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "3, because 27 = 3³; the exponent that makes |27|₃ = 3⁻³.", "more": "The 3-adic valuation of 27, which is 3 because 27 = 3³. It is the integer that the absolute value |27|₃ = 3^(−3) is built from, and it sits in the denominator beside that absolute value’s reciprocal so that both faces of the p-adic ruler are on show: the exponent and the size. A positive integer, so it keeps the denominator away from zero.", "cites": [ "r-25.2", "r-25.5" ], "links_to_cards": [ "adicabs" ] }, { "id": "prodformula", "url": "https://www.teclamairme.com/#g-prodformula", "name": "the product formula, saying nothing", "symbol": "|27|3 · |27|∞ − 1", "symbol_markup": "|27|_{3} · |27|_{∞} − 1", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "All the absolute values of a nonzero rational number, multiplied together, give 1, so this is 0: (1/27)·27 − 1.", "more": "The product formula: for any nonzero rational x, the product of |x|_p over every prime p, times the ordinary absolute value |x|_∞, is exactly 1, because each prime’s absolute value cancels that prime’s contribution to |x|_∞. For x = 27 only the prime 3 contributes, so |27|₃ · |27|_∞ = (1/27) · 27 = 1 and the rider is exactly 0. The formula is classical for ℚ and was made an axiom of global fields by Artin and Whaples in 1945; in that setting the real absolute value is also called the infinite prime. An integer 0.", "cites": [ "r-25.4", "r-25.5" ], "links_to_cards": [ "absp", "adicabs" ] }, { "id": "threeadic", "url": "https://www.teclamairme.com/#g-threeadic", "name": "the all-twos 3-adic integer", "symbol": "…2223", "symbol_markup": "…222_{3}", "chapter": 25, "type": "obj", "badge": "Non-real object", "quick": "Equal to −1 in ℤ₃, since Σ 2·3ᵏ = 2/(1 − 3).", "more": "The 3-adic integer with every digit 2. Kurt Hensel introduced the p-adic numbers in 1897, and in the 3-adic distance, where numbers are close when their difference is divisible by a high power of 3, the series 2 + 2·3 + 2·3² + … converges to 2/(1 − 3) = −1. It is a non-real object, which is why it is tinted as one: it lives in ℤ₃, not on the real line, and only the result of adding 1 to it comes home.", "cites": [ "r-25.1", "r-25.3" ], "links_to_cards": [ "Zp", "reals" ] }, { "id": "sqrtm1_5", "url": "https://www.teclamairme.com/#g-sqrtm1_5", "name": "a 5-adic square root of minus one", "symbol": "√−1 ∈ ℤ₅", "chapter": 25, "type": "obj", "badge": "Non-real object", "quick": "Exists because 5 ≡ 1 mod 4, so −1 is a square modulo 5 and Hensel lifts it.", "more": "A square root of −1 in the 5-adic integers. It exists because 5 ≡ 1 (mod 4), so −1 is a square modulo 5 (2² = 4 ≡ −1), and Hensel’s lemma lifts it: one root begins 2 + 1·5 + 2·5² + …. Named on the Hensel’s lemma card beside the 7-adic square root of 2, to show that which square roots exist depends on the prime; not in the formula. A non-real object.", "cites": [ "r-25.8", "r-25.9" ], "links_to_cards": [ "Zp", "hensel" ] }, { "id": "vp", "url": "https://www.teclamairme.com/#g-vp", "name": "the p-adic valuation", "symbol": "vp(r)", "symbol_markup": "v_{p}(r)", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "The exponent of the prime p in a nonzero r: the index of the first nonzero digit of its p-adic expansion.", "more": "The p-adic valuation v_p(r) is the exponent of p in the factorization of a nonzero rational r, equivalently the index of the first nonzero digit in its p-adic expansion; so v₃(27) = 3 and v₂(12) = 2. It is additive, v_p(rs) = v_p(r) + v_p(s), and obeys v_p(r + s) ≥ min(v_p(r), v_p(s)). Kurt Hensel introduced the p-adic numbers in 1897 and the valuation is the ruler he measured them with; the word valuation (Bewertung) comes from Josef Kürschák, who announced his first results at Cambridge in 1912 and published his axioms the year after; what he axiomatized is the absolute value rather than the exponent. A tool of the framework, not a value.", "cites": [ "r-25.1", "r-25.18", "r-25.2" ], "links_to_cards": [ "padicexp", "val3_27", "vp0" ] }, { "id": "vp0", "url": "https://www.teclamairme.com/#g-vp0", "name": "the valuation of zero", "symbol": "vp(0) = ∞", "symbol_markup": "v_{p}(0) = ∞", "chapter": 25, "type": "inf", "badge": "Infinity or limit device", "quick": "Zero has no first nonzero digit, so by convention its valuation is infinite, which makes |0|_p = p^(−∞) = 0 as it should be.", "more": "The convention v_p(0) = ∞: the p-adic expansion of 0 has no nonzero digit, so no index exists, and setting the valuation to infinity is the only choice that keeps the rules v_p(rs) = v_p(r) + v_p(s) and |r|_p = p^(−v_p(r)) true for r = 0, since then |0|_p = 0. It is a convention of the extended integers, not a number, which is why it wears the infinity tint, and it never enters the formula.", "cites": [ "r-25.11", "r-25.2" ], "links_to_cards": [ "absp", "padicexp" ] }, { "id": "absp", "url": "https://www.teclamairme.com/#g-absp", "name": "the p-adic absolute value", "symbol": "|r|p = p−vp(r)", "symbol_markup": "|r|_{p} = p^{−v_{p}(r)}", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "Size measured by divisibility: the more powers of p divide r, the smaller r is.", "more": "The p-adic absolute value, |r|_p = p^(−v_p(r)) for r ≠ 0 and |0|_p = 0. It satisfies |rs|_p = |r|_p |s|_p and the ultrametric inequality, and it makes the numbers divisible by high powers of p the small ones, which is the whole inversion of intuition behind this chapter. Kürschák axiomatized it, announcing his first results at Cambridge in 1912 and printing his axioms the year after. Completing the rationals with respect to it gives ℚ_p, exactly as completing with the ordinary absolute value gives ℝ. A tool of the framework.", "cites": [ "r-25.11", "r-25.18", "r-25.2" ], "links_to_cards": [ "Qp", "reals", "ultra" ] }, { "id": "ultra", "url": "https://www.teclamairme.com/#g-ultra", "name": "the ultrametric inequality", "symbol": "|x + y|p ≤ max(|x|p, |y|p)", "symbol_markup": "|x + y|_{p} ≤ max(|x|_{p}, |y|_{p})", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "The strong triangle inequality: every triangle in the p-adics is isosceles, and every point inside a ball is one of its centers.", "more": "The ultrametric, or strong triangle, inequality: |x + y|_p never exceeds the larger of |x|_p and |y|_p, with equality whenever the two differ. It replaces the ordinary triangle inequality and produces the strange geometry of ℚ_p: every triangle is isosceles, two balls are either nested or disjoint, and a series converges exactly when its terms tend to 0, which is why …222₃ has a sum at all. Marc Krasner introduced the name in a note in the Comptes Rendus in 1944. A rule of the framework, not a value.", "cites": [ "r-25.11", "r-25.19" ], "links_to_cards": [ "Qp", "absp", "threeadic" ] }, { "id": "Qp", "url": "https://www.teclamairme.com/#g-Qp", "name": "the field of p-adic numbers", "symbol": "ℚp", "symbol_markup": "ℚ_{p}", "chapter": 25, "type": "obj", "badge": "Non-real object", "quick": "The completion of the rationals under |·|_p, one field for each prime, as ℝ is the completion under the ordinary absolute value.", "more": "The field of p-adic numbers, ℚ_p: the completion of ℚ with respect to |·|_p, introduced by Kurt Hensel in 1897 by analogy with power series. Its elements are the expansions Σ aₖ pᵏ with finitely many negative k and digits 0 ≤ aₖ < p. By Ostrowski’s theorem of 1916, ℝ and the fields ℚ_p for each prime p are all the completions the rationals have, so each prime gets its own number line. A non-real object.", "cites": [ "r-25.1", "r-25.10", "r-25.12" ], "links_to_cards": [ "absp", "ostrowski", "reals", "sqrtm1_5" ] }, { "id": "Zp", "url": "https://www.teclamairme.com/#g-Zp", "name": "the ring of p-adic integers", "symbol": "ℤp", "symbol_markup": "ℤ_{p}", "chapter": 25, "type": "obj", "badge": "Non-real object", "quick": "The p-adic numbers with no negative powers: the closed unit ball |x|_p ≤ 1, where …222₃ lives.", "more": "The ring of p-adic integers, ℤ_p, the elements of ℚ_p with |x|_p ≤ 1, equivalently the expansions Σ aₖ pᵏ with k ≥ 0. It contains ℤ densely, it is compact, and every element of ℤ_p not divisible by p is invertible, so 1/2 is a 3-adic integer (…1112₃) while 1/3 is not. For a prime p it is an integral domain: the absolute value is multiplicative, so a product of nonzero elements is nonzero, and the only idempotents are 0 and 1. That is why the chapter’s automorphic pair lives in ℤ₁₀, where 10 is composite, and not here. It should not be confused with the integers modulo p, which are ℤ_p’s residue field; ℤ_p is the limit of ℤ/pᵏℤ over all k. A non-real object.", "cites": [ "r-25.11", "r-25.12", "r-25.6" ], "links_to_cards": [ "Qp", "absp", "threeadic" ] }, { "id": "padicexp", "url": "https://www.teclamairme.com/#g-padicexp", "name": "the p-adic expansion", "symbol": "Σ ak pk", "symbol_markup": "Σ a_{k} p^{k}", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "Digits running left forever instead of right: every p-adic number is a base-p numeral with infinitely many digits before the point.", "more": "The p-adic expansion, r = Σ aₖ pᵏ with 0 ≤ aₖ < p, the sum starting at k = v_p(r) and running to +∞. In the ordinary reals a base-p numeral runs off to the right, toward small powers; in ℚ_p it runs off to the left, toward large powers, which converge because |pᵏ|_p = p^(−k) → 0. The expansion is unique: unlike decimals, where 0.999… and 1 are two spellings of one number, no p-adic number has two expansions. Hensel’s 1897 construction was precisely this analogy with power series in a variable. A notation of the framework.", "cites": [ "r-25.1", "r-25.2" ], "links_to_cards": [ "Qp", "nines", "reals" ] }, { "id": "hensel", "url": "https://www.teclamairme.com/#g-hensel", "name": "Hensel’s lemma", "symbol": "Hensel’s lemma", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "A simple root modulo p lifts to an exact root in ℤ_p, which is why √2 exists in ℤ₇ and √−1 in ℤ₅.", "more": "Hensel’s lemma: if a polynomial with p-adic integer coefficients has a root modulo p at which its derivative is not zero modulo p, that root lifts to a unique root in ℤ_p, digit by digit, by the same Newton iteration used for real numbers. Kurt Hensel stated a special case of it, for monic polynomials, in Neue Grundlagen der Arithmetik in 1904, and the monicity restriction came off in his Theorie der algebraischen Zahlen of 1908. It is the tool that puts √2 in ℤ₇ (from 3² ≡ 2 mod 7) and √−1 in ℤ₅ (from 2² ≡ −1 mod 5). A tool of the framework.", "cites": [ "r-25.8", "r-25.9" ], "links_to_cards": [ "Zp", "reals", "sqrt2" ] }, { "id": "ostrowski", "url": "https://www.teclamairme.com/#g-ostrowski", "name": "Ostrowski’s theorem", "symbol": "Ostrowski’s theorem", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "Every nontrivial absolute value on the rationals is equivalent to the ordinary one or to one |·|_p: the reals and the p-adics are the only completions there are.", "more": "Ostrowski’s theorem, proved by Alexander Ostrowski in 1916: every nontrivial absolute value on ℚ is equivalent either to the usual absolute value or to a p-adic absolute value for exactly one prime p. So the rationals have exactly one completion per prime, plus ℝ, and no others; the primes together with the point at infinity are the places of ℚ. A theorem of the framework.", "cites": [ "r-25.10", "r-25.12" ], "links_to_cards": [ "absp", "reals" ] }, { "id": "hasse", "url": "https://www.teclamairme.com/#g-hasse", "name": "the local–global principle", "symbol": "Hasse–Minkowski", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "A quadratic form with rational coefficients has a zero other than the origin exactly when it has one in ℝ and in every ℚ_p: check every prime and infinity, and the rationals follow.", "more": "The Hasse–Minkowski theorem, the local-to-global principle for quadratic forms: a quadratic form with rational coefficients represents zero over ℚ if and only if it does so over ℝ and over every ℚ_p. Helmut Hasse proved it for ℚ in his 1921 thesis, extending Hermann Minkowski’s work, and published the extension to all number fields in 1924. It was among the first results to show the wider mathematical world what the p-adic numbers were for. A theorem of the framework.", "cites": [ "r-25.13", "r-25.20" ], "links_to_cards": [ "Qp", "reals" ] }, { "id": "monsky", "url": "https://www.teclamairme.com/#g-monsky", "name": "Monsky’s theorem", "symbol": "Monsky’s theorem", "chapter": 25, "type": "fn", "badge": "Function or operator", "quick": "A square cannot be cut into an odd number of triangles of equal area, and the only known proof uses the 2-adic valuation.", "more": "Monsky’s theorem, proved by Paul Monsky in 1970 after Fred Richman posed the problem in 1967: a square cannot be dissected into an odd number of triangles of equal area. The proof colors the plane by the 2-adic valuations of the coordinates, applies Sperner’s lemma, and finds a triangle whose area has negative 2-adic valuation, which 1/n cannot have when n is odd. No proof avoiding the p-adics is known, which makes it the cleanest example of the 2-adic ruler settling a question about ordinary geometry. A theorem of the framework.", "cites": [ "r-25.14", "r-25.21" ], "links_to_cards": [] }, { "id": "nines10", "url": "https://www.teclamairme.com/#g-nines10", "name": "the 10-adic all-nines, plus one", "symbol": "…99910 + 1", "symbol_markup": "…999_{10} + 1", "chapter": 25, "type": "int", "badge": "Integer or rational", "quick": "The same carry as the numerator, run in base ten: …999 is −1 in ℤ₁₀, so this is 0.", "more": "The 10-adic number with every digit 9, plus 1: the carry runs left forever and leaves all zeros, so …999 = −1 in ℤ₁₀, the mirror image of 0.999… = 1 in the reals. It is the base-ten twin of the chapter’s numerator and is here as a key entry so a reader can see the trick in the base they already know. Its value is the integer 0.", "cites": [ "r-25.3" ], "links_to_cards": [ "nines", "reals" ] }, { "id": "epsomega", "url": "https://www.teclamairme.com/#g-epsomega", "name": "the infinite and the infinitesimal, canceled", "symbol": "ε·ω − 1", "chapter": 26, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0, because the surreals are a field.", "more": "The numerator: ε times ω, minus 1. Because the surreal numbers are a field and ε = 1/ω, the product is exactly 1 and the difference exactly 0. The value is the integer 0, which is why it is tinted as one even though it was computed among the surreals.", "cites": [ "r-26.3", "r-26.5" ], "links_to_cards": [ "No" ] }, { "id": "omm1", "url": "https://www.teclamairme.com/#g-omm1", "name": "one less than infinity", "symbol": "ω − 1", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "ω minus one, still infinite: a surreal that ordinal arithmetic cannot even write down.", "more": "One less than infinity: {0, 1, 2, … | ω}, born the day after ω. Among the ordinals there is no such thing, because ω has no immediate predecessor; among the surreals it is an ordinary infinite number, and ω − 1 + 1 = ω. A non-real object, and positive.", "cites": [ "r-26.4", "r-26.6" ], "links_to_cards": [ "No", "ordarith" ] }, { "id": "sqrtomega", "url": "https://www.teclamairme.com/#g-sqrtomega", "name": "the square root of omega", "symbol": "√ω", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "A surreal whose square is ω.", "more": "A surreal whose square is ω, which exists because Conway’s field is real-closed: every positive surreal has a square root. It is infinite but smaller than ω/n for every n, and it has no counterpart among the ordinals. A non-real object.", "cites": [ "r-26.3", "r-26.6" ], "links_to_cards": [ "ordarith" ] }, { "id": "omom", "url": "https://www.teclamairme.com/#g-omom", "name": "omega to the omega", "symbol": "ωω", "symbol_markup": "ω^{ω}", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "ω raised to the power ω in the surreals: still a number, still nonzero.", "more": "ω raised to the power ω in the surreals, an infinite number vastly larger than ω, still a member of the same ordered field with a reciprocal of its own. The ordinal ω^ω exists too, in Chapter ᚚ(P)’s sense, but only the surreal one can be divided by. A non-real object, positive.", "cites": [ "r-26.3", "r-26.6" ], "links_to_cards": [ "No" ] }, { "id": "halfS", "url": "https://www.teclamairme.com/#g-halfS", "name": "one half, born on day two", "symbol": "{0 | 1}", "chapter": 26, "type": "int", "badge": "Integer or rational", "quick": "The simplest number between 0 and 1.", "more": "One half, written as Conway writes it: {0 | 1}, the simplest number strictly between 0 and 1, born on day two. It is the same ½ as everywhere else, which is why it is tinted as a rational; it sits in the denominator to show that the surreal field contains the ordinary numbers as well as the strange ones.", "cites": [ "r-26.2" ], "links_to_cards": [ "No", "sbirthday" ] }, { "id": "omhalf", "url": "https://www.teclamairme.com/#g-omhalf", "name": "half of infinity", "symbol": "ω/2", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "Half of ω: a surreal that is still infinite.", "more": "Half of ω, {0, 1, 2, … | ω, ω − 1, ω − 2, …}, an infinite surreal that is exactly half of the first infinite number. Ordinal arithmetic has no ordinal that added to itself gives ω; surreal arithmetic can, because it is a field. A non-real object, positive.", "cites": [ "r-26.3", "r-26.6" ], "links_to_cards": [ "omm1", "ordarith" ] }, { "id": "surzero", "url": "https://www.teclamairme.com/#g-surzero", "name": "the surreal zero, born on day zero", "symbol": "{ | }", "chapter": 26, "type": "int", "badge": "Integer or rational", "quick": "The first surreal number, with nothing on either side; every other number is built from it, and it is 0.", "more": "The surreal number { | }, born on day 0: the empty set on the left and the empty set on the right. Conway’s construction, found around 1970 and set out in Conway’s book of 1976, makes every number from two sets of earlier numbers, so on the first day there are no earlier numbers and the only possibility is this one, which behaves as 0 (x + { | } = x for every x). It is the framework’s own zero, exactly 0, and it rides in the numerator as such.", "cites": [ "r-26.18", "r-26.2" ], "links_to_cards": [ "sbirthday" ] }, { "id": "surgap", "url": "https://www.teclamairme.com/#g-surgap", "name": "a gap in the surreals", "symbol": "", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "A cut {L | R} whose two sides use up the whole of No: at least one of them is then a proper class, so a gap is not a surreal number at all, and Conway’s ∞ is one of them.", "more": "Conway defines a gap in No as a pair {L | R} in which every member of L is less than every member of R and the two sides between them exhaust the surreal numbers; because at least one side is then a proper class, a gap is not a number. Gaps are where the surreals’ missing least upper bounds live. There is no least positive infinite surreal, but the gap between the surreals that lie below some whole number and those above every whole number is greater than every real and less than every positive infinite surreal, and it is the least upper bound of the reals in the completion of No; On = {No | } is the gap above every surreal, and ∞ is named among the gaps as well. That is the distinction this chapter turns on: ω is a surreal number and divides like any other, while Conway’s ∞ is a gap, not a number, and the extended-real +∞ of the book’s left side is an adjoined symbol, not a field element. A non-real object, and not a number.", "cites": [ "r-26.14", "r-26.19" ], "links_to_cards": [ "No", "lrnotation", "reals" ] }, { "id": "hyper", "url": "https://www.teclamairme.com/#g-hyper", "name": "the hyperreal standard part", "symbol": "st(1/H)", "chapter": 26, "type": "int", "badge": "Integer or rational", "quick": "For an infinite hyperreal H, 1/H is infinitesimal and its standard part is 0; this H is a hyperreal, not the letter of a chapter.", "more": "The standard part of 1/H for an infinite hyperreal H. Abraham Robinson announced nonstandard analysis in a 1961 paper and gave the full treatment in his 1966 book Non-standard Analysis, making Leibniz’s infinitesimals rigorous nearly three centuries later; the hyperreals *ℝ are built as an ultrapower of ℝ, and Jerzy Łoś’s theorem of 1955 is what transfers first-order truth between ℝ and *ℝ, once the quantifiers are read as ranging over internal sets. 1/H is infinitesimal and the standard-part map st rounds every finite hyperreal to its nearest real, so st(1/H) = 0; this is how nonstandard analysis defines a derivative, as st((f(x + dx) − f(x))/dx). The value is the integer 0.", "cites": [ "r-26.7", "r-26.8", "r-26.9" ], "links_to_cards": [ "hyperreals", "infinitesimal", "logicsyms", "reals", "stpart" ] }, { "id": "hyperH", "url": "https://www.teclamairme.com/#g-hyperH", "name": "a positive infinite hyperreal", "symbol": "H", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "A fixed infinite element of the hyperreal field *ℝ; nonzero and invertible there, and not the extended-real symbol ∞.", "more": "H denotes a fixed positive infinite hyperreal, an element of *ℝ larger than every ordinary real, such as the class of (1, 2, 3, …) in the ultrapower construction. It is an ordinary nonzero element of an ordered field and so has a reciprocal 1/H, itself a positive infinitesimal; taking the standard part of that reciprocal is where this chapter’s second zero comes from. H is not the extended-real ∞, which is an adjoined symbol, not an element of a field, and it is not the letter of any chapter. A non-real object.", "cites": [ "r-26.14", "r-26.7" ], "links_to_cards": [ "hyperreals", "infinitesimal", "stpart" ] }, { "id": "omegaS", "url": "https://www.teclamairme.com/#g-omegaS", "name": "surreal omega", "symbol": "ω", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "{0, 1, 2, … | }, the first infinite surreal number.", "more": "The first infinite surreal, ω = {0, 1, 2, … | }, born on day ω of Conway’s construction, which he found around 1970 while analyzing Go endgames and published in On Numbers and Games in 1976; Knuth’s 1974 novelette named the numbers surreal. Every natural number is on its left and nothing on its right, so it exceeds them all. Unlike the ordinal ω of Chapter ᚚ(P) it has a reciprocal, a square root, a half, and a predecessor, because the surreals form a field. A non-real object.", "cites": [ "r-26.1", "r-26.18", "r-26.2", "r-7.16" ], "links_to_cards": [ "No", "sbirthday" ] }, { "id": "epsS", "url": "https://www.teclamairme.com/#g-epsS", "name": "surreal infinitesimal", "symbol": "ε", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "{0 | 1, ½, ¼, …} = 1/ω, smaller than every positive real but not zero.", "more": "The surreal infinitesimal ε = {0 | 1, ½, ¼, …}, born on day ω like its partner, greater than 0 and less than every positive real. Conway showed it equals 1/ω exactly, because the surreals form an ordered field, so ε·ω = 1. The letter ε is reused across the book: here a surreal, in Chapter ᚔ(I) a dual number with ε² = 0. A non-real object.", "cites": [ "r-26.10", "r-26.5" ], "links_to_cards": [ "No", "dual", "infinitesimal", "sbirthday" ] }, { "id": "lrnotation", "url": "https://www.teclamairme.com/#g-lrnotation", "name": "Conway’s bracket", "symbol": "{ L | R }", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "Every surreal number is a pair of sets of earlier surreals, a left set and a right set, with every left member less than every right member.", "more": "Conway’s notation {L | R}: a surreal number is defined by a left set L and a right set R of previously created numbers with no member of L greater than or equal to any member of R; the number is the simplest one lying between them. So {0 | } = 1, { | 0} = −1, {0 | 1} = ½, and {0, 1, 2, … | } = ω. Set out by Conway in On Numbers and Games (1976); the construction first appeared in print in Knuth’s 1974 novelette, which presents the same two rules carved on a stone found on a beach. A notation of the framework.", "cites": [ "r-26.2", "r-26.21", "r-26.24" ], "links_to_cards": [ "No", "halfS", "omegaS", "signexp" ] }, { "id": "sbirthday", "url": "https://www.teclamairme.com/#g-sbirthday", "name": "the birthday of a surreal", "symbol": "b(x)", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "The first day on which a number can be written: b(0) = 0, b(±1) = 1, b(½) = 2, b(ω) = ω.", "more": "The birthday b(x) of a surreal number, the ordinal on which it is first created; numbers born earlier are simpler, and the number defined by {L | R} is always the earliest-born number between L and R. The naturals are born on their own day, dyadic rationals on finite days, every other real on day ω, and ω and ε on day ω as well; ω − 1 and ω/2 wait for later days. The word as Conway’s On Numbers and Games of 1976 uses it. A function of the framework.", "cites": [ "r-26.2", "r-26.24", "r-26.6" ], "links_to_cards": [ "lrnotation", "omhalf", "omm1", "ordarith" ] }, { "id": "No", "url": "https://www.teclamairme.com/#g-No", "name": "the class of all surreal numbers", "symbol": "𝐍𝐨", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "Too big to be a set: it contains every real, every ordinal, and the field they generate.", "more": "Conway’s name for the surreal numbers as a whole, No, which is a proper class rather than a set because it contains every ordinal. It is a real-closed ordered field—strictly a Field, capital F, since it is a proper class—and in von Neumann–Bernays–Gödel set theory with global choice it is universal: every ordered field whose universe is a set or a proper class embeds in it, and up to isomorphism No is the only homogeneous universal ordered field. The ordinals sit inside it as the numbers with empty right set. Conway 1976. A non-real object.", "cites": [ "r-26.11", "r-26.16", "r-26.24" ], "links_to_cards": [] }, { "id": "omnific", "url": "https://www.teclamairme.com/#g-omnific", "name": "the omnific integers", "symbol": "𝐎𝐳", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "Conway’s integers among the surreals: ω and ω − 1 are integers here, while ½ and ε are not.", "more": "The omnific integers, Oz, first defined by Simon Norton and set out by Conway, are the generalization of the integers to the surreals: x is an omnific integer when x = {x − 1 | x + 1}. Every ordinal is one, so ω and ω^ω are integers in this sense—and so is ω − 1, which is no ordinal at all. Every surreal is the ratio of two of them, just as every rational is a ratio of two integers, so No is the field of fractions of Oz. Conway 1976. A non-real object.", "cites": [ "r-26.17", "r-26.24" ], "links_to_cards": [ "No", "ints", "omm1", "omom", "ordarith" ] }, { "id": "cnf", "url": "https://www.teclamairme.com/#g-cnf", "name": "Conway normal form", "symbol": "Σ ωyi ri", "symbol_markup": "Σ ω^{y_{i}} r_{i}", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "Every surreal is a decreasing sum of powers of ω with real coefficients, Cantor’s normal form for ordinals extended to a whole field.", "more": "The Conway normal form: every surreal number can be written uniquely as a sum of terms ω^y·r over a decreasing sequence of surreal exponents y with nonzero real coefficients r, possibly transfinitely many. It generalizes Cantor’s normal form for ordinals of 1897, and it is what makes ω − 1, ω/2, and √ω (which is ω^½) routine to write down. Conway 1976. A notation of the framework.", "cites": [ "r-26.11", "r-26.22", "r-26.24", "r-27.20" ], "links_to_cards": [ "omhalf", "omm1", "ordarith", "sqrtomega" ] }, { "id": "signexp", "url": "https://www.teclamairme.com/#g-signexp", "name": "the sign expansion of a surreal", "symbol": "sign expansion", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "A surreal number as a transfinite string of + and −: ½ is +−, ω is + repeated ω times, ε is + followed by ω minuses.", "more": "The sign expansion: every surreal number corresponds to a sequence of plus and minus signs indexed by an ordinal, its birthday being the length of the sequence; ½ is (+, −), ω is ω pluses, and ε is a plus followed by ω minuses. It is the surreal analogue of a binary expansion, and Conway’s rule of simplicity says {L | R} is the number whose sign expansion is the shortest one between L and R. Conway 1976; developed by Gonshor in 1986. A notation of the framework.", "cites": [ "r-26.12", "r-26.24" ], "links_to_cards": [ "lrnotation", "ordarith", "sbirthday" ] }, { "id": "hyperreals", "url": "https://www.teclamairme.com/#g-hyperreals", "name": "the hyperreal numbers", "symbol": "*ℝ", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "Robinson’s extension of the reals with infinitesimals and infinite numbers, built as an ultrapower and obeying the same first-order laws as the reals once the quantifiers are read as ranging over internal sets.", "more": "The hyperreal field *ℝ of Abraham Robinson’s nonstandard analysis (paper 1961, book 1966): an ordered field extending ℝ that contains infinitesimals and their infinite reciprocals. It is constructed as an ultrapower of ℝ, sequences of reals modulo a nonprincipal ultrafilter—whose existence needs the ultrafilter lemma, a weak form of the axiom of choice, and which cannot be constructed explicitly—an idea going back to Hewitt (1948) and Łoś (1955). Unlike the surreals it is a set, and, once the quantifiers are read as ranging over internal sets, it has the same first-order theory as ℝ by the transfer principle. A non-real object.", "cites": [ "r-26.11", "r-26.24", "r-26.7", "r-26.8" ], "links_to_cards": [ "No", "ac", "infinitesimal", "reals", "transfer" ] }, { "id": "transfer", "url": "https://www.teclamairme.com/#g-transfer", "name": "the transfer principle", "symbol": "transfer principle", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "A first-order statement holds of the reals exactly when its starred version holds of the hyperreals, quantifiers ranging over internal sets: that is what lets a proof with infinitesimals count as a proof about ℝ.", "more": "The transfer principle of nonstandard analysis: a formula holds in ℝ if and only if its starred transform holds in *ℝ, with quantification bounded and ranging over internal sets. The qualification is not a formality. Read without it, the principle would make *ℝ Archimedean, since “every positive number exceeds 1/n for some positive integer n” is true in ℝ; what carries over is the version in which n may be a hyperinteger. It is a consequence of Łoś’s theorem on ultraproducts (1955) and was formulated by Robinson in 1961 as the rigorous version of Leibniz’s law of continuity. It is the license under which st(1/H) = 0 turns into ordinary calculus. A theorem of the framework.", "cites": [ "r-26.7", "r-26.8" ], "links_to_cards": [ "hyper", "hyperreals", "infinitesimal", "reals" ] }, { "id": "stpart", "url": "https://www.teclamairme.com/#g-stpart", "name": "the standard part", "symbol": "st(x)", "chapter": 26, "type": "fn", "badge": "Function or operator", "quick": "Rounds every finite hyperreal to the unique real infinitely close to it; st(1/H) = 0 for infinite H.", "more": "The standard-part function st: every finite hyperreal x is infinitely close to exactly one real number, and st(x) is that real. It is the map that finishes a nonstandard calculation, discarding the infinitesimal error, so that the derivative is st((f(x + dx) − f(x))/dx). Robinson 1961–1966. A function of the framework.", "cites": [ "r-26.13", "r-26.7", "r-26.9" ], "links_to_cards": [ "hyper", "hyperreals", "infinitesimal" ] }, { "id": "infinitesimal", "url": "https://www.teclamairme.com/#g-infinitesimal", "name": "an infinitesimal", "symbol": "dx", "chapter": 26, "type": "obj", "badge": "Non-real object", "quick": "A number greater than 0 but smaller than every positive real: Leibniz’s dx, banished by Weierstrass and readmitted by Robinson.", "more": "An infinitesimal: a positive number smaller than 1/n for every natural n. Leibniz was using them by 1675, when he first wrote ∫ f(x)dx, and Euler freely; Weierstrass’s epsilon-delta limits, written down in his Berlin lectures of 1861 and anticipated by Bolzano in 1817 and Cauchy in the 1820s, removed them from analysis. Robinson’s hyperreals of 1961 and Conway’s surreals of 1976 both contain them legitimately, the surreal ε = 1/ω being one. The reals contain none, by the Archimedean property. A non-real object.", "cites": [ "r-26.10", "r-26.15", "r-26.20", "r-26.24", "r-26.7" ], "links_to_cards": [ "No", "hyperreals", "reals" ] }, { "id": "empty", "url": "https://www.teclamairme.com/#g-empty", "name": "cardinality of the empty set", "symbol": "|∅|", "chapter": 27, "type": "int", "badge": "Integer or rational", "quick": "0: the empty set has no members, so there is nothing to count.", "more": "The empty set has cardinality 0. Cantor gave his theory of cardinal numbers its systematic presentation and introduced aleph notation in 1895; André Weil later recalled choosing ∅ for Bourbaki after the Norwegian letter Ø. In von Neumann’s 1923 construction it is also the ordinal 0. The integer 0, tinted as one.", "cites": [ "r-27.1", "r-27.11", "r-27.20", "r-27.9" ], "links_to_cards": [ "vnzero" ] }, { "id": "aleph0", "url": "https://www.teclamairme.com/#g-aleph0", "name": "cardinality of the natural numbers", "symbol": "ℵ0", "symbol_markup": "ℵ_{0}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The smallest infinite size: the size of the counting numbers.", "more": "Aleph-null is the cardinality of the natural numbers and the smallest infinite cardinal. Cantor introduced aleph notation in 1895. A non-real object: a cardinal number, not a point on the real line.", "cites": [ "r-27.2", "r-27.20" ], "links_to_cards": [ "nats", "reals" ] }, { "id": "aleph1", "url": "https://www.teclamairme.com/#g-aleph1", "name": "the first uncountable cardinal", "symbol": "ℵ1", "symbol_markup": "ℵ_{1}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The next size after ℵ₀.", "more": "Aleph-one is the first uncountable cardinal, the next size after ℵ₀. It is the size of Cantor’s second number class, the countably infinite ordinals; whether it is the size of the real line is the continuum hypothesis. A non-real object.", "cites": [ "r-27.2", "r-27.20", "r-27.4" ], "links_to_cards": [ "aleph0", "ch", "reals" ] }, { "id": "cont", "url": "https://www.teclamairme.com/#g-cont", "name": "cardinality of the continuum", "symbol": "𝔠", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "2^ℵ₀, the size of the real line; if the usual axioms are consistent they neither prove nor refute 𝔠 = ℵ₁ (Gödel 1938, in full 1940; Cohen 1963).", "more": "The continuum has cardinality 𝔠 = 2^ℵ₀ and is the size of the real line and of the power set of the natural numbers. Cantor proved that it exceeds ℵ₀ and conjectured in 1878 that it equals ℵ₁. Gödel announced the relative consistency of that conjecture in the Proceedings of the National Academy of Sciences in 1938. He presented the proof in full in a Princeton monograph in 1940. Cohen established the relative consistency of its negation in 1963, making the continuum hypothesis independent of ZFC. A non-real object.", "cites": [ "r-27.20", "r-27.22", "r-27.3", "r-27.4", "r-8.11" ], "links_to_cards": [ "aleph0", "aleph1", "axioms", "ch", "chindep", "powerset", "reals" ] }, { "id": "bethomega", "url": "https://www.teclamairme.com/#g-bethomega", "name": "the ω-th beth number", "symbol": "ℶω", "symbol_markup": "ℶ_{ω}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The limit of ℶ₀, ℶ₁, ℶ₂, …, each the size of the power set of the one before.", "more": "The ω-th beth number is the limit of ℶ₀ = ℵ₀, ℶ₁ = 2^ℵ₀, ℶ₂ = 2^ℶ₁, …, each the size of the power set of the one before. Charles Sanders Peirce proposed the beth notation in a December 1900 letter to Cantor. The cardinal ℶ_{ω} is the smallest strong-limit cardinal above ℵ₀. A non-real object.", "cites": [ "r-27.19", "r-27.5" ], "links_to_cards": [ "aleph0", "cont", "ordomega", "powerset" ] }, { "id": "alephomega", "url": "https://www.teclamairme.com/#g-alephomega", "name": "the first singular cardinal", "symbol": "ℵω", "symbol_markup": "ℵ_{ω}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "In ZFC, the limit of ℵ₀, ℵ₁, ℵ₂, … and the smallest infinity that is the union of countably many smaller infinities.", "more": "Aleph-omega is the limit of the finite-indexed alephs and the first singular cardinal, because it is the union of countably many smaller cardinals. ZFC proves that ℵ_{ω} exists; assuming ZFC is consistent, it cannot prove that an inaccessible cardinal exists. König’s theorem implies 2^ℵ₀ ≠ ℵ_{ω}. A non-real object.", "cites": [ "r-27.17", "r-27.23", "r-27.6" ], "links_to_cards": [ "aleph0", "aleph1", "cont", "inaccessible", "zfc" ] }, { "id": "bijection", "url": "https://www.teclamairme.com/#g-bijection", "name": "a one-to-one correspondence", "symbol": "A ≈ B", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "Cantor’s test for same size: pair the two sets off with nothing left over on either side.", "more": "Two sets have the same cardinality exactly when a bijection pairs their members off with nothing left over on either side. On infinite sets the test behaves strangely: the square numbers are a proper part of the whole numbers and can still be paired with them one for one, which is Galileo’s paradox, and Dedekind made that property the definition of an infinite set; the even numbers pair off the same way, by doubling. It is the rule the whole chapter rests on. Type: a tool of the framework.", "cites": [ "r-27.1", "r-27.26" ], "links_to_cards": [] }, { "id": "ordzero", "url": "https://www.teclamairme.com/#g-ordzero", "name": "ordinal arithmetic’s zero", "symbol": "ω·0", "chapter": 27, "type": "int", "badge": "Integer or rational", "quick": "Ordinal multiplication by 0 gives 0 even for ω; this ω is the ordinal, not the surreal.", "more": "Ordinal multiplication gives ω·0 = 0: the right factor counts zero copies of ω. Cantor introduced the transfinite ordinals in 1883 and presented their arithmetic systematically in 1895–1897. Ordinal arithmetic is not commutative: 2·ω = ω but ω·2 = ω + ω. This ω is the ordinal used in this chapter. The value is the integer 0.", "cites": [ "r-27.20", "r-27.8" ], "links_to_cards": [ "ordomega" ] }, { "id": "vnzero", "url": "https://www.teclamairme.com/#g-vnzero", "name": "von Neumann’s zero", "symbol": "0 = ∅", "chapter": 27, "type": "int", "badge": "Integer or rational", "quick": "The ordinal 0 is the empty set itself, 1 is {∅}, 2 is {∅, {∅}}: every ordinal is the set of the ordinals before it.", "more": "In von Neumann’s 1923 definition, each ordinal is the set of all smaller ordinals, so 0 = ∅, 1 = {0}, 2 = {0, 1}, and ω = {0, 1, 2, …}. Zero is therefore not just of size zero but literally the empty set. It rides beside ω·0 as the brace’s other route to zero, and its value is the integer 0.", "cites": [ "r-27.9" ], "links_to_cards": [ "ordomega", "ordzero" ] }, { "id": "wellorder", "url": "https://www.teclamairme.com/#g-wellorder", "name": "a well-order and its order type", "symbol": "min T exists", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "An order in which every non-empty part has a first member. Two such orders have the same order type when they can be matched up in order—and an ordinal is that order type.", "more": "A well-order is a total order in which every non-empty subset has a least element. That single condition is what licenses induction and recursion past the finite: there is always a first counterexample to argue about. Every element except possibly a greatest one has an immediate successor, but an element other than the first may have no predecessor at all — ω is the standard case, and that is what makes it a limit. Two ordered sets have the same order type when they can be matched one for one in a way that preserves the order, and an ordinal simply is the order type of a well-ordered set. This is why ω + 1 and 1 + ω differ: adding a point at the end gives an order with a last element, adding one at the front does not change the shape at all. A property of orders, uncolored.", "cites": [ "r-27.27" ], "links_to_cards": [ "ordarith", "ordomega" ] }, { "id": "pset0", "url": "https://www.teclamairme.com/#g-pset0", "name": "the power set of nothing", "symbol": "|𝒫(∅)| − 1", "chapter": 27, "type": "int", "badge": "Integer or rational", "quick": "0: the empty set has exactly one subset, itself, so its power set has one member (2⁰ = 1), and 1 − 1 = 0.", "more": "The power set 𝒫(X) is the set of all subsets of X; for the empty set it has the single member ∅, so |𝒫(∅)| = 2⁰ = 1. Cantor’s theorem says that every power set is strictly larger than its base set, and in von Neumann’s construction 𝒫(∅) = {∅} is the ordinal 1. Subtracting 1 gives the integer 0.", "cites": [ "r-27.12", "r-27.3", "r-27.9" ], "links_to_cards": [ "cantorthm", "powerset" ] }, { "id": "powerset", "url": "https://www.teclamairme.com/#g-powerset", "name": "the power set", "symbol": "𝒫(X)", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "The set of all subsets of X; Cantor’s theorem says it is strictly bigger than X, whatever X is.", "more": "The power set 𝒫(X) is the set of all subsets of X, and for the empty set it has the single member ∅. Cantor’s theorem says 𝒫(X) cannot be put in bijection with X, so it is strictly larger for every X, which is what makes the beth ladder climb without end. Taking it of nothing gives a set with one member; subtracting 1 gives the chapter’s third zero. Type: a tool of the framework.", "cites": [ "r-27.12", "r-27.3" ], "links_to_cards": [ "bijection", "cantorthm" ] }, { "id": "initialhom", "url": "https://www.teclamairme.com/#g-initialhom", "name": "the unique map from the empty set", "symbol": "|HomSet(∅, X)|−1", "symbol_markup": "|Hom_{Set}(∅, X)|−1", "chapter": 27, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: for every set X there is one and only one function from ∅ to X, so the count is 1 and 1 − 1 = 0.", "more": "In the category Set, the empty set is initial: for every set X there is exactly one morphism ∅ → X, the empty function. Therefore |Hom_{Set}(∅, X)| − 1 = 0. Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper. An integer identity carrying a framework.", "cites": [ "r-27.13", "r-27.14" ], "links_to_cards": [ "category", "emptyfn" ] }, { "id": "ordomega", "url": "https://www.teclamairme.com/#g-ordomega", "name": "the first infinite ordinal", "symbol": "ω", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The order type of the natural numbers; it has no ordinal predecessor, while Chapter ᚙ(X)’s surreal ω − 1 is not an ordinal.", "more": "The first transfinite ordinal ω is the order type of 0, 1, 2, … and has no immediate predecessor. Ordinal addition gives 1 + ω = ω but ω + 1 > ω. Chapter ᚙ(X) instead uses surreal arithmetic, in which ω − 1 exists but is not an ordinal. A non-real object.", "cites": [ "r-27.10", "r-27.20", "r-27.8" ], "links_to_cards": [ "nats", "omm1", "ordarith" ] }, { "id": "ordarith", "url": "https://www.teclamairme.com/#g-ordarith", "name": "the ordinals after omega", "symbol": "ω + 1, ω·2, ω2, ωω", "symbol_markup": "ω + 1, ω·2, ω^{2}, ω^{ω}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "Counting past infinity: after all the naturals come ω, ω + 1, …, then ω·2, then ω², ω^ω and beyond, each a different well-ordering.", "more": "Ordinal arithmetic gives ω + 1 as ω followed by one more element, ω·2 as two copies of ω, ω² as ω copies of ω, and ω^ω as the limit of ω, ω², ω³, …. Cantor developed these order types from 1883 through his 1895–1897 papers. Addition and multiplication are not commutative because they describe orderings rather than sizes; all of these examples are countable. A collection of non-real objects.", "cites": [ "r-27.20", "r-27.8" ], "links_to_cards": [ "ordomega" ] }, { "id": "epsilon0ord", "url": "https://www.teclamairme.com/#g-epsilon0ord", "name": "epsilon nought", "symbol": "ε0", "symbol_markup": "ε_{0}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The first ordinal α with ω^α = α, the limit of ω, ω^ω, ω^(ω^ω), …; Gentzen used induction up to it in his consistency proof for arithmetic.", "more": "The ordinal ε₀ is the least α > 0 satisfying ω^α = α, equivalently the limit of the tower ω, ω^ω, ω^(ω^ω), …; it is countable. Gentzen used transfinite induction up to ε₀ in his 1936 consistency proof for Peano arithmetic. This ε is unrelated to Chapter ᚙ(X)’s infinitesimal. A non-real object.", "cites": [ "r-27.18" ], "links_to_cards": [ "infinitesimal", "ordarith", "ordomega" ] }, { "id": "omega1", "url": "https://www.teclamairme.com/#g-omega1", "name": "the first uncountable ordinal", "symbol": "ω1", "symbol_markup": "ω_{1}", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The set of all countable ordinals, which is itself uncountable; its cardinality is ℵ₁.", "more": "The first uncountable ordinal ω₁ is the set of all countable ordinals; its infinite members form Cantor’s second number class. Every ordinal below it is countable, no countable sequence of smaller ordinals reaches it, and its cardinality is ℵ₁. A non-real object.", "cites": [ "r-27.2", "r-27.20", "r-27.34" ], "links_to_cards": [ "aleph1", "ordarith" ] }, { "id": "cantorthm", "url": "https://www.teclamairme.com/#g-cantorthm", "name": "Cantor’s theorem", "symbol": "2κ > κ", "symbol_markup": "2^{κ} > κ", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "The power set of any set is strictly bigger than the set, so there is no largest infinity.", "more": "Cantor’s theorem says that every set X has fewer members than its power set: 2^|X| > |X|. Cantor proved it with the diagonal method in an 1891/92 paper. It creates the endless beth ladder and rules out a largest cardinal. A theorem of the framework.", "cites": [ "r-27.20", "r-27.3" ], "links_to_cards": [ "diagonal", "powerset" ] }, { "id": "diagonal", "url": "https://www.teclamairme.com/#g-diagonal", "name": "the diagonal argument", "symbol": "Cantor’s diagonal argument", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "List the reals however you like; change the n-th digit of the n-th number and you have built a real not on the list.", "more": "Cantor’s diagonal argument, published at the start of the 1890s, builds from any list of infinite binary sequences a sequence that differs from its n-th entry in the n-th place. The new sequence is absent from the list, so the binary sequences—and hence the reals—are uncountable. A method of the framework.", "cites": [ "r-27.20", "r-27.3" ], "links_to_cards": [ "reals" ] }, { "id": "cardarith", "url": "https://www.teclamairme.com/#g-cardarith", "name": "cardinal arithmetic", "symbol": "κ + λ = κ·λ = max(κ, λ)", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "For infinite cardinals in ZFC, addition and multiplication return the larger size; among the basic operations, only exponentiation can make a larger one.", "more": "In ZFC, if κ and λ are infinite cardinals, then κ + λ = κ·λ = max(κ, λ). Among addition, multiplication, and exponentiation, only exponentiation can produce a larger infinite cardinal; also 0·κ = 0. Cardinal multiplication has no general inverse, which is why this chapter’s formula adds its braces instead of dividing them. A rule of the framework.", "cites": [ "r-27.7" ], "links_to_cards": [ "zfc" ] }, { "id": "ch", "url": "https://www.teclamairme.com/#g-ch", "name": "the continuum hypothesis", "symbol": "𝔠 = ℵ1", "symbol_markup": "𝔠 = ℵ_{1}", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "Is the size of the real line the very next infinity after the naturals? If the usual axioms are consistent, they neither prove it nor refute it.", "more": "The continuum hypothesis is Cantor’s conjecture of 1878, in modern notation 2^ℵ₀ = ℵ₁, and became Hilbert’s first problem in 1900. Gödel’s 1940 monograph showed that it cannot be refuted from ZFC if ZFC is consistent, and Cohen showed in 1963 that it cannot be proved if ZFC is consistent. A statement of the framework, independent of ZFC.", "cites": [ "r-27.20", "r-27.22", "r-27.28", "r-27.4", "r-8.13" ], "links_to_cards": [ "aleph1", "axioms", "chindep", "cont", "reals", "zfc" ] }, { "id": "zfc", "url": "https://www.teclamairme.com/#g-zfc", "name": "the standard axioms of set theory", "symbol": "ZFC", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "The standard first-order set theory: the Zermelo–Fraenkel axioms plus Choice.", "more": "Zermelo–Fraenkel set theory with Choice, ZFC, grew from Zermelo’s 1908 axiomatization through work by Fraenkel, Skolem, and von Neumann. It is the standard first-order framework relative to which the continuum hypothesis is, if ZFC is consistent, independent. A framework, not a value.", "cites": [ "r-27.15", "r-27.28", "r-27.4" ], "links_to_cards": [ "axioms", "ch" ] }, { "id": "ac", "url": "https://www.teclamairme.com/#g-ac", "name": "the axiom of choice", "symbol": "AC", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "Given any collection of nonempty sets, you may pick one element from each; harmless for finitely many, powerful and strange for infinitely many.", "more": "The axiom of choice says that a choice function exists for every collection of nonempty sets. Zermelo isolated it in his 1904 proof that every set can be well ordered; Gödel and Cohen later showed that it is independent of ZF, assuming ZF is consistent. A statement of the framework.", "cites": [ "r-27.15" ], "links_to_cards": [] }, { "id": "russell", "url": "https://www.teclamairme.com/#g-russell", "name": "Russell’s paradox", "symbol": "{x | x ∉ x}", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "The set of all sets that are not members of themselves: it is a member of itself exactly when it is not, so no such set exists.", "more": "Russell’s paradox asks for the collection R of all sets not members of themselves; then R ∈ R exactly when R ∉ R. The contradiction helped force set theory into an axiomatic form and shows why some collections are proper classes rather than sets. A contradiction of the naive framework.", "cites": [ "r-27.16" ], "links_to_cards": [] }, { "id": "inaccessible", "url": "https://www.teclamairme.com/#g-inaccessible", "name": "large cardinals", "symbol": "κ inaccessible", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "Assuming ZFC is consistent, it cannot prove that these cardinals exist; a strongly inaccessible cardinal is uncountable, regular, and a strong limit.", "more": "A cardinal κ is strongly inaccessible when it is uncountable, regular (not the union of fewer than κ smaller cardinals), and a strong limit: 2^λ < κ for every λ < κ. Assuming ZFC is consistent, Gödel’s second incompleteness theorem prevents ZFC from proving that such a cardinal exists. That is why none stands in the product. A non-real object, if it exists at all.", "cites": [ "r-27.17" ], "links_to_cards": [ "godel", "zfc" ] }, { "id": "category", "url": "https://www.teclamairme.com/#g-category", "name": "a category", "symbol": "ob(C), hom(C)", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "Objects and the arrows between them, with a composition rule and an identity arrow on every object.", "more": "A category is a class of objects, a class of morphisms or arrows between them, an associative composition ∘: hom(b, c) × hom(a, b) → hom(a, c), and an identity arrow on every object. The objects are atomic in the definition—nothing is assumed about what is inside them—so they can only be characterized by their relations to other objects, which is exactly what the arrows record. Eilenberg and Mac Lane introduced categories, functors, and natural transformations in their 1945 paper. Type: a tool of the framework.", "cites": [ "r-27.13", "r-27.24" ], "links_to_cards": [] }, { "id": "initialobj", "url": "https://www.teclamairme.com/#g-initialobj", "name": "an initial object", "symbol": "I → X", "chapter": 27, "type": "fn", "badge": "Function or operator", "quick": "An object with exactly one arrow to every object of its category.", "more": "An initial object of a category C is an object I such that for every object X of C there is precisely one morphism I → X; a terminal object has exactly one morphism the other way. Two initial objects of the same category are related by a unique isomorphism. In Set the empty set is the unique initial object, and its single arrow to each X is the empty function, which is where this chapter’s fourth zero comes from. Type: a tool of the framework.", "cites": [ "r-27.14", "r-27.25" ], "links_to_cards": [ "category", "emptyfn" ] }, { "id": "emptyfn", "url": "https://www.teclamairme.com/#g-emptyfn", "name": "the empty function", "symbol": "∅ → X", "chapter": 27, "type": "obj", "badge": "Non-real object", "quick": "The empty function is the unique function whose domain is the empty set, one for every possible codomain X, since there are no elements to assign anywhere and so nothing that could go wrong.", "more": "For any set X there is exactly one function ∅ → X, vacuously satisfying the definition of a function because it has no inputs to map incorrectly; this is why the empty set is the initial object of the category Set, with the empty function as its single outgoing arrow to every object. It is a genuine function even though its graph is the empty set. In the other direction there is usually no function at all: a function X → ∅ exists only when X is itself empty. The count |Hom(∅, X)| = 1 for every X is what makes the empty set initial; the terminal objects of Set, with exactly one arrow in from every set, are the one-element sets, so no object of Set is both initial and terminal.", "cites": [ "r-27.35" ], "links_to_cards": [ "category", "initialobj" ] }, { "id": "dbn", "url": "https://www.teclamairme.com/#g-dbn", "name": "de Bruijn–Newman constant", "symbol": "ΛdBN", "symbol_markup": "Λ_{dBN}", "chapter": 28, "type": "open", "badge": "Status open", "quick": "A definite real number nobody knows: it is 0 exactly when the Riemann Hypothesis is true, and otherwise lies in (0, 0.2].", "more": "The de Bruijn–Newman constant. Nicolaas de Bruijn introduced in 1950 a one-parameter heat-flow deformation of the Riemann ξ function and showed that for parameters ≥ ½ all its zeros are real; Charles Newman proved in 1976 that there is a single real constant Λ such that the zeros are real exactly when the parameter is ≥ Λ, and conjectured Λ ≥ 0, remarking that the Riemann Hypothesis, if true, is only barely so. The Riemann Hypothesis is equivalent to Λ ≤ 0. Brad Rodgers and Terence Tao proved Λ ≥ 0 (arXiv 2018), and Platt and Trudgian’s verification of the hypothesis up to height 3·10¹² gives Λ ≤ 0.2 (preprint 2020, published 2021). So Λ is a definite real in [0, 0.2] that is exactly 0 if and only if the hypothesis holds; its status is open, which is its color, and because it may be zero it can never sit in a denominator.", "cites": [ "r-28.3", "r-28.4" ], "links_to_cards": [ "rh", "xifn" ] }, { "id": "lE37", "url": "https://www.teclamairme.com/#g-lE37", "name": "the L-function of the rank-one curve at one", "symbol": "L(E₃₇, 1)", "chapter": 28, "type": "int", "badge": "Integer or rational", "quick": "Exactly 0: the curve’s root number is −1; LMFDB records analytic rank 1 and algebraic rank 1, so the rank equality predicted by Birch and Swinnerton-Dyer holds for this curve.", "more": "The L-function of the elliptic curve of conductor 37, y² + y = x³ − x (LMFDB 37.a1), the smallest-conductor curve of rank one, evaluated at s = 1. Its root number is −1, so L(E, s) vanishes at s = 1 to odd order and L(E₃₇, 1) is exactly 0. LMFDB records analytic rank 1 and algebraic rank 1 for the curve, so the equality of analytic and algebraic ranks predicted by Birch and Swinnerton-Dyer holds here. The value is the integer 0, which is its color, and it is the zero that makes the whole epilogue vanish.", "cites": [ "r-19.8", "r-28.5" ], "links_to_cards": [ "E37", "LfuncE", "dirichletL", "rankE" ] }, { "id": "poincare", "url": "https://www.teclamairme.com/#g-poincare", "name": "the Poincaré conjecture, as a bit", "symbol": "[every simply connected closed 3-manifold is S³]", "chapter": 28, "type": "int", "badge": "Integer or rational", "quick": "Conjectured in 1904, proven by Perelman in 2002–2003, the only Millennium Problem solved; the bracket is 1.", "more": "The Poincaré conjecture as an Iverson bracket: 1 if every simply connected closed 3-manifold is the 3-sphere, 0 otherwise. Henri Poincaré posed it in 1904 after finding a counterexample to his own earlier version; Grigori Perelman proved it in 2002–2003 using Hamilton’s Ricci flow, in three papers posted online and never sent to a journal, and declined the 2006 Fields Medal and the 2010 Clay prize. The bracket is the integer 1, the only Millennium bit whose value is known.", "cites": [ "r-28.6" ], "links_to_cards": [ "iverson", "ricci", "threemfld" ] }, { "id": "pnp", "url": "https://www.teclamairme.com/#g-pnp", "name": "the P versus NP question, as a bit", "symbol": "[P = NP]", "chapter": 28, "type": "int", "badge": "Integer or rational", "quick": "A definite integer, 0 or 1, that nobody knows: 1 if every problem whose answer can be checked quickly can also be solved quickly, 0 if not. Most complexity theorists expect 0.", "more": "The P versus NP question as an Iverson bracket. Stephen Cook formulated it in 1971, Leonid Levin independently in work published in 1973, and Richard Karp showed in 1972 that twenty-one natural problems are all equally hard, which is why the question matters: P is what can be solved quickly, NP what can be checked quickly, and [P = NP] is 1 if they coincide and 0 if not. It is a definite integer nobody knows, so it is colored as an integer rather than as an open real; finite either way, it multiplies in safely and can never sit in a denominator. Most complexity theorists expect 0, in which case this brace joins L(E₃₇, 1) in killing the product; if 1, it is a factor of one and changes nothing.", "cites": [ "r-28.17", "r-28.7" ], "links_to_cards": [ "iverson", "lE37" ] }, { "id": "hodge", "url": "https://www.teclamairme.com/#g-hodge", "name": "the Hodge conjecture, as a bit", "symbol": "[every Hodge class is algebraic]", "chapter": 28, "type": "int", "badge": "Integer or rational", "quick": "A definite integer, 0 or 1, that nobody knows: 1 if, on every smooth projective complex variety, every rational cohomology class of type (p, p) comes from algebraic subvarieties, 0 if some class does not.", "more": "The Hodge conjecture as an Iverson bracket. W. V. D. Hodge stated it in 1950: on a smooth projective complex variety, every rational cohomology class of type (p, p) is a rational combination of classes of algebraic subvarieties. Atiyah and Hirzebruch showed in work published in 1962 that the version with integer coefficients is false, which is why the bracket must say rational. The natural alternative, a count of the Hodge classes that are not algebraic taken over all varieties, could be infinite if the conjecture fails, and the piece forbids an infinity beside a zero; the bracket is the only shape that obeys the rules. A definite integer, 0 or 1.", "cites": [ "r-28.21", "r-28.8" ], "links_to_cards": [ "cohomology", "hodgeclass", "iverson", "smoothproj" ] }, { "id": "nsrate", "url": "https://www.teclamairme.com/#g-nsrate", "name": "the Navier–Stokes blow-up rate", "symbol": "1/T∗", "symbol_markup": "1/T_{∗}", "chapter": 28, "type": "open", "badge": "Status open", "quick": "A definite real number, zero or positive, that nobody knows: the least upper bound of the reciprocal smooth lifetimes of unforced three-dimensional flows started from data of enstrophy at most one, and 0 exactly when no such flow ever breaks down.", "more": "The Navier–Stokes blow-up rate, a number built for this book from the Clay problem. Claude-Louis Navier (1822) and George Stokes (1845) wrote the equations of viscous flow; Jean Leray proved in 1934 that smooth solutions exist for a short time and weak solutions exist forever, and the prize asks whether smooth solutions can always continue for all time in three dimensions. Definition: take every smooth, divergence-free, rapidly decaying starting velocity on ℝ³ with unit viscosity and enstrophy at most one (the integral of the squared velocity gradient); let T(u₀) be how long the smooth solution from u₀ lasts; set 1/T∗ := sup 1/T(u₀), reading 1/∞ as 0. Leray’s local theory gives every such flow a lifetime of at least a universal c > 0, so the number lies in [0, 1/c] and is finite. The equations’ scaling symmetry shrinks enstrophy while stretching time, so any blow-up anywhere rescales to one inside the unit ball, and, by this book’s own argument, 1/T∗ = 0 if and only if the Clay statement of existence and smoothness holds on ℝ³; without the unit-enstrophy normalization the same scaling would push the earliest blow-up time to 0 and the rate to infinity, which is why the definition must be stated carefully. Written as a supremum of reciprocals so that T∗, which may be infinite, is never divided by. In September 2026 a finite-time singularity was announced for a forced flow and Clay marked the problem’s page active; the official problem is four separate statements, and that result bears on the forced ones, while this number is a supremum over flows with no force term. Status open, like its twin Λ_dBN.", "cites": [ "r-28.10", "r-28.22", "r-28.26", "r-28.9" ], "links_to_cards": [ "dbn", "enstrophy", "weaksol" ] }, { "id": "ymbit", "url": "https://www.teclamairme.com/#g-ymbit", "name": "the complete Yang–Mills problem, as a bit", "symbol": "[a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has Δ>0]", "chapter": 28, "type": "int", "badge": "Integer or rational", "quick": "A definite integer, 0 or 1, that nobody knows: it includes both the rigorous existence of a non-trivial theory on ℝ⁴ for every compact simple gauge group and a positive mass gap, so it is never undefined.", "more": "The official Clay problem asks for a non-trivial quantum Yang–Mills theory on ℝ⁴ for every compact simple gauge group, satisfying rigorous axioms, and for a mass gap Δ>0. A raw Δ presupposes the theory’s construction; without it a quotient such as Δ/m_{p} would be undefined. The Iverson bracket is total: 1 if the complete statement is true, 0 otherwise. Its value remains unknown, but it is always a finite integer and never appears in a denominator.", "cites": [ "r-28.11", "r-28.12" ], "links_to_cards": [ "iverson", "mp", "ymgap" ] }, { "id": "ymgap", "url": "https://www.teclamairme.com/#g-ymgap", "name": "the Yang–Mills mass gap", "symbol": "Δ", "chapter": 28, "type": "open", "badge": "Status open", "quick": "The mass of the lightest particle of a pure Yang–Mills theory in four dimensions; the conjecture is that it is positive, and lattice computations put it near 1.7 GeV for SU(3).", "more": "The Yang–Mills mass gap. Chen Ning Yang and Robert Mills wrote the non-abelian gauge theory in 1954. The Clay problem asks for a rigorous construction of a non-trivial quantum Yang–Mills theory for every compact simple gauge group and a proof of a positive gap Δ. A numerical Δ presupposes that construction, so the book keeps Δ in prose and represents the complete prize problem with a total Iverson bit. Lattice glueball computations remain physical motivation, not a proof of the Clay statement. Status open.", "cites": [ "r-28.11", "r-28.19", "r-28.26" ], "links_to_cards": [ "glueball", "iverson" ] }, { "id": "mp", "url": "https://www.teclamairme.com/#g-mp", "name": "proton mass", "symbol": "mp", "symbol_markup": "m_{p}", "chapter": 28, "type": "pmeas", "badge": "Physics: measured", "quick": "The measured rest mass of the proton, 938.272 MeV/c² (CODATA 2022). It enters the formula only through the proton-to-electron mass ratio mp/me; against the Yang–Mills mass gap Δ it is a comparison in prose, never a divisor, since no Δ has been shown to exist.", "more": "The proton’s rest mass is 938.27208943(29) MeV/c² in CODATA 2022. The book does not divide an unconstructed mass gap by this value; the proton remains a useful physical comparison in prose. Physics: measured.", "cites": [ "r-28.13" ], "links_to_cards": [ "ymgap" ] }, { "id": "critline", "url": "https://www.teclamairme.com/#g-critline", "name": "the critical line", "symbol": "Re(s) = ½", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "Where the Riemann Hypothesis says every nontrivial zero of ζ lies; more than 10¹² of them have been checked and all are there.", "more": "The critical line Re(s) = ½ in the complex plane, on which Riemann conjectured in 1859 that every nontrivial zero of ζ(s) lies. Platt and Trudgian rigorously verified (preprint 2020, published 2021) that every zero up to height 3·10¹² is on it. The de Bruijn–Newman constant is 0 exactly when all the rest lie on it too. A locus of the framework, not a value.", "cites": [ "r-18.17", "r-18.28", "r-28.14", "r-28.26" ], "links_to_cards": [ "cplane", "dbn", "rh", "zeta" ] }, { "id": "rankE", "url": "https://www.teclamairme.com/#g-rankE", "name": "the rank of an elliptic curve", "symbol": "rank(E)", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "How many independent rational points of infinite order a curve has; the Birch and Swinnerton-Dyer conjecture ties it to the L-function.", "more": "The rank of an elliptic curve over ℚ, the number of independent rational points of infinite order in its Mordell–Weil group, which Mordell proved finitely generated in 1922. The curve of conductor 37 has rank 1. The Birch and Swinnerton-Dyer conjecture says the rank equals the order of vanishing of L(E, s) at s = 1. An integer-valued invariant of the framework.", "cites": [ "r-19.19", "r-28.15" ], "links_to_cards": [ "E37", "LfuncE", "bsdconj", "mordellweil" ] }, { "id": "E37", "url": "https://www.teclamairme.com/#g-E37", "name": "the curve of conductor 37", "symbol": "y² + y = x³ − x", "chapter": 28, "type": "obj", "badge": "Non-real object", "quick": "The smallest-conductor elliptic curve of rank one, LMFDB 37.a1, whose L-function vanishes at 1.", "more": "The elliptic curve y² + y = x³ − x, Cremona’s 37a1 and LMFDB’s 37.a1: conductor 37, discriminant 37, rank 1, no torsion, generated by the point (0, 0). It is the first curve in conductor order whose rank is positive; LMFDB lists its invariants for the Birch and Swinnerton-Dyer conjecture, and it is the curve this epilogue brings back from Chapter ᚒ(U). A non-real object.", "cites": [ "r-19.8", "r-28.15", "r-28.5" ], "links_to_cards": [ "LfuncE", "bsdconj", "rankE" ] }, { "id": "ricci", "url": "https://www.teclamairme.com/#g-ricci", "name": "Ricci flow", "symbol": "∂g/∂t = −2 Ric", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "Hamilton’s 1982 equation that lets a shape’s metric evolve like heat; Perelman drove it through its singularities to prove Poincaré.", "more": "Ricci flow, the equation ∂g/∂t = −2 Ric(g) introduced by Richard Hamilton in 1982, which evolves the metric of a manifold so that curvature spreads out like heat. Hamilton’s program was to use the flow, which usually deforms a manifold towards a rounder shape, to prove the conjecture for every simply connected closed 3-manifold; Perelman’s three papers of 2002–2003 controlled the singularities with his entropy and surgery and completed it, proving both the Poincaré conjecture and Thurston’s geometrization. An operator of the framework.", "cites": [ "r-28.16", "r-28.26" ], "links_to_cards": [ "poincare", "threemfld" ] }, { "id": "npcomplete", "url": "https://www.teclamairme.com/#g-npcomplete", "name": "the hardest problems in NP", "symbol": "NP-complete", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "Problems, such as satisfiability, to which every problem in NP reduces in polynomial time: solve one quickly and P = NP.", "more": "The NP-complete problems: those in NP to which every other NP problem reduces in polynomial time, so a fast algorithm for any one of them would give P = NP. Cook’s 1971 result established Boolean satisfiability as NP-complete; Karp used it in 1972 to show twenty other natural problems NP-complete, including graph coloring, knapsack, and the Hamiltonian circuit. P is the class solvable in polynomial time, NP the class checkable in polynomial time; the bracket [P = NP] is 1 exactly when the two coincide. A class of the framework.", "cites": [ "r-28.17", "r-28.26" ], "links_to_cards": [ "pnp" ] }, { "id": "hodgeclass", "url": "https://www.teclamairme.com/#g-hodgeclass", "name": "a Hodge class", "symbol": "Hp,p(X) ∩ H2p(X, ℚ)", "symbol_markup": "H^{p,p}(X) ∩ H^{2p}(X, ℚ)", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "On a smooth projective complex variety, a rational cohomology class of type (p, p): algebraic subvarieties always produce such classes, and the conjecture says every one of them is a rational combination of those.", "more": "A Hodge class on a smooth projective complex variety X is a rational cohomology class in H^(2p)(X, ℚ) whose complexification has Hodge type (p, p). The fundamental class of an algebraic subvariety of codimension p is always of type (p, p), and the Hodge conjecture says every rational (p, p)-class is a rational combination of such fundamental classes. An object of the framework.", "cites": [ "r-28.18" ], "links_to_cards": [ "cohomology", "hodge", "smoothproj" ] }, { "id": "weaksol", "url": "https://www.teclamairme.com/#g-weaksol", "name": "a weak solution of Navier–Stokes", "symbol": "Leray–Hopf solution", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "A flow that satisfies the equations in an averaged sense and, from finite-energy data, exists forever (Leray 1934); whether it is also smooth is the prize.", "more": "A weak, or Leray–Hopf, solution of the Navier–Stokes equations: a divergence-free velocity field of finite energy that satisfies the equations when tested against smooth functions and obeys the energy inequality. Jean Leray proved in 1934 that such a solution exists for all time from finite-energy data on ℝ³. Smooth solutions have been proved to exist only for a short time, or for all time when the data are small; the Clay problem asks whether the weak solutions are in fact smooth, or whether some smooth flow blows up. An object of the framework.", "cites": [ "r-28.9" ], "links_to_cards": [] }, { "id": "enstrophy", "url": "https://www.teclamairme.com/#g-enstrophy", "name": "enstrophy", "symbol": "∫|∇u|² dx", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "The integral of the squared velocity gradient, the quantity whose growth signals a flow tearing itself apart; the rate 1/T∗ is defined over flows with enstrophy at most one.", "more": "Enstrophy, the integral of the squared velocity gradient, equal for divergence-free flows to the integral of the squared vorticity. Its rate of growth controls whether a Navier–Stokes solution stays smooth: Leray showed in 1934 that a smooth solution lasts at least a time proportional to one over the square of the initial enstrophy (with the viscosity set to 1), and that the enstrophy must blow up if the solution does. The blow-up rate 1/T∗ is normalized to starting flows of enstrophy at most one for exactly this reason. A quantity of the framework.", "cites": [ "r-28.10", "r-28.9" ], "links_to_cards": [ "nsrate" ] }, { "id": "glueball", "url": "https://www.teclamairme.com/#g-glueball", "name": "the lightest glueball", "symbol": "m(0++) ≈ 1.7 GeV", "symbol_markup": "m(0^{++}) ≈ 1.7 GeV", "chapter": 28, "type": "open", "badge": "Status open", "quick": "The lightest particle of pure SU(3) gauge theory on the lattice, about 1.7 GeV (1730 ± 50 ± 80 and 1710 ± 50 ± 80 MeV); the numerical face of the mass gap Δ.", "more": "The scalar glueball, the lightest state of pure SU(3) Yang–Mills theory, a bound state of gluons with no quarks. Lattice computations give its mass as 1730 ± 50 ± 80 MeV (Morningstar and Peardon 1999) and 1710 ± 50 ± 80 MeV (Chen et al. 2006), so the lightest glueball weighs about 1.8 times the proton (m_p, about 938 MeV), a comparison of the book’s own. No glueball has been unambiguously identified in experiment, where glueballs mix with ordinary mesons, which is why the value is a computation rather than a measurement. Status open, like Δ itself.", "cites": [ "r-28.19", "r-28.26" ], "links_to_cards": [ "mp", "ymgap" ] }, { "id": "clay", "url": "https://www.teclamairme.com/#g-clay", "name": "the Millennium Prize Problems", "symbol": "Clay Mathematics Institute, 2000", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "Seven problems announced in Paris on 24 May 2000, a million dollars each; one has been solved and the prize declined.", "more": "The Millennium Prize Problems, announced by the Clay Mathematics Institute at the Collège de France on 24 May 2000, a century after Hilbert’s Paris address, with a million-dollar prize for each: the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP, the Poincaré conjecture, the Riemann Hypothesis, and Yang–Mills existence and mass gap. Perelman solved Poincaré in 2002–2003 and declined the prize in 2010; the other six are open. An event of the framework.", "cites": [ "r-28.1", "r-28.6" ], "links_to_cards": [ "bsdconj", "glueball", "hodge", "pnp", "poincare", "rh" ] }, { "id": "hilbert23", "url": "https://www.teclamairme.com/#g-hilbert23", "name": "Hilbert’s problems of 1900", "symbol": "Hilbert’s 23 problems", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "The list David Hilbert launched in Paris in August 1900 that shaped a century; the Riemann Hypothesis is the only problem on both Hilbert’s list and the Clay Millennium list.", "more": "Hilbert’s twenty-three problems: ten of them were read out at the International Congress of Mathematicians in Paris on 8 August 1900, and all twenty-three were printed soon after. The first was the continuum hypothesis (Chapter ᚚ(P)), the eighth contains the Riemann Hypothesis and Goldbach’s conjecture, the seventh asked whether 2^√2 is transcendental (Chapter ᚂ(L)), the tenth whether Diophantine equations are decidable (answered no by Matiyasevich in 1970). The Riemann Hypothesis is the only problem on both Hilbert’s list and the Clay list. An event of the framework.", "cites": [ "r-28.1", "r-28.2", "r-28.20", "r-28.30" ], "links_to_cards": [ "ch", "gelfond", "rh" ] }, { "id": "threemfld", "url": "https://www.teclamairme.com/#g-threemfld", "name": "a closed three-manifold", "symbol": "M³, ∂M = ∅", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "A space that looks like ordinary three-dimensional space at every point, with no edge and no room to run off to infinity.", "more": "A three-manifold is a space in which every point has a neighborhood that looks like ordinary three-dimensional space; closed adds that it is compact and has no boundary—no edge to fall off and nowhere to escape to. The three-sphere S³ is the model example, the three-dimensional counterpart of the surface of a ball. Poincaré’s question was whether being closed and having every loop contractible is enough to force a three-manifold to be the three-sphere, and the answer, after a century, is yes: Perelman’s proof settled the one Clay problem that has been solved. A class of spaces, uncolored.", "cites": [ "r-28.23" ], "links_to_cards": [ "poincare" ] }, { "id": "smoothproj", "url": "https://www.teclamairme.com/#g-smoothproj", "name": "a smooth projective variety", "symbol": "X ⊂ ℙⁿ(ℂ)", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "The common zeros of homogeneous polynomials in complex projective space, with no singular points—the setting in which the Hodge conjecture is stated.", "more": "A projective variety is the set of common zeros, inside complex projective space, of a family of homogeneous polynomials; smooth, or non-singular, means it has no corners or self-crossings, so it is a complex manifold as well as an algebraic object. Living in projective space makes it compact, and that is what keeps its cohomology finite and well behaved. This double nature—defined by equations, studied by topology—is what the Hodge conjecture trades on: it asks which classes in the cohomology of such a variety are combinations of the classes cut out by its own subvarieties. A class of spaces, uncolored.", "cites": [ "r-28.24" ], "links_to_cards": [ "cohomology", "hodge" ] }, { "id": "gauge", "url": "https://www.teclamairme.com/#g-gauge", "name": "gauge symmetry", "symbol": "ψ ↦ e^(iθ(x))ψ", "chapter": 28, "type": "fn", "badge": "Function or operator", "quick": "A symmetry you may apply differently at every point of space and time, and which the physics cannot see. Weyl named it in 1918, after cask sizes.", "more": "A gauge symmetry is a transformation that may be chosen independently at every point of spacetime and that leaves the Lagrangian unchanged; insisting on such a symmetry is what forces a force field into existence. Hermann Weyl introduced both the idea and the word in 1918, trying to fold electromagnetism into general relativity by letting the scale of length vary from point to point—Eichinvarianz, after the standard sizes of casks, the same root as railway gauge. The scale idea failed, but once quantum mechanics arrived Weyl, Fock, and London replaced the scale factor with a phase, and the surviving U(1) symmetry is electromagnetism. Yang–Mills theory is the same construction with a nonabelian group, and the Clay problem asks for it to be built rigorously for every compact simple gauge group. A principle, uncolored.", "cites": [ "r-28.25" ], "links_to_cards": [] }, { "id": "bsdconj", "url": "https://www.teclamairme.com/#g-bsdconj", "name": "the Birch and Swinnerton-Dyer conjecture", "symbol": "rank(E) = ords=1 L(E, s)", "symbol_markup": "rank(E) = ord_{s=1} L(E, s)", "chapter": 28, "type": "open", "badge": "Status open", "quick": "The Birch and Swinnerton-Dyer conjecture proposes that an elliptic curve’s rank, the number of independent rational points of infinite order it has, equals the order of vanishing of its L-function at s = 1.", "more": "Bryan Birch and Peter Swinnerton-Dyer formulated the conjecture in the early 1960s from numerical computations of curves’ point counts modulo many primes, tying the purely algebraic rank, from the Mordell–Weil theorem’s finitely generated group of rational points, to the analytic behavior of L(E, s), an Euler product built from those same point counts. It is one of the seven Millennium Prize Problems, and the strongest evidence for it comes from ranks 0 and 1, including the curve of conductor 37 whose L-function vanishes to order exactly 1 at s = 1, matching its rank of 1. No proof yet covers curves of rank greater than 1, and the general statement remains open.", "cites": [ "r-28.27" ], "links_to_cards": [ "E37", "LfuncE", "clay", "mordellweil" ] } ], "references": [ { "id": "r-1.1", "url": "https://www.teclamairme.com/#r-1.1", "chapter": 1, "status": "confirmed", "claim": "In the rich fixed mathematical setting described on D’s card, reals uniquely definable without parameters in the countable original language form an externally countable set. Every rational, every algebraic, and every computable real is definable there; a real outside that set is irrational, transcendental, and uncomputable. The author proposes Ruis numbers as a name for the entire collection outside D. The notation ᚏ_{b} := x/b presents a chosen member using a nonzero rational label b and a shared x within one formula; this book holds x fixed throughout. Being countable, D has Lebesgue measure zero. Chaitin’s fixed-machine Ω, though uncomputable, is definable. The Ogham letter Ruis is glossed as the elder; Ogham was also used, less often, to write Pictish.", "sources": [ { "title": "Definable real number", "url": "https://en.wikipedia.org/wiki/Definable_real_number", "own_work": false, "note": "defines definable reals by unique specification in a formal language, states there are only countably many formulas, and states that the reals are uncountable. Re-read raw on 17 September 2026 for the containment the chapter’s inference rests on: “All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, π, e, et cetera, along with all algebraic numbers.”", "fetched": [ "2026-09-09" ] }, { "title": "Cantor’s diagonal argument", "url": "https://en.wikipedia.org/wiki/Cantor's_diagonal_argument", "own_work": false, "note": "gives Cantor’s 1891 diagonal proof that the real numbers are uncountable: “Georg Cantor published this proof in 1891, but it was not his first proof of the uncountability of the real numbers, which appeared in 1874.”", "fetched": [ "2026-09-09" ] }, { "title": "Author’s declaration and own computation", "url": null, "own_work": true, "note": "The notation dates to V17, 11 September 2026; labels were widened from positive to nonzero rationals on 17 September 2026. Clarified 30 September 2026: “Ruis numbers” names ℝ ∖ D, while ᚏ_{b} := x/b presents a chosen member. Labels used together in one formula share x, and this book keeps that choice fixed throughout; a formula outside this book may choose another x. If x/b were definable without parameters in the original setting, multiplying by the definable rational b would define x there. Also x ≠ 0, so x/b = x/c implies b = c. Thus distinct rational labels sharing x give distinct undefinable values. These calculations establish relationships without specifying those values.", "fetched": [] }, { "title": "Null set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Null_set", "own_work": false, "note": "“For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null.”", "fetched": [ "2026-09-22" ] }, { "title": "Chaitin’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chaitin%27s_constant", "own_work": false, "note": "states that Omega is definable though uncomputable: “Ω is among the few definable algorithmically random numbers and is the best-known algorithmically random number, but it is not at all typical of all algorithmically random numbers.”", "fetched": [ "2026-09-22" ] }, { "title": "Ogham, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ogham", "own_work": false, "note": "“Ruis, Old Irish Ruis means ‘red’ or ‘redness’, glossed as trom ‘elder’.”; “It was also used much less frequently to record the Pictish language in what is now Scotland.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.2", "url": "https://www.teclamairme.com/#r-1.2", "chapter": 1, "status": "confirmed", "claim": "Definability without parameters depends on the formal language and model. Pointwise definability is not a first-order property; if ZFC is consistent, there are models of ZFC in which every real is definable without parameters, and such models are countable.", "sources": [ { "title": "Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory", "url": "https://arxiv.org/abs/1105.4597", "own_work": false, "note": "abstract and bibliographic record confirm the pointwise-definable-model result; posted to arXiv in 2011 and published in The Journal of Symbolic Logic 78(1) (2013), 139–156. Introduction, pp. 1–3: definitions are relative to a language and structure and use no parameters. Page 2 discusses ⟨V_{ω+ω}, ∈⟩ as a rich background for classical mathematical definitions. This book adopts that fixed set structure and counts its definable reals externally; this choice is not the paper’s definition of “Ruis.” The pointwise-definable-model result explains why the argument cannot be transferred indiscriminately to every model. p. 2: “In any fixed structure M in a countable language, including the higher-order set-theoretic structures ⟨Vα, ∈⟩, the math-tea argument seems fine”; p. 3: “Clearly, pointwise definable models in a countable language must be countable, since there are only countably many definitions.” and “The first two claims have fairly soft proofs and might be considered to be a part of the mathematical folklore. Statement (1) could be credited to Myhill [Myh52].”", "fetched": [ "2026-09-30" ] }, { "title": "Definable real number", "url": "https://en.wikipedia.org/wiki/Definable_real_number", "own_work": false, "note": "lines on model-relative definability, non-expressibility in ZFC, and countable models in which all reals are definable: “there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-1.3", "url": "https://www.teclamairme.com/#r-1.3", "chapter": 1, "status": "confirmed", "claim": "The real numbers form the complete ordered field and are commonly denoted by ℝ; the inclusion of naturals, integers, rationals, and complex numbers is standard notation. Two independent definitions were published in 1872, Dedekind’s by cuts and Cantor’s by equivalence classes of Cauchy sequences.", "sources": [ { "title": "Real number", "url": "https://en.wikipedia.org/wiki/Real_number", "own_work": false, "note": "confirms the real-number system, its order/completeness presentation, and standard notation. Re-read raw on 17 September 2026 for the dates: “Beginning with Richard Dedekind in 1858, several mathematicians worked on the definition of the real numbers … leading to the publication in 1872 of two independent definitions of real numbers, one by Dedekind, as Dedekind cuts, and the other one by Georg Cantor, as equivalence classes of Cauchy sequences.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-1.4", "url": "https://www.teclamairme.com/#r-1.4", "chapter": 1, "status": "confirmed", "claim": "The extended real line is ℝ ∪ {−∞,+∞}, with finite x divided by either infinity conventionally equal to 0 and indeterminate expressions such as ∞−∞ left undefined. Measure theory sets 0·∞ = 0, a convention chosen for integration. In the min-plus semiring the additive identity, its zero, is +∞.", "sources": [ { "title": "Extended real number line", "url": "https://en.wikipedia.org/wiki/Extended_real_number_line", "own_work": false, "note": "confirms adjoining the two infinities and the standard arithmetic conventions and undefined forms.", "fetched": [ "2026-09-09" ] }, { "title": "A. Sengupta, Lecture 2. Measure and Integration, Math 7312 notes, Louisiana State University", "url": "https://www.math.lsu.edu/~sengupta/7312s02/msreinteg.pdf", "own_work": false, "note": "“0 · ∞ = 0” among the conventions; “The convention 0·∞ = 0 might seem like an arbitrary choice, but it is the really the right one : it will be useful in integration and there it will appear as the natural choice.”", "fetched": [ "2026-09-22" ] }, { "title": "Tropical semiring, Wikipedia", "url": "https://en.wikipedia.org/wiki/Tropical_semiring", "own_work": false, "note": "min-plus semiring: “The identity element for ⊕ is +∞, and the identity element for ⊗ is 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.5", "url": "https://www.teclamairme.com/#r-1.5", "chapter": 1, "status": "confirmed", "claim": "In the max-plus tropical semiring, addition is maximum, multiplication is ordinary addition, the additive identity is −∞, and the multiplicative identity is 0; min-plus uses minimum and +∞. In the extended reals a finite number divided by +∞ or −∞ is 0. Tropical addition, like the addition of every semiring, is associative and commutative. Tropical multiplication distributes over tropical addition, the tropical semiring is a semifield (every nonzero element invertible), and tropical polynomials are piecewise-linear functions.", "sources": [ { "title": "Tropical semiring", "url": "https://en.wikipedia.org/wiki/Tropical_semiring", "own_work": false, "note": "gives both min-plus and max-plus operations and their identities, and identifies tropical geometry as an application.", "fetched": [ "2026-09-09" ] }, { "title": "Extended real number line, Wikipedia", "url": "https://en.wikipedia.org/wiki/Extended_real_number_line", "own_work": false, "note": "“The arithmetic operations of ℝ can be partially extended to ℝ̄ as follows:” including a/±∞ = 0 for a ∈ ℝ.", "fetched": [ "2026-09-22" ] }, { "title": "Semiring, Wikipedia", "url": "https://en.wikipedia.org/wiki/Semiring", "own_work": false, "note": "definition: “(R, +) is a commutative monoid with an identity element called 0,” listing (a + b) + c = a + (b + c) and a + b = b + a.", "fetched": [ "2026-09-22" ] }, { "title": "Tropical mathematics, David Speyer and Bernd Sturmfels, arXiv", "url": "https://arxiv.org/abs/math/0408099", "own_work": false, "note": "states distributivity (min-plus convention): “The distributive law holds for tropical addition and tropical multiplication: x ⊙ (y ⊕ z) = x ⊙ y ⊕ x ⊙ z.”", "fetched": [ "2026-09-22" ] }, { "title": "Semifield, Wikipedia", "url": "https://en.wikipedia.org/wiki/Semifield", "own_work": false, "note": "“a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse.” and “(R, max, +)” ... “These are called the tropical semiring.”", "fetched": [ "2026-09-22" ] }, { "title": "Tropical geometry, Wikipedia", "url": "https://en.wikipedia.org/wiki/Tropical_geometry", "own_work": false, "note": "“Thus a tropical polynomial F is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.6", "url": "https://www.teclamairme.com/#r-1.6", "chapter": 1, "status": "confirmed", "claim": "Imre Simon introduced the min-plus semiring in 1978, in the context of automata theory, in Limited subsets of a free monoid, Proc. 19th Annual Symposium on Foundations of Computer Science, IEEE, 1978, pp. 143–150. The adjective tropical was coined afterwards by French colleagues in his honor — he lived and worked in Brazil — with Jean-Éric Pin crediting Dominique Perrin and Simon himself crediting Christian Choffrut. Tropical mathematics has applications including scheduling and shortest paths, and tropical geometry consolidated as a field from the late 1990s and took off around 2000, with Kontsevich, Mikhalkin, Speyer, and Sturmfels among those who built it. Tropical (idempotent) mathematics, the setting of max-plus algebra, dates back to the early 1960s.", "sources": [ { "title": "Tropical semiring", "url": "https://en.wikipedia.org/wiki/Tropical_semiring", "own_work": false, "note": "attributes the name to Imre Simon and states applications and the connection with tropical geometry.", "fetched": [ "2026-09-09" ] }, { "title": "Tropical geometry", "url": "https://en.wikipedia.org/wiki/Tropical_geometry", "own_work": false, "note": "dates the consolidation of the field to the late 1990s, names Mikhalkin and Kontsevich, and records that Jean-Éric Pin credits the adjective to Dominique Perrin while Simon credited it to Christian Choffrut, and describes Simon as “the Hungarian-born Brazilian computer scientist.”", "fetched": [ "2026-09-09" ] }, { "title": "Jean-Éric Pin, Tropical Semirings, in J. Gunawardena (ed.), Idempotency, Publications of the Newton Institute 11, Cambridge University Press, 1998, pp. 50–69", "url": "https://www.irif.fr/~jep/PDF/Tropical.pdf", "own_work": false, "note": "read as extracted PDF text. §2.2: “The Min-Plus semiring is M = (N ∪ {∞}, min, +) … This semiring was introduced by Simon [41] in the context of automata theory (it is also a familiar semiring in Operations Research).” Its reference [41] is “I. Simon, Limited subsets of a free monoid, in Proc. 19th Annual Symposium on Foundations of Computer Science, Piscataway, N.J., 1978, pp. 143–150, IEEE,” which Crossref confirms at doi 10.1109/sfcs.1978.21, October 1978. §1: “These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our brazilian colleague and friend Imre Simon, but are also commonly known as (min, +)-semirings.”", "fetched": [ "2026-09-17" ] }, { "title": "David Speyer and Bernd Sturmfels, Tropical mathematics, Mathematics Magazine 82 (2009), 163–173", "url": "https://arxiv.org/abs/math/0408099", "own_work": false, "note": "the expository paper behind the chapter’s naming of Sturmfels among the founders; posted to the arXiv in 2004 and published in 2009. Its introduction (arXiv version, read as extracted PDF text): “The adjective ‘tropical’ was coined by French mathematicians, including Jean-Eric Pin, in the honor of their Brazilian colleague Imre Simon … There is no deeper meaning in the adjective ‘tropical’. It simply stands for the French view of Brazil.”", "fetched": [ "2026-09-17" ] }, { "title": "N. Krivulin, Algebraic solutions of tropical optimization problems, arXiv", "url": "https://arxiv.org/abs/1406.1777", "own_work": false, "note": "introduction: “Tropical (idempotent) mathematics, which is concerned with the theory and applications of semirings with idempotent addition, dates back to the early 1960’s, when a few innovative works by Pandit [47], Cuninghame-Green [11], Giffler [24], Hoffman [29], Vorob’ev [51] and Romanovskiĭ [48] made their appearance.”", "fetched": [ "2026-09-22" ] }, { "title": "Bernd Sturmfels, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bernd_Sturmfels", "own_work": false, "note": "“Bernd Sturmfels has made contributions to a variety of areas of mathematics, including algebraic geometry, commutative algebra, discrete geometry, Gröbner bases, toric varieties, tropical geometry, algebraic statistics, and computational biology.”", "fetched": [ "2026-09-22" ] }, { "title": "Min-plus matrix multiplication, Wikipedia", "url": "https://en.wikipedia.org/wiki/Min-plus_matrix_multiplication", "own_work": false, "note": "read as raw wikitext: the distance product “is standard matrix multiplication for the semi-ring of tropical numbers in the min convention,” and “This operation is closely related to the shortest path problem.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The chapter also dates max-plus algebra, as distinct from Simon’s 1978 min-plus semiring, to the early 1960s. We keep that: Krivulin’s survey dates tropical (idempotent) mathematics to the early 1960s. Extended 17 September 2026. Until then the 1978 date was asserted in four places — the chapter prose, #tropical, #tropplus, and #minplus — and appeared in no claim line and in no cited source; Pin’s survey now carries the year, the min-plus attribution, and the automata context in one sentence. The chapter’s “took off around 2000 with Mikhalkin, Sturmfels, and others” and this entry’s “late 1990s” were never in conflict, but the entry named different people from the page; since the compendium is the source, the entry widened to it rather than the other way round." }, { "id": "r-1.7", "url": "https://www.teclamairme.com/#r-1.7", "chapter": 1, "status": "confirmed", "claim": "Richard’s paradox uses a list of defining phrases and diagonalization; Richard’s observation dates to 1905. Its resolution is that the property of defining a unique real number is not expressible in the formal language itself and belongs to the metatheory, outside the language.", "sources": [ { "title": "Richard’s paradox", "url": "https://en.wikipedia.org/wiki/Richard's_paradox", "own_work": false, "note": "confirms the diagonal construction and Jules Richard’s 1905 publication. Re-read on 29 September 2026 for the resolution the chapter now states: “The property D(φ) that a formula φ of ZFC defines a unique real number is not itself expressible by ZFC, but must be considered as part of the metatheory used to formalize ZFC.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-1.8", "url": "https://www.teclamairme.com/#r-1.8", "chapter": 1, "status": "confirmed", "claim": "The notation of the governing sentence has the meanings used in the chapter: the set-theoretic symbols ∖, ∈, and ∪, and := for a definition rather than an asserted equality. The forms ≜, ≝, and := each abbreviate the phrase “let x = E.” The definable reals D form a countable subset of ℝ.", "sources": [ { "title": "Set (mathematics)", "url": "https://en.wikipedia.org/wiki/Set_(mathematics)", "own_work": false, "note": "confirms the standard set-membership and set-operation meanings.", "fetched": [ "2026-09-09" ] }, { "title": "Set difference", "url": "https://en.wikipedia.org/wiki/Set_difference", "own_work": false, "note": "confirms A∖B as the elements of A not in B.", "fetched": [ "2026-09-09" ] }, { "title": "Defined, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/Defined.html", "own_work": false, "note": "gives A := B as one of the standard notations for A is defined to be B.", "fetched": [ "2026-09-09" ] }, { "title": "Glossary of mathematical symbols, Wikipedia", "url": "https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols", "own_work": false, "note": "entry for ≜, ≝ (=def), and :=: “Any of these is sometimes used for naming a mathematical object.” The forms x ≜ E, x ≝ E, x := E and E =: x “are each an abbreviation of the phrase “let x = E.””", "fetched": [ "2026-09-22" ] }, { "title": "Definable real number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Definable_real_number", "own_work": false, "note": "“there are only countably many formulas, and so only countably many elements of M can be definable over M.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.9", "url": "https://www.teclamairme.com/#r-1.9", "chapter": 1, "status": "confirmed", "claim": "In ordinal arithmetic α·β consists of β copies of α, while the surreal numbers contain the ordinals and have different arithmetic; in particular the surreal reading uses 1+ω=ω+1.", "sources": [ { "title": "Ordinal arithmetic", "url": "https://en.wikipedia.org/wiki/Ordinal_arithmetic", "own_work": false, "note": "confirms ordinal multiplication as repeated copies and the noncommutative behavior of ordinal arithmetic.", "fetched": [ "2026-09-09" ] }, { "title": "Surreal number", "url": "https://en.wikipedia.org/wiki/Surreal_number", "own_work": false, "note": "confirms that ordinals embed in the surreals and that surreal arithmetic differs from ordinal arithmetic.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-1.10", "url": "https://www.teclamairme.com/#r-1.10", "chapter": 1, "status": "confirmed", "claim": "Read in the surreal numbers under the piece’s own convention ε = 1/ω, taking −∞ as −ω makes (x/4.16)/(−ω) = −(x/4.16)·ε: an infinitesimal whose size depends on x, and not ε itself.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Using the glossary’s ε = 1/ω, (x/4.16)/(−ω) = −(x/4.16)·(1/ω) = −(x/4.16)·ε. Equality with ε would force x = −4.16, which is definable and therefore excluded by the governing sentence.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-1.11", "url": "https://www.teclamairme.com/#r-1.11", "chapter": 1, "status": "confirmed", "claim": "Berry’s paradox was first published by Bertrand Russell in 1906 and credited by him to G. G. Berry, a junior librarian at the Bodleian. The wording usually quoted, in which the least integer not nameable in fewer than nineteen syllables is itself a name of eighteen syllables, is Russell’s of 1908. The letters version quoted on the card, the smallest positive integer not definable in under sixty letters, has fifty-seven letters. Chaitin built a proof of Gödel’s incompleteness theorem on the paradox in 1971.", "sources": [ { "title": "Berry paradox", "url": "https://en.wikipedia.org/wiki/Berry_paradox", "own_work": false, "note": "G. G. Berry (1867–1928), junior librarian at the Bodleian; Russell the first to discuss the paradox in print; cites Russell, Les paradoxes de la logique, Revue de métaphysique et de morale 14 (1906) 627–650.", "fetched": [ "2026-09-09" ] }, { "title": "Russell, Mathematical Logic as Based on the Theory of Types (1908), p. 223", "url": "https://en.wikisource.org/wiki/Page:Mathematical_Logic_as_Based_on_the_Theory_of_Types_-_Bertrand_Russell_(1908).pdf/3", "own_work": false, "note": "the least integer not nameable in fewer than nineteen syllables denotes 111,777 and is itself a name consisting of eighteen syllables, credited to Berry in a footnote.", "fetched": [ "2026-09-09" ] }, { "title": "Berry Paradox, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/BerryParadox.html", "own_work": false, "note": "gives the nineteen/eighteen syllable form and attributes it to Russell, American Journal of Mathematics 30 (1908), 223.", "fetched": [ "2026-09-09" ] }, { "title": "G. J. Chaitin, Computational Complexity and Gödel’s Incompleteness Theorem, ACM SIGACT News No. 9 (April 1971), pp. 11–12 (PDF, Charles University)", "url": "https://www.karlin.mff.cuni.cz/~krajicek/chaitin71.pdf", "own_work": false, "note": "header “ACM SIGACT News, No. 9 (April 1971), pp. 11–12”; abstract: “The proof resembles Berry’s paradox, not the Epimenides nor Richard paradoxes.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The card had given the nineteen-syllable wording as Russell’s of 1906. Russell published the paradox in 1906 in a different wording and gave the quoted one in 1908; the card now says both. The fifty-seven letters are countable from the phrase itself." }, { "id": "r-1.12", "url": "https://www.teclamairme.com/#r-1.12", "chapter": 1, "status": "confirmed", "claim": "Peano introduced ∪ and ∩ in 1888 and ∈ in 1889; := is the assignment operator of Algol, from Algol 58 onward.", "sources": [ { "title": "Earliest Uses of Symbols of Set Theory and Logic (Jeff Miller, MacTutor)", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/", "own_work": false, "note": "∈ in Peano, Arithmetices principia nova methodo exposita (1889), as an abbreviation for est — re-read raw on 17 September 2026, the page says “He stated that the symbol was an abbreviation for est; the entire work is in Latin” and “Peano’s symbol for membership appears to be a lunate (or uncial) epsilon, and not the stylized epsilon that is now used”; ∪ and ∩ in Peano, Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann (1888).", "fetched": [ "2026-09-09" ] }, { "title": "Assignment (computer science)", "url": "https://en.wikipedia.org/wiki/Assignment_(computer_science)", "own_work": false, "note": "x := expr is given as originally Algol 1958, popularized by Pascal.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The card for := had credited the notation to Algol 60 in 1960. The notation is Algol 58’s; Algol 60 carried it on, and the card now says so." }, { "id": "r-1.13", "url": "https://www.teclamairme.com/#r-1.13", "chapter": 1, "status": "confirmed", "claim": "The chapter epigraph is Cantor’s, from the Grundlagen einer allgemeinen Mannigfaltigkeitslehre of 1883: the essence of mathematics lies in its freedom.", "sources": [ { "title": "Georg Cantor, Wikiquote", "url": "https://en.wikiquote.org/wiki/Georg_Cantor", "own_work": false, "note": "read as raw HTML: under the heading “Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)” it gives “The essence of mathematics lies entirely in its freedom,” with the variant translation “The essence of mathematics is in its freedom.”", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. One wording note: the German is “Das Wesen der Mathematik liegt gerade in ihrer Freiheit,” and the page prints “lies precisely in its freedom” where this source gives “lies entirely in its freedom.” Both are current renderings of gerade; the page’s is the more usual one and is kept." }, { "id": "r-1.14", "url": "https://www.teclamairme.com/#r-1.14", "chapter": 1, "status": "confirmed", "claim": "Defines the integers as the natural numbers with negatives adjoined, a ring closed under addition, subtraction, and multiplication.", "sources": [ { "title": "Integer, Wikipedia", "url": "https://en.wikipedia.org/wiki/Integer", "own_work": false, "note": "defines the integers as the natural numbers together with their negatives and zero, forming a ring, and gives the blackboard-bold notation Z from German Zahlen; “The integers form the smallest group and the smallest ring containing the natural numbers.”", "fetched": [ "2026-09-22" ] }, { "title": "Semiring, Wikipedia", "url": "https://en.wikipedia.org/wiki/Semiring", "own_work": false, "note": "carries the ring/semiring distinction the card draws: “Any field is also a ring, which in turn is a semiring in which also additive inverses exist.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.15", "url": "https://www.teclamairme.com/#r-1.15", "chapter": 1, "status": "confirmed", "claim": "Defines the natural numbers as the counting numbers and notes Peano’s 1889 axiomatization.", "sources": [ { "title": "Natural number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Natural_number", "own_work": false, "note": "describes the natural numbers as the counting numbers, notes the convention split on whether 0 is included, and dates Peano’s axioms to 1889.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.16", "url": "https://www.teclamairme.com/#r-1.16", "chapter": 1, "status": "confirmed", "claim": "Defines the rational numbers as ratios of integers forming a field, gives them as exactly the reals whose decimal expansion terminates or eventually repeats, states that between any two rationals there is another, and states their countability.", "sources": [ { "title": "Rational number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rational_number", "own_work": false, "note": "defines the rationals as quotients of integers forming a field and states that the set is countably infinite; the set is “closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations”; “The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over,” and “The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.”", "fetched": [ "2026-09-22" ] }, { "title": "Definable real number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Definable_real_number", "own_work": false, "note": "carries the countability of the definable reals: “This follows from the fact that there are only countably many formulas, and so only countably many elements of M can be definable over M.”", "fetched": [ "2026-09-22" ] }, { "title": "Earliest Uses of Symbols of Number Theory, MacTutor (Jeff Miller)", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/", "own_work": false, "note": "carries the origin of the letter Q: “Q for the set of rational numbers and Z for the set of integers are apparently due to N. Bourbaki. ... The letters stand for the German Quotient and Zahlen.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-1.17", "url": "https://www.teclamairme.com/#r-1.17", "chapter": 1, "status": "confirmed", "claim": "In the book’s adopted setting, D is countable. Under a uniform continuous draw from [0, 1], a countable set has probability zero, so its complement has probability 1. A million languages with finite descriptions and interpretations fixed independently of the target still supply only countably many describable targets in total. These conclusions are the author’s deductions from the counting and measure facts below; the sources do not introduce Ruis notation. “Normal” in the chapter means typical in everyday speech, not the technical digit-frequency property.", "sources": [ { "title": "Hamkins, Linetsky and Reitz, Pointwise Definable Models of Set Theory", "url": "https://arxiv.org/pdf/1105.4597", "own_work": false, "note": "introduction, pp. 1–3: fixed languages/structures, parameter-free definitions, and the external viewpoint.", "fetched": [ "2026-09-30" ] }, { "title": "Curtis McMullen, Harvard Math 114 Real Analysis notes", "url": "https://people.math.harvard.edu/~ctm/home/text/class/harvard/114/07/html/home/course/course.pdf", "own_work": false, "note": "§1.1 gives an analogous countable-set probability argument for algebraic numbers; §1.2 develops measure and countable additivity.", "fetched": [ "2026-09-30" ] }, { "title": "Becher and Figueira, An example of a computable absolutely normal number", "url": "https://glyc.dc.uba.ar/santiago/papers/absnor.pdf", "own_work": false, "note": "abstract and introduction: normality concerns digit-block frequencies and can hold for a computable number.", "fetched": [ "2026-09-30" ] } ], "editorial_note": "Probability 1 is exact in this ideal model and does not make a definable outcome logically impossible. A physical dart or a computer selecting finite-precision outputs uses a different model." }, { "id": "r-1.18", "url": "https://www.teclamairme.com/#r-1.18", "chapter": 1, "status": "confirmed", "claim": "“Unnameable” as descriptive vocabulary.", "sources": [ { "title": "Asaf Karagila, Name that number, 21 September 2015", "url": "https://karagila.org/2015/name-that-number/", "own_work": false, "note": "the paragraph beginning “And we can prove, now…” uses unnameable alongside undefinable. The later discussion distinguishes intuitive nameability from mathematical definability and explains dependence on the language and setting. This supports the vocabulary, not an absolute category beyond every possible language.", "fetched": [ "2026-09-30" ] } ], "editorial_note": "" }, { "id": "r-1.19", "url": "https://www.teclamairme.com/#r-1.19", "chapter": 1, "status": "confirmed", "claim": "“Indefinables” and the dark-matter analogy.", "sources": [ { "title": "Anne Newstead and James Franklin, On the Reality of the Continuum, Philosophy 83 (2008), pp. 117–128", "url": "https://web.maths.unsw.edu.au/~jim/newsteadcontinuum.pdf", "own_work": false, "note": "printed p. 121 / fifth PDF page uses indefinables and compares them to the dark matter of mathematics. The latter is an analogy, not a formal class called “dark numbers.” This citation supports those descriptions; the book’s mathematical definition remains the one declared on D’s card.", "fetched": [ "2026-09-30" ] } ], "editorial_note": "" }, { "id": "r-2.1", "url": "https://www.teclamairme.com/#r-2.1", "chapter": 2, "status": "confirmed", "claim": "The current Oxford dating of the Bakhshali manuscript places the sampled folios in the late eighth to early eleventh centuries, with the manuscript modeled at about 799–1102 AD; the earlier Folio 16 result was inaccurate and Folio 17 was excluded as an outlier.", "sources": [ { "title": "Chivall, Lladó-Buisán, Howell and Evison, Radiocarbon dating of the Bakhshālī manuscript, University of Oxford (2024)", "url": "https://ora.ox.ac.uk/objects/uuid:5a6d1dd7-f20c-4209-adb6-33849f5b08f4", "own_work": false, "note": "and the report PDF downloaded and read: Table 1 gives the five folios at 95.4% confidence as 773–986 (15), 901–1032 (16), 890–1014 (23), 885–995 (33), and 774–991 (55) cal AD; the OxCal phase model gives 799–892 cal AD for the start of production and 900–1102 cal AD for the end; the second determination on Folio 16 “demonstrated that the initial determination was inaccurate” and Folio 17 was excluded as an outlier.", "fetched": [ "2026-09-09" ] }, { "title": "Bakhshali manuscript, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bakhshali_manuscript", "own_work": false, "note": "summarizes the 14 October 2024 revision. Its table prints Folio 16 as 931–1032; the Oxford report itself gives 901–1032, and the report is the primary source, so the chapter follows the report. It also gives the superseded 2017 dates: “224–383 CE for folio 16 (Indo-Scythian), 680–779 CE for folio 17 (Turk Shahis), and 885–993 CE for folio 33 (Saffarid dynasty).”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-2.2", "url": "https://www.teclamairme.com/#r-2.2", "chapter": 2, "status": "confirmed", "claim": "The Bakhshali manuscript is a 70-leaf birch-bark mathematical text found near Peshawar in 1881 and held by the Bodleian Library.", "sources": [ { "title": "Bakhshali manuscript, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Bakhshali_manuscript/", "own_work": false, "note": "confirms the 1881 discovery, birch bark, approximately 70 surviving leaves, and presentation to the Bodleian in 1902.", "fetched": [ "2026-09-09" ] }, { "title": "Bakhshali manuscript, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bakhshali_manuscript", "own_work": false, "note": "confirms the 70-leaf description, discovery near Peshawar, and Bodleian location.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-2.3", "url": "https://www.teclamairme.com/#r-2.3", "chapter": 2, "status": "confirmed", "claim": "Babylonian sexagesimal notation used blank places and later two wedge marks as a placeholder; Ptolemy used a barred circle as a sexagesimal placeholder around 130–150 AD. The derivation of that sign from the first letter of οὐδέν is commonly supposed and has been doubted (Neugebauer dismissed it, since omicron already stood for 70); in ninth-century Byzantine manuscripts the sign is still a full-size uncial omicron with a macron, and the form alters markedly only in some late medieval manuscripts. The placeholder signs were never used alone. The Sumerians counted in base sixty in the third millennium BC, and the Babylonians inherited it. In the Seleucid period, from 300 BC, Babylonian astronomical texts used a zero sign, never in final position.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "describes Babylonian base 60, the two-wedge placeholder around 400 BC, and Ptolemy’s sexagesimal placeholder around 130 AD; on the omicron derivation: “Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70.”", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Uses of Symbols for Constants, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/", "own_work": false, "note": "Burton: “there is no evidence that Ptolemy regarded the symbol as a number by itself that could enter into computations with other numbers”; Berggren: “Papyri from the period when Ptolemy lived show a small ‘o’ with a bar over it as the symbol for 0”; and “Babylonians in the Seleucid period (300 BC onward) used a symbol for zero, mainly in astronomical texts, and never in final positions.”", "fetched": [ "2026-09-09" ] }, { "title": "Raymond Mercier, Ancient Greek Numbers: A Greek Zero Sign, Unicode Technical Committee document L2/04-054", "url": "https://www.unicode.org/L2/L2004/04054-greek-zero.pdf", "own_work": false, "note": "read as extracted PDF text. Under ORIGIN OF THE SYMBOL: “It is commonly supposed that these forms arise from an abbreviation of οὐδέν, but given the rather arbitrary and opaque character of the majority of the symbols, one might well doubt this rather simple explanation.” Under Ninth century: “In ninth century Byzantine manuscripts the character for zero differs little from the papyri … the zero consists of a full size uncial omicron with a macron.” Under Fourteenth century: “In the some late medieval manuscripts the form of the symbol has altered considerably.”", "fetched": [ "2026-09-17" ] }, { "title": "Wikipedia, Greek numerals, § Zero", "url": "https://en.wikipedia.org/wiki/Greek_numerals", "own_work": false, "note": "read as raw HTML: Hellenistic astronomers limited each sexagesimal position to 59, “which uses only the letters up through nu (ν) and included otherwise unused omicron (ο) as a special symbol for zero,” a system “probably adapted from Babylonian numerals by Hipparchus c. 140 BCE” and then used by Ptolemy.", "fetched": [ "2026-09-17" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "the Babylonian placeholder never stood alone: “Significantly, however, these placeholder signs were not considered a numerical value, they were never used alone”", "fetched": [ "2026-09-22" ] }, { "title": "Sexagesimal, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sexagesimal", "own_work": false, "note": "carries the Sumerian origin and the Babylonian inheritance: “It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Extended 17 September 2026 for the #greekzero card, which carried the origin dispute and a Byzantine transmission claim on no citation at all. The card’s sentence that the sign “had turned into a plain omicron” in tenth-century Byzantine copies did not survive the check and was replaced by what Mercier’s survey shows. The card names Otto Neugebauer as the scholar who rejected the οὐδέν derivation; that is the standard attribution and Wikipedia cites Neugebauer, The Exact Sciences in Antiquity, 2nd ed., Dover 1969, pp. 13–14 and plate 2, for the Greek zero — but those pages were not read in this pass, so the doubt is cited here to Mercier, who states it in terms, and the Neugebauer attribution is flagged for a later pass to confirm." }, { "id": "r-2.4", "url": "https://www.teclamairme.com/#r-2.4", "chapter": 2, "status": "confirmed", "claim": "The Maya used base-twenty notation and a shell zero; positional use is attested by the seventh century and earlier Maya zero artifacts by the fourth century. The Long Count calendar in which the earliest dates are written is positional, and its two oldest surviving dates, Stela 2 at Chiapa de Corzo (36 BC) and Stela C at Tres Zapotes (32 BC), are Epi-Olmec rather than Maya. Maya numerals write one as a dot and five as a bar; Tres Zapotes Stela C carries the second-oldest Long Count date, 32 BC, in Epi-Olmec style. Several Mesoamerican cultures counted in base twenty, almost certainly from fingers and toes; Maya numerals use a dot for one, a bar for five, and a shell for zero, and the third place counts 360s rather than 400s.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "confirms Maya base 20, a zero symbol by 665, and use of zero before the place-valued system.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Uses of Symbols for Constants, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/", "own_work": false, "note": "dates the oldest listed Maya zero artifact without positional notation to AD 357 and the oldest listed positional zero to AD 665.", "fetched": [ "2026-09-09" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "gives Stela 2 at Chiapa de Corzo with a Long Count date of 36 BC as the oldest, and says “it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs.”", "fetched": [ "2026-09-09" ] }, { "title": "Maya numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Maya_numerals", "own_work": false, "note": "the dot-and-bar notation: “The numerals are made up of three symbols: zero (a shell), one (a dot) and five (a bar).”", "fetched": [ "2026-09-22" ] }, { "title": "Mesoamerican Long Count calendar, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar", "own_work": false, "note": "the Epi-Olmec earliest dates and Stela C: “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.” and “This is the second oldest Long Count date yet discovered. The numerals 7.16.6.16.18 translate to September 1, 32 BCE (Gregorian).”", "fetched": [ "2026-09-22" ] }, { "title": "Mayan mathematics, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Mayan_mathematics/", "own_work": false, "note": "“Almost certainly the reason for base 20 arose from ancient people who counted on both their fingers and their toes.”; “it changes in the third place and this denotes the number of 360’s up to 19 instead of the number of 400’s.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-2.5", "url": "https://www.teclamairme.com/#r-2.5", "chapter": 2, "status": "confirmed", "claim": "Brahmagupta’s Brāhmasphuṭasiddhānta was composed in 628 and gave rules treating zero as a number, including the incorrect rule 0/0 = 0. Brahmagupta gave rules for negative numbers as debts; the Sanskrit name for zero, śūnya, is the source of cipher and zero.", "sources": [ { "title": "Brahmagupta, Wikipedia", "url": "https://en.wikipedia.org/wiki/Brahmagupta", "own_work": false, "note": "confirms composition in 628 and identifies the work as the first book giving arithmetic rules applying to zero and negative numbers.", "fetched": [ "2026-09-09" ] }, { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "describes Brahmagupta’s rules and his treatment of division by zero.", "fetched": [ "2026-09-09" ] }, { "title": "Brahmagupta, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Brahmagupta/", "own_work": false, "note": "negatives as debts: “He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers)”", "fetched": [ "2026-09-22" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "the Sanskrit name: “Sifr evolved to mean zero when it was used to translate śūnya (Sanskrit: शून्य) from India.”", "fetched": [ "2026-09-22" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/z/", "own_work": false, "note": "the descent of the names: “ZERO, like its synonym CIPHER, derives from the Sanskrit sunya.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-2.6", "url": "https://www.teclamairme.com/#r-2.6", "chapter": 2, "status": "confirmed", "claim": "The Khmer inscription K-127 contains a positional zero and is dated to 683 AD; an Old Malay inscription at Kedukan Bukit in Sumatra carries a zero in the same Śaka year, and the two cannot be ranked against each other. Sources differ on the Kedukan Bukit Śaka year: Cœdès read 605 in 1930, and the MAA article reads 604, which is why no Śaka year is printed. Corrections welcome. The Gwalior inscription is dated to 876 AD, records zeros in 270 and 50, and is the oldest Indian zero whose date and genuineness are not disputed; older Indian candidates exist, among them inscriptions of AD 672, 683, and 684, but their dates or their authenticity are questioned. The Gwalior inscription is at the Chaturbhuj Temple. The inscription is in the Chaturbhuj temple in Gwalior Fort. Cœdès’s 1930 paper is his reading of the Sumatran stone; his publication of the Sambor inscription K-127 is the 1931 paper of 2.14, so the card’s 1931 and the MAA article’s 1930 are two publications, not a dispute.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "confirms the 876 Gwalior inscription (“The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876”), the garden dimensions and the garlands; the 683 Khmer zero is on the Earliest Uses line.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Uses of Symbols for Constants, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/", "own_work": false, "note": "confirms the 683 Khmer point and discusses the Gwalior date and its priority claims.", "fetched": [ "2026-09-09" ] }, { "title": "Frank J. Swetz, Shaharir bin Mohamad Zain and Iwan Pranoto, Mathematical Treasure: Indonesian Zeros, MAA Convergence", "url": "https://web.archive.org/web/2023/https://www.maa.org/press/periodicals/convergence/mathematical-treasure-indonesian-zeros", "own_work": false, "note": "through the Internet Archive, the live MAA address now returning HTTP 403. Three Sriwijayan stones, “known as Kedukan Bukit, Talang Tuo, and Kota Kapur, bear the numerals 604, 606, and 608 respectively, denoting years as reckoned in the Hindu Saka era calendar, which translate into our Common Era calendar as the dates 683, 684, and 686,” and “the stone’s zero symbol thus corroborates the evidence of an early zero.” On the reading of the date: in his 1930 paper Coеdes “hesitantly interpreted the numerical symbol as 605” with a footnote saying that the units digit “ne peut être que 4 ou 5, mais 5 est plus probable,” and he read the Sambor inscription as 605 as well.", "fetched": [ "2026-09-17" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "the temple: “a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated AD 876.”", "fetched": [ "2026-09-22" ] }, { "title": "Chaturbhuj Temple, Gwalior, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chaturbhuj_Temple,_Gwalior", "own_work": false, "note": "the temple “is a Hindu temple excavated in a rock face in the Gwalior Fort”; “One of the temple’s inscriptions contains the earliest known inscription of the circular symbol “O,” to represent zero.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Gwalior is often called the oldest Indian zero outright. MacTutor is more careful: it is “the first record of the Indian use of zero which is dated and agreed by all to be genuine”. Older Indian candidates exist — Michael Closs lists inscriptions of AD 672, 683, and 684, and there are copper plates that have been read as sixth-century — but their dates or their authenticity are questioned. So the chapter says undisputed rather than oldest. The Khmer stone is not uniquely oldest either: Miller lists a small circle for zero in an Old Malay inscription at Kedukan Bukit, Sumatra, in the same year, 683, and the two cannot be ranked, because K-127 records only the Śaka year and the fifth day of the waning moon, with no month. The chapter therefore calls Sambor one of the two oldest rather than the oldest." }, { "id": "r-2.7", "url": "https://www.teclamairme.com/#r-2.7", "chapter": 2, "status": "confirmed", "claim": "Qin Jiushao’s 1247 Mathematical Treatise in Nine Sections uses a circle for zero, and Fibonacci used zephirum in Liber Abaci (1202); ṣifr is the source of cipher and zero. Al-Khwarizmi’s treatise on Indian reckoning, written about 825, is what carried the digit westward. 〇 also writes zero in the Suzhou numerals, the only surviving form of the rod numerals, and is still used for zero in Chinese and in Japanese kanji notation; one hypothesis derives it from □, the mark for a missing character.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "confirms Qin Jiushao’s 1247 circle and Fibonacci’s use of zero in Liber Abaci around 1200.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics (Z), MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/z/", "own_work": false, "note": "traces Sanskrit śūnya through Arabic ṣifr to Latin zephirum in Fibonacci’s 1202 Liber Abaci and later cipher/zero forms.", "fetched": [ "2026-09-09" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "“Qín Jiǔsháo’s 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol ‘〇’ for zero”, and “the origin of this symbol is unknown; it may have been produced by modifying a square symbol”.", "fetched": [ "2026-09-09" ] }, { "title": "On the First Printed Edition of Mathematical Book in Nine Chapters (1842), Springer", "url": "https://link.springer.com/chapter/10.1007/978-3-031-49617-2_13", "own_work": false, "note": "the 1842 Yijiatang Congshu edition is the first printed edition of Qin Jiushao’s treatise, which until then circulated in manuscript.", "fetched": [ "2026-09-09" ] }, { "title": "Suzhou numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Suzhou_numerals", "own_work": false, "note": "“The Suzhou numeral system is the only surviving variation of the rod numeral system.” and “Zero is represented by the character for zero (〇).”", "fetched": [ "2026-09-22" ] }, { "title": "〇, Wiktionary", "url": "https://en.wiktionary.org/wiki/%E3%80%87", "own_work": false, "note": "Chinese: “It is considered a placeholder for a missing Chinese character. Used to express a zero digit in a number since at least the (c.) 12th century, perhaps even earlier than the concept “zero” appeared in the Chinese language. Also written as □ in early literature.” Japanese: “Used to write zero in kanji positional notation.”", "fetched": [ "2026-09-22" ] }, { "title": "Chinese numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chinese_numerals", "own_work": false, "note": "present-day use: “Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-2.8", "url": "https://www.teclamairme.com/#r-2.8", "chapter": 2, "status": "confirmed", "claim": "Dionysius Exiguus constructed an Easter table in 525, which introduced the Anno Domini numbering of years, and Bede extended that tradition in 725; Bede’s N represents nulla or nihil. Roman numerals had no zero; Bede’s N stood for 0 in a table of epacts, in De temporum ratione, written in Northumbria in 725.", "sources": [ { "title": "Dionysius Exiguus’ Easter table, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dionysius_Exiguus%27_Easter_table", "own_work": false, "note": "confirms the table was constructed in 525 and that Bede’s Easter cycle developed from it; “This table introduced the first use of the modern system of numbering years,” its first year being “Anno Domini 532,” and “With his Easter table Dionysius introduced the Christian era.”", "fetched": [ "2026-09-09" ] }, { "title": "Bede’s Easter cycle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bede%27s_Easter_cycle", "own_work": false, "note": "confirms Bede’s 725 extension of Dionysius’s Easter table.", "fetched": [ "2026-09-09" ] }, { "title": "0, Wikipedia", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "“The initial ‘N’ was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725”, and gives Dionysius Exiguus’s 525 table with the Latin nulla (“none”); it does not say which word the N stands for (the Roman numerals line: “the initial of nulla or of nihil”).", "fetched": [ "2026-09-09" ] }, { "title": "Roman numerals, Wikipedia, § Zero", "url": "https://en.wikipedia.org/wiki/Roman_numerals", "own_work": false, "note": "the Roman system “lacked a numeral for the number zero itself”; “Dionysius Exiguus used nulla alongside Roman numerals in a manuscript from 525 AD. About 725, Bede or one of his colleagues used the letter N, the initial of nulla or of nihil (the Latin word for “nothing”) for 0, in a table of epacts, all written in Roman numerals.”", "fetched": [ "2026-09-22" ] }, { "title": "The Reckoning of Time, Wikipedia", "url": "https://en.wikipedia.org/wiki/The_Reckoning_of_Time", "own_work": false, "note": "“The Reckoning of Time (Latin: De temporum ratione, CPL 2320) is a Medieval Latin treatise written by the Northumbrian monk Bede in 725.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The N is attributed to Bede here, as the chapter attributes it; the fullest source available says “by Bede—or his colleagues—”, and no source seen names the hand that wrote it." }, { "id": "r-2.9", "url": "https://www.teclamairme.com/#r-2.9", "chapter": 2, "status": "judgement", "claim": "Zero arose more than once: the Maya development was independent of the Old World, while historians differ on how far the Indian zero owed something to Babylonian practice through Greek astronomy; and the chapter’s classification of placeholder, digit, and number is a historical synthesis.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "emphasizes that zero has distinct placeholder and number uses and describes Babylonian, Maya, Indian, and Arabic developments without endorsing a fixed count of independent inventions.", "fetched": [ "2026-09-09" ] }, { "title": "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’, Bodleian Libraries", "url": "https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhshali-manuscript-contains-oldest-recorded-origins-symbol-zero", "own_work": false, "note": "“the use of zero as a placeholder was seen in several different ancient cultures, such as among the ancient Mayans and Babylonians,” but “it was only in India that this zero developed into a number in its own right.”", "fetched": [ "2026-09-09" ] }, { "title": "Indian numerals, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/", "own_work": false, "note": "“Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "The chapter’s three-part framing (placeholder, digit, number) is an interpretive synthesis. MacTutor: “many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers”; of the Maya zero, “This is a remarkable achievement but sadly did not influence other peoples.”" }, { "id": "r-2.10", "url": "https://www.teclamairme.com/#r-2.10", "chapter": 2, "status": "contradicted", "claim": "The Bodleian’s 2017 radiocarbon result for the Bakhshali manuscript, which dated one leaf to the third or fourth century AD, has been withdrawn by the laboratory that produced it: a second sample from the same leaf, taken in 2018 and published on 14 October 2024, showed the first determination to be inaccurate.", "sources": [ { "title": "Chivall, Lladó-Buisán, Howell and Evison, Radiocarbon dating of the Bakhshālī manuscript, University of Oxford (2024)", "url": "https://ora.ox.ac.uk/objects/uuid:5a6d1dd7-f20c-4209-adb6-33849f5b08f4", "own_work": false, "note": "current Oxford record for the revised radiocarbon study.", "fetched": [ "2026-09-09" ] }, { "title": "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol ‘zero’, Bodleian Libraries (2017)", "url": "https://www.glam.ox.ac.uk/article/carbon-dating-finds-bakhshali-manuscript-contains-oldest-recorded-origins-symbol-zero", "own_work": false, "note": "records the superseded 2017 announcement: the manuscript “dates from as early as the 3rd or 4th century” and “is in fact composed of material from at least three different periods”; the dating is withdrawn, but not its description of the text as “a seminal mathematical text which contains hundreds of zeroes,” the dot that “can be seen throughout the Bakhshali manuscript.”", "fetched": [ "2026-09-09" ] }, { "title": "Plofker, Keller, Hayashi, Montelle and Wujastyk, The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating, History of Science in South Asia 5.1 (2017), 134–150", "url": "https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22", "own_work": false, "note": "the five scholars argue that a manuscript is dated by its latest datable folio rather than its earliest, and that the manuscript’s zeros work as numbers in their own right and not only as placeholders. Oxford’s 2024 re-dating bears out the first point.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Both datings are on the page. The 2024 report is stated as the fact, because it is a re-measurement by the same laboratory rather than a restatement; the 2017 result is kept as the history it corrects, since without it the scholarly objection and the correction make no sense." }, { "id": "r-2.11", "url": "https://www.teclamairme.com/#r-2.11", "chapter": 2, "status": "confirmed", "claim": "Each displayed glyph denotes zero in its own notation, so multiplying the thirteen factors gives zero.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "The displayed product contains thirteen zero-valued factors; 0 × 0 × … × 0 = 0, with no denominator present.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-2.12", "url": "https://www.teclamairme.com/#r-2.12", "chapter": 2, "status": "confirmed", "claim": "The character 零 is read rei in Japanese and líng in Chinese and is used as a word for zero in both languages. In Japanese the preferred spoken reading of 0 is the English loanword zero (ゼロ).", "sources": [ { "title": "零, Kanji Data Base (Kanjipedia)", "url": "https://www.kanjipedia.jp/kanji/0007256600", "own_work": false, "note": "confirms the character and its Japanese readings/meanings.", "fetched": [ "2026-09-09" ] }, { "title": "Zero, Treccani", "url": "https://www.treccani.it/enciclopedia/zero_%28Enciclopedia-Italiana%29/", "own_work": false, "note": "the history of the word, not the Chinese or Japanese one: “Leonardo latinizzò in zephirum il vocabolo ṣifr usato dagli Arabi in corrispondenza al sanscrito śūnya (vuoto)”; it has nothing on 零.", "fetched": [ "2026-09-09" ] }, { "title": "零, Wiktionary", "url": "https://en.wiktionary.org/wiki/%E9%9B%B6", "own_work": false, "note": "a phono-semantic compound of 雨 (rain) with 令; the original sense is a light rain or drizzle, with the extended senses of a remnant or fraction, and thence zero.", "fetched": [ "2026-09-09" ] }, { "title": "Japanese numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Japanese_numerals", "own_work": false, "note": "its table gives for 0 (零 / 〇) the on reading rei and, as preferred reading, “zero (ゼロ) (loanword from English, gairaigo)”; and “as a number, it is only written as 0 or rei (零).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-2.13", "url": "https://www.teclamairme.com/#r-2.13", "chapter": 2, "status": "confirmed", "claim": "The feather mark ᚛ used in this Pictish Ogham inscription is punctuation rather than a letter; it can mark the beginning and reading direction. The shared alphabet has twenty-six Unicode letter characters, of which this book reserves Ruis for the left-hand side, leaving twenty-five for its twenty-five lettered chapters.", "sources": [ { "title": "The Unicode Standard 16.0, Chapter 8: Europe-II — Ogham", "url": "https://www.unicode.org/versions/Unicode16.0.0/core-spec/chapter-8/", "own_work": false, "note": "“U+169B OGHAM FEATHER MARK and U+169C OGHAM REVERSED FEATHER MARK are used at the beginning and the end of Ogham text, particularly in manuscript Ogham,” and “In some cases, only the Ogham feather mark is used, which can indicate the direction of the text.”", "fetched": [ "2026-09-10" ] }, { "title": "UnicodeData.txt, Unicode Character Database", "url": "https://www.unicode.org/Public/UCD/latest/ucd/UnicodeData.txt", "own_work": false, "note": "downloaded and read: 169B OGHAM FEATHER MARK carries General_Category Ps, open punctuation, and 169C OGHAM REVERSED FEATHER MARK carries Pe, while the twenty-six Ogham letters from 1681 OGHAM LETTER BEITH to 169A OGHAM LETTER PEITH all carry Lo. The feather mark is therefore not one of the letters and spends none.", "fetched": [ "2026-09-10" ] }, { "title": "Own enumeration", "url": null, "own_work": true, "note": "the twenty-five-letters claim was enumerated against the book itself and holds. The twenty-six Ogham letters are U+1681 to U+169A; the glossary’s letters section reserves Ruis for the left-hand side, and the chapters carrying the remaining twenty-five run from Chapter ᚁ(B) to Chapter ᚚ(P), which is exactly twenty-five, with the prologue and the epilogue taking the two non-letter feather marks U+169B and U+169C.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "The prologue’s first two sentences describe the feather mark and what it does, and its closing paragraph says the mark spends no letter and that the piece has twenty-five letters for twenty-five chapters. None of the three carried a footnote, and the V10 pass recorded the letter count as unchecked. The Unicode Standard supplies the first two; the third is an enumeration of the book’s own chapters, and it is right." }, { "id": "r-2.14", "url": "https://www.teclamairme.com/#r-2.14", "chapter": 2, "status": "confirmed", "claim": "The Sambor inscription K-127 was published by George Cœdès in 1931, disappeared during the Khmer Rouge period, and was found again in a shed at Angkor Conservation in Siem Reap in January 2013. Aczel told the story in his book Finding Zero (2015).", "sources": [ { "title": "G. Cœdès, À propos de l’origine des chiffres arabes, Bulletin of the School of Oriental Studies 6/2 (1931), 323–328", "url": "https://www.cambridge.org/core/journals/bulletin-of-the-school-of-oriental-and-african-studies/article/abs/a-propos-de-lorigine-des-chiffres-arabes/E6EE00E7C7811888DF9E074114A0EBDD", "own_work": false, "note": "the journal record gives author G. Cœdès, volume 6, issue 2, year 1931, pages 323–328, and DOI 10.1017/S0041977X00092806, and its abstract describes the paper as answering G. R. Kaye’s objections to an Indian origin for positional notation. The full text was not read in this pass, so this source is cited for the 1931 publication itself and not for what the paper argues about K-127.", "fetched": [ "2026-09-10" ] }, { "title": "Amir D. Aczel, Searching for the World’s First Zero, HuffPost (15 May 2013)", "url": "https://www.huffpost.com/amir-aczel/worlds-first-zero_b_3276709.html", "own_work": false, "note": "the finder’s own first-person account: Georges Cœdès published the K-127 inscription in 1931; it was lost when the Khmer Rouge stole or destroyed close to 10,000 artifacts; and on 2 January 2013 Aczel found K-127 in the Angkor Conservation compound near Siem Reap. (First cited here: Aczel, The Origin of the Number Zero, Smithsonian Magazine, December 2014, which refuses automated readers).", "fetched": [ "2026-09-22" ] }, { "title": "Amir Aczel, Wikipedia", "url": "https://en.wikipedia.org/wiki/Amir_Aczel", "own_work": false, "note": "the book: “was awarded a Sloan Foundation grant to research his book Finding Zero (2015)”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The prose sentence naming Cœdès, 1931, the Khmer Rouge years, and the 2013 rediscovery carried no footnote of its own, and neither of r-2.6’s two sources mentions Cœdès, 1931, the Khmer Rouge, or 2013; both were fetched and read in full during the V14 prose pass. Aczel’s article is first-hand for the 2013 find and secondary for the 1931 publication, which the Cambridge record confirms independently." }, { "id": "r-2.15", "url": "https://www.teclamairme.com/#r-2.15", "chapter": 2, "status": "confirmed", "claim": "At most eight of the twenty primary ogham letter names are the names of trees. Beith is ‘birch’, fern ‘alder’, sail ‘willow’ (cognate with Latin salix), dair ‘oak’, coll ‘hazel’, and onn ‘ash’; idad is probably an artificial form of ‘yew’, a sense McManus finds the kennings support, and ailm may mean ‘pine’. The rest mean other things: luis ‘blaze’ or ‘herb’, nin ‘fork’ or ‘loft’, úath ‘horror, fear’, tinne ‘bar of metal, ingot’ (probably cognate with Old Irish tend, ‘strong’, or tind, ‘brilliant’), cert ‘bush’, muin ‘neck’, ‘ruse’ or ‘love’, gort ‘field’, gétal ‘killing’, straif ‘sulphur’, ruis ‘red’, and úr ‘earth, clay, soil’, while edad is of unknown meaning; edad and idad are a pair, idad perhaps altered to match. Of the supplementary letters, ór is ‘gold’, uilleand ‘elbow’, emancholl ‘twin of coll’, pín probably from Latin spina, and ébad unknown. On attestation, the distinction this book has to make because its own script is Pictish: úath is unattested in the orthodox Irish inscriptions, which is the corpus McManus is describing, but the letter is common in the Scottish inscriptions of the Pictish area — it occurs in a dozen or so words across eight or nine of them, singly at the start or end of a word and doubled as HH in the middle, the Bressay cross slab in Shetland among them. Fern is from Primitive Irish *wernā, so the letter’s original value was [w].", "sources": [ { "title": "Ogham, section Letter names, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ogham", "own_work": false, "note": "gives the twenty primary names above letter by letter, and in its letters table óir ‘gold’, uilleann ‘elbow’, pín ‘spine; thorn’, eamhancholl ‘twin of coll’, and éabhadh ‘unknown’; it attributes the etymologies to McManus, naming A Guide to Ogam §3.15 for the etymologies of all twenty names, McManus 1988 p. 164 for idad as an artificial form of iubhar and for the eighth-century poem behind ailm, and McManus 1986 for the suggested value [y] of úath. It states flatly that “out of twenty letter names, only eight at most are the names of trees”; for Beith: “Old Irish Beithe means ‘birch-tree’, cognate to Middle Welsh bedw. Latin betula is considered a borrowing from the Gaulish cognate.”; for Sail: “Old Irish Sail means ‘willow-tree’, cognate to Latin salix”; for Fearn: “Old Irish Fern means ‘alder-tree’, Primitive Irish *wernā, so that the original value of the letter was [w]”; for Dair: “Old Irish Dair means ‘oak’ (Proto-Indo-European: *doru-)”; for Coll: “Old Irish Coll meant ‘hazel-tree’, cognate with Welsh collen,” and “Latin corulus or corylus is cognate”; for Uilleann: “Old Irish Uilleand ‘elbow’; the arboreal tradition has edleand ‘honeysuckle’”; for nGéadal: “Old Irish Gétal from the kennings has a meaning of ‘killing’, maybe cognate to gonid ‘slays’,” and “The value of the letter in Primitive Irish, then, was a voiced labiovelar”; for Straif: “Straif, Old Irish Straiph means ‘sulphur’. The Primitive Irish letter value is uncertain, it may have been a sibilant different from s, which is taken by sail, maybe a reflex of /st/ or /sw/”; and in its notes: “With regard to ailm, in the “King and Hermit” poem the hermit Marban says “caine ailmi ardom-peitet” – “beautiful are the pines that make music for me.” This is a reference to the idea that pine makes a pleasing, soothing sound as the wind passes through its needles.”; for Ceirt: “Old Irish Cert is cognate with Welsh perth ‘bush’, Latin quercus ‘oak’ (Proto-Indo-European: *perkwos). It was confused with Old Irish ceirt ‘rag’, reflected in the kennings.”; for idad: “McManus (1988, p. 164) also believes that the name idad is probably an artificial form of iubhar ‘yew’, as the kennings support that meaning.”", "fetched": [ "2026-09-17" ] }, { "title": "Bríatharogam, Wikipedia", "url": "https://en.wikipedia.org/wiki/Br%C3%ADatharogam", "own_work": false, "note": "“edad and idhad are paired names of unknown meaning, although idhad may be a form of ‘yew-tree’, altered to make a pairing.”; “gort means “field” (cognate to garden).”; for tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant.””", "fetched": [ "2026-09-24" ] }, { "title": "Forfeda, Wikipedia", "url": "https://en.wikipedia.org/wiki/Forfeda", "own_work": false, "note": "for pín: “McManus states that the name Pín was probably influenced by Latin pinus (‘pine’), but a more likely explanation is that it derives from Latin spina (‘thorn’), as the kennings indicate a tree or shrub with sweet tasting fruit (therefore not a pine)”; for Éabhadh: “The name appears modelled after Eadhadh and Iodhadh.”", "fetched": [ "2026-09-24" ] }, { "title": "BabelStone Blog, The Ogham Stones of Scotland (8 June 2013)", "url": "https://www.babelstone.co.uk/Blog/2013/06/ogham-stones-of-scotland.html", "own_work": false, "note": "“The letter uath ᚆ, which corresponds to H in the manuscript tradition, is not normally found in Ogham inscriptions outside Scotland, as it appears not to have been needed for writing Primitive Irish”; “However, this letter is very common in Scottish inscriptions from the Pictish area, occuring in 12 or 13 words in eight or nine inscriptions, usually written singly at the start or end of a word, and doubled as HH ᚆᚆ in the middle of words,” its list including “BREAY/1 : NAHHTVVDDADDS,” the “decorated cross slab from Bressay in Shetland.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "The letters table used to give the book’s own meaning for each letter in its Represents column, which duplicated the Chapter column and had gone stale in places. The column now gives the letter name’s own meaning, so it needed an entry behind it. The primary scholarship is McManus; what was read here is the encyclopedia article that reproduces him, and this entry says so rather than implying the books were opened." }, { "id": "r-2.16", "url": "https://www.teclamairme.com/#r-2.16", "chapter": 2, "status": "confirmed", "claim": "Three lists of Bríatharogaim, or word-oghams, survive from the Old Irish period, ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn. Each gives a two-word kenning for every letter name. The first two are in all three copies of the Ogam Tract; the Cú Chulainn list is absent from the Book of Ballymote and is known only from sixteenth- and seventeenth-century manuscripts, and it has no entry for four of the five supplementary letters.", "sources": [ { "title": "Bríatharogam, Wikipedia", "url": "https://en.wikipedia.org/wiki/Br%C3%ADatharogam", "own_work": false, "note": "prints all three lists in full “as edited (in normalized Old Irish) and translated by McManus (1988),” which is the text quoted on the letter cards. Beithe: féochos foltchain, glaisem cnis, maise malach. Luis: lí súla, carae cethrae, lúth cethrae. Fern: airenach fían, comét lachta, dín cridi. Sail: lí ambi, lúth bech, tosach mela. Nin: costud síde, bág ban, bág maise. Úath: condál cúan, bánad gnúise, ansam aidche. Dair: ardam dosae, grés soír, slechtam soíre. Tinne: trian roith, smiur gúaile, trian n-airm. Coll: caíniu fedaib, carae blóesc, milsem fedo. Cert: clithar baiscill, bríg anduini, dígu fethail. Muin: tressam fedmae, árusc n-airlig, conar gotha. Gort: milsiu féraib, ined erc, sásad ile. Gétal: lúth lego, étiud midach, tosach n-échto. Straif: tressam rúamnai, mórad rún, saigid nél. Ruis: tindem rucci, rúamnae drech, bruth fergae. Ailm: ardam íachta, tosach frecrai, tosach garmae. Onn: congnaid ech, féthem soíre, lúth fían. Úr: úaraib adbaib, sílad cland, forbbaid ambí. Edad: érgnaid fid, commaín carat, bráthair bethi (?). Idad: sinem fedo, caínem sen, lúth lobair (?). The English renderings on the cards are this page’s, and the question marks are its own; the Auraicept line for Dair: “arirde dossaib duir “higher than bushes is an oak””; for Tinne, “the word is probably cognate to Old Irish tend “strong” or tind “brilliant””; for Iphín, the table gives “milsem fedo “sweetest tree”” and “amram mlais “most wonderful taste”” (milsem fedo is also one of Coll’s). For Muin, “the kennings connect this name to three different words, muin “neck, upper part of the back,” muin “wile, ruse,” and muin “love, esteem.”” For Ébad the table gives “snámchaín feda “fair-swimming letter,”” “cosc lobair “[admonishing?] of an infirm person”” and “caínem éco “fairest fish.””", "fetched": [ "2026-09-17" ] }, { "title": "Forfeda, Wikipedia", "url": "https://en.wikipedia.org/wiki/Forfeda", "own_work": false, "note": "prints the same three columns for the five supplementary letters and shows the Cú Chulainn column empty for all but Ébad (caínem éco).", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Twenty-eight letter cards quote these kennings, so the quotation needed one entry of its own rather than a footnote each. The translations are McManus’s as the encyclopedia reproduces them; where that page marks a reading uncertain, the cards mark it uncertain too." }, { "id": "r-2.17", "url": "https://www.teclamairme.com/#r-2.17", "chapter": 2, "status": "confirmed", "claim": "Reading the ogham letter names as names of trees belongs to the medieval manuscript tradition, the Auraicept na nÉces and In Lebor Ogaim, and postdates the naming of the letters. The Auraicept itself finds only five of the twenty names transparent without glossing — beith, fearn, saille, duir, and coll — and concedes that some of its trees are not known. McManus dates the tree-alphabet idea to the Old Irish period and traces it to the letters themselves being called feda, trees, and nin, forking branches, from their shape. McManus counts onn, ‘ash’, as a further definite tree name and says the Auraicept wrongly glosses it ‘furze’. Modern Irish giolcach means reeds, and also broom. Ó Dónaill’s dictionary gives peith as ‘dwarf elder’ and as the letter P of the later ogham alphabet.", "sources": [ { "title": "Ogham, section Letter names, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ogham", "own_work": false, "note": "quotes the Auraicept conceding “Now all these are wood names such as are found in the Ogham Book of Woods, and are not derived from men” while admitting “some of these trees are not known today”; names the five transparent tree names; reports McManus that the idea “dates to the Old Irish period (say, 10th century), but it postdates the Primitive Irish period” and arose because the letters were called feda or nin for their shape; and names ruis, úath, and gort as letter names the glossators reinterpreted “more or less forcefully” as epithets of trees. It also reports that McManus “adds one other definite tree name: onn ‘ash’ (the Auraicept wrongly has ‘furze’).”", "fetched": [ "2026-09-17" ] }, { "title": "Bríatharogam, Wikipedia", "url": "https://en.wikipedia.org/wiki/Br%C3%ADatharogam", "own_work": false, "note": "prints the Auraicept gloss for each name with the Irish plant word — caertheand rowan, uinnius ash, sce whitethorn, cuileand holly, aball apple, finemhain vine, edind ivy, cilcach broom or fern, draighin blackthorn, trom elder, crand giuis .i. ochtach fir or pine, aiten furze, fraech heath, crand fir no crithach test-tree or aspen, ibhar yew — together with the short medieval sentences quoted on the cards; for cert: “The Auraicept glosses aball “apple,” clithar boaiscille .i. elit gelt quert (.i. aball) “shelter of a boiscill, i.e. a wild hind is queirt, i.e. an apple tree””; and of the forfeda, “oir with feorus no edind “spindle-tree or ivy.””", "fetched": [ "2026-09-17" ] }, { "title": "Forfeda, Wikipedia", "url": "https://en.wikipedia.org/wiki/Forfeda", "own_work": false, "note": "“According to Kelly (1976) the name spín … appears in the Old Irish tree lists as meaning either gooseberry or thorn” (Fergus Kelly, The Old Irish Tree-list, Celtica 11 (1976) 107–124; no free copy of the article).", "fetched": [ "2026-09-22" ] }, { "title": "giolcach, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie", "url": "https://www.teanglann.ie/en/fgb/giolcach", "own_work": false, "note": "“giolcach … 1. Coll: Reeds. Páirc na giolcaí, the reed field … 2. Reed; cane. 3. ~ (shléibhe), broom.”", "fetched": [ "2026-09-22" ] }, { "title": "peith, Foclóir Gaeilge–Béarla (Ó Dónaill), teanglann.ie", "url": "https://www.teanglann.ie/en/fgb/peith", "own_work": false, "note": "“Lit: 1. Dwarf elder. 2. Letter P in later ogham alphabet.”", "fetched": [ "2026-09-24" ] }, { "title": "úathmar, eDIL (Electronic Dictionary of the Irish Language)", "url": "https://dil.ie/42831", "own_work": false, "note": "“úathmar … adj o-ā (úath) dreadful; terrifying; ‘terrible, horrible, awful, direful’.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The glossary intro already said the plant column records “later medieval and modern arboreal associations” and is not the secure original meaning of every letter name. That sentence had no entry behind it, and twenty-eight cards now rest on it, so it has one. The three layers the cards keep apart are exactly the layers this entry separates: the name, the medieval plant gloss, and anything later." }, { "id": "r-2.18", "url": "https://www.teclamairme.com/#r-2.18", "chapter": 2, "status": "confirmed", "claim": "The five aicme forfeda were invented in the Old Irish period, centuries after the peak of ogham use, and the sounds they stood for are uncertain: the Forfeda page says they “appear to have represented sounds felt to be missing from the original alphabet, maybe” é(o), ó(i), ú(i), p and ch. Apart from Ébad they barely appear in the orthodox inscriptions: Ór twice (once in Ireland, once on the Formaston stone), Uilleann once, Pín in two or three, Emancholl not at all, though some later scholastic inscriptions use all five. Later oghamists reassigned all five to diphthongs, which changed the values of Pín, renamed Ifín, and of Emancholl, and left the alphabet with no letter for p, so Peith — beithe bog, soft beithe, also peithbog — was invented, and appears in manuscripts only. Sources differ on the name Pín: Latin spina (the Forfeda page, reporting McManus) or Latin pinus (Hayden and Stifter, OG(H)AM, 2022); corrections welcome.", "sources": [ { "title": "Forfeda, Wikipedia", "url": "https://en.wikipedia.org/wiki/Forfeda", "own_work": false, "note": "states that the five aicme forfeda “were invented in the Old Irish period, several centuries after the peak of Ogham usage”; that four are glossed with plants in the Auraicept and Emancholl is not; that Emancholl is named for its shape, ᚙ resembling two ᚉ; that Ébad is frequent in inscriptions inside the formula word KOI, meaning roughly ‘here’ and answering to Latin hic iacet; that Ór appears at Killogrone in County Kerry and on the Formaston cross slab in Aberdeenshire, Uilleann once at Teeromoyle in County Kerry, and Pín at Cool East, Crickowel, and probably Margam; and, under the heading Peith, that [p] “had again to be expressed as a modification of [b], called peithe, after beithe, also called beithe bog,” (the Ogham line carries “appears in the manuscripts only”). It also reports McManus on pín and Latin spina, and Kelly on spín in the tree lists. Of Ébad it reads: “Four of these names are glossed in the Auraicept with tree names, ebad as crithach “aspen””; “The kennings for Éabhadh point to the sound éo or é, which is also the word for “salmon””; and “Éabhadh was in fact frequently used as part of the formula word KOI ᚕᚑᚔ, but with the value /k/ or /x/”; its table of “The forfeda letter names and their kennings, as edited (in normalized Old Irish) and translated by McManus (1988)” has rows for Éabhadh, Óir, Uilleann, Ifín, and Eamhancholl and none for Peith, and the four Auraicept tree glosses it gives are for ebad, oir, uilleand, and iphin.", "fetched": [ "2026-09-17" ] }, { "title": "Ogham, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ogham", "own_work": false, "note": "gives the same reassignment and the invention of peithboc, ‘soft B’, which “appears in the manuscripts only”; and “Unlike orthodox ogham, some medieval inscriptions feature all five Forfeda.”", "fetched": [ "2026-09-17" ] }, { "title": "Deborah Hayden and David Stifter, Ogam and Trees, OG(H)AM project, University of Glasgow (20 December 2022)", "url": "https://ogham.glasgow.ac.uk/index.php/2022/12/20/ogam-and-trees/", "own_work": false, "note": "“Iphin, Pín is probably called from Lat. pīnus ‘pine, fir’, after the model of the other tree names in the core ogam set”; and of Gétal, “Medieval scholars explain it as gilcach ‘reed’”; and “Ór means ‘gold’, a loan from Latin aurum.”; of the forfedae: “Ébad, allegedly ‘aspen’ (we have seen that one before) or ‘elecampane’ (real name: elenn). Ébad is probably just a meaningless variation on edad.”", "fetched": [ "2026-09-22" ] }, { "title": "Formaston Stone, Elphinstone Institute, University of Aberdeen", "url": "https://www.abdn.ac.uk/elphinstone/resources/bbaf/category/3/180/", "own_work": false, "note": "“A fragment of a class II Pictish Symbol Stone,” with “two vertical Ogam inscriptions”; Aboyne, Aberdeenshire, dated 800–1000 AD.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Six of the book’s twenty-eight terms are supplementary letters or later inventions — Ébad, Ór, Uilleann, Ifín, Eamhancholl, and Peith — and their cards have to say that the ground under them is softer than under Beith or Dair. Ruling D21 applies: the uncertainty goes on the card." }, { "id": "r-2.19", "url": "https://www.teclamairme.com/#r-2.19", "chapter": 2, "status": "confirmed", "claim": "The tree calendar and the divinatory meanings attached to the ogham letters today are twentieth-century. Robert Graves, in The White Goddess of 1948, read the letter order as an ancient seasonal calendar of tree magic and reduced the alphabet to thirteen consonant months, in which Ruis, the elder, is the thirteenth and last, 25 November to 22 December. Celtic scholarship rejects the reconstruction, and the letter order Graves followed with it. It remains the main source of modern ogham tree lore and divination.", "sources": [ { "title": "Ogham, section Neopaganism, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ogham#Neopaganism", "own_work": false, "note": "states that modern New Age and neopagan approaches to ogham “largely derive from the now-discredited theories of Robert Graves in his book The White Goddess”; that Graves explored the tree folklore of the letter names and proposed that their order formed an ancient “seasonal calendar of tree magic” (citing Graves 1961 p. 165); that his theories have been “discredited and discarded by modern scholars,” Macalister included; that he followed Macalister’s BLNFS letter order, which scholars reject; and that the meanings used in modern divination “are usually based on the tree ogham,” with Graves still a major influence. (Read again on 23 September 2026: the article does not cite Hutton’s The Triumph of the Moon, which an earlier note said it did.)", "fetched": [ "2026-09-17" ] }, { "title": "Celtic Tree Calendar, Mary Jones, Jones’s Celtic Encyclopedia", "url": "https://www.maryjones.us/jce/celtictreecalendar.html", "own_work": false, "note": "“there is no evidence that the Celts ever named their months after trees”; lists the thirteen months and gives “Ruis (Elder) November 25 to December 22,” with 23 December ruled by no tree; describes the scheme as Davies and Graves “picking and choosing the facts to suit their purpose,” and records that Graves ignored Macalister’s advice not to take the tree alphabet too seriously.", "fetched": [ "2026-09-17" ] }, { "title": "Brian Walsh, Robert Graves’ fabrication of the Celtic tree calendar, Spiral Nature, 21 August 2013", "url": "https://www.spiralnature.com/spirituality/graves-frabrication-of-the-celtic-tree-calendar/", "own_work": false, "note": "calls the calendar “an invention unprecedented in the original lore” and concludes that “the result is a calendar that bears no resemblance to any traditional Celtic time keeping system we have on record,” setting out how Graves rearranged the Song of Amergin and cut the alphabet to thirteen consonants to do it.", "fetched": [ "2026-09-17" ] }, { "title": "The White Goddess, Wikipedia", "url": "https://en.wikipedia.org/wiki/The_White_Goddess", "own_work": false, "note": "identifies Robert Graves’s book-length essay The White Goddess: a Historical Grammar of Poetic Myth, “First published in 1948,” with revised editions in 1952 and 1961.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The guardrail on this column is that nothing modern may be presented as ancient. Every letter card that mentions present-day tree lore carries this marker, and the wording is always that the association is modern. Chapter ᚏ(R) says Ruis is associated with the elder and with transitions: the elder is medieval and well sourced, the transition is not — it comes from Graves putting Ruis last in the year. That split is stated on the Ruis letter card." }, { "id": "r-2.20", "url": "https://www.teclamairme.com/#r-2.20", "chapter": 2, "status": "confirmed", "claim": "Elder folklore, which is where the modern reading of Ruis as a letter of endings and thresholds comes from, is late and regional: the tree is a guardian to be asked before it is cut, on pain of the Elder Mother’s revenge; it was auspicious growing near a dwelling; there was a taboo on burning its wood; Midsummer’s Eve under an elder was the time to meet the fairy court; and legend made it the tree Judas hanged himself from. Scottish Gaelic ruis, one of its names, occurs only rarely in Scottish place names.", "sources": [ { "title": "Elder, Trees for Life", "url": "https://treesforlife.org.uk/into-the-forest/trees-plants-animals/trees/elder/", "own_work": false, "note": "“The best time to encounter faeries was under an elder on Midsummer’s Eve, when the Faery King and Queen and their train would pass by”; “Like rowan, people saw elder as being a protective tree. It was auspicious if it was growing near one’s dwelling, especially if it had seeded itself there”; “Legend has it that it was the tree from which Judas Iscariot hanged himself”; the belief that burning it released the Devil “reinforced the taboo against burning the wood”; and the note that the Gaelic names ruis or droman occur only rarely in Scottish place names.", "fetched": [ "2026-09-17" ] }, { "title": "Sambucus, section In culture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sambucus#In_culture", "own_work": false, "note": "“If an elder tree was cut down, a spirit known as the Elder Mother would be released and take her revenge,” and the tree “could only safely be cut while chanting a rhyme to the Elder Mother”; records that some traditions have the elder warding off witches and others have witches congregating under it, and that Romani people hold burning elder wood to bring bad luck. Cites Royal Botanic Gardens, Kew, and Howard, Traditional Folk Remedies (1987), 134–5.", "fetched": [ "2026-09-17" ] }, { "title": "Elder Mother, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elder_Mother", "own_work": false, "note": "read as raw wikitext: “to take wood from the elder tree one would have to ask the Elder Mother first, or else ill luck would befall the woodsman,” and “The spirit is said to haunt or torment people who build from elder wood unless they ask permission first.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "This is folklore, cited as folklore. It is the layer the Ruis letter card labels modern, and it is kept apart from the Auraicept gloss trom, elder, which is medieval and sits under r-2.17. Neither of these sources connects the folklore to the ogham letter; the Trees for Life page mentions ruis only as a Gaelic plant name. That is why the card says what modern lore says about the tree, and does not claim a medieval letter of transition." }, { "id": "r-2.21", "url": "https://www.teclamairme.com/#r-2.21", "chapter": 2, "status": "confirmed", "claim": "The chapter epigraph is Whitehead’s, from An Introduction to Mathematics of 1911: nobody goes out to buy zero fish, and zero is the most civilized of the cardinals.", "sources": [ { "title": "A. N. Whitehead, An Introduction to Mathematics (1911), full text at the Internet Archive", "url": "https://archive.org/stream/anintroductionto00whituoft/anintroductionto00whituoft_djvu.txt", "own_work": false, "note": "read as extracted page text on the chapter on zero: “The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought..”", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. **The epigraph drops three words.** Whitehead wrote “It is in a way the most civilized of all the cardinals” and the page prints “It is the most civilized of all the cardinals.” The sense is unchanged and the hedge is Whitehead’s; restoring it is a prose question and is recorded as such." }, { "id": "r-2.22", "url": "https://www.teclamairme.com/#r-2.22", "chapter": 2, "status": "confirmed", "claim": "In positional notation a digit’s value depends on the column it stands in, so a fixed set of signs writes every number. The Babylonian base-60 system is credited as the first positional system, and it lacked a true zero: a space, and later a punctuation sign, marked an empty column. Two comparable sources disagree on the date of the sign and are recorded here rather than silently reconciled: Wikipedia’s Positional notation puts a space or punctuation symbol “such as two slanted wedges” at about 700 BC, while MacTutor’s history of Zero distinguishes them, dating a tablet using three hooks to about 700 BC and the two wedges to about 400 BC, while Wikipedia’s 0 has two slanted wedges repurposed as a placeholder by 300 BC. The Babylonian placeholder card and the chapter write ~300 BC, the latest of the three dates, by which all three sources have the wedges in use, and that card says that the sources differ. Corrections welcome. Either way the placeholder was never written alone or at the end of a number, so it could not distinguish a trailing empty place. The Indians were the first to develop a base-10 positional system, and our place-value system descends from it.", "sources": [ { "title": "Positional notation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Positional_notation", "own_work": false, "note": "read as raw HTML: “the Babylonian numeral system, credited as the first positional numeral system, was base-60. However, it lacked a real zero. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a ‘space’ or a ‘punctuation symbol’ (such as two slanted wedges) between numerals. It was a placeholder…,” and contrasts positional with sign-value systems such as Roman numerals.", "fetched": [ "2026-09-17" ] }, { "title": "Indian numerals, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Indian_numerals/", "own_work": false, "note": "read as raw HTML: “The Babylonians had a place-value system as early as the 19th century BC but the Babylonian systems were to base 60. The Indians were the first to develop a base 10 positional system,” and “our place-value system is a direct descendant of the Indian system.”", "fetched": [ "2026-09-17" ] }, { "title": "0, Wikipedia, § Ancient Near East", "url": "https://en.wikipedia.org/wiki/0", "own_work": false, "note": "“By 300 BC, a punctuation symbol (two slanted wedges) was repurposed as a placeholder.”", "fetched": [ "2026-09-22" ] }, { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "“It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero.”", "fetched": [ "2026-09-22" ] }, { "title": "Babylonian numerals, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Babylonian_numerals/", "own_work": false, "note": "“The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-2.23", "url": "https://www.teclamairme.com/#r-2.23", "chapter": 2, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: A tablet from Kish of perhaps 700 BC marks the empty sexagesimal place with three hooks.", "sources": [ { "title": "Zero, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Zero/", "own_work": false, "note": "read raw: “This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-2.24", "url": "https://www.teclamairme.com/#r-2.24", "chapter": 2, "status": "confirmed", "claim": "The Long Count calendar is a positional, mostly base-twenty numbering system for dates, attested in Epi-Olmec and Maya inscriptions from the first centuries BC forward, and it needed a placeholder zero to work as a place-value count. Its earliest date yet discovered, Stela 2 at Chiapa de Corzo, and the second oldest, Stela C at Tres Zapotes, are in Epi-Olmec style, and a shell glyph stood for zero.", "sources": [ { "title": "Mesoamerican Long Count calendar, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mesoamerican_Long_Count_calendar", "own_work": false, "note": "describes the positional base-20 (with one base-18 place) system and its oldest surviving dates: “The earliest contemporaneous Long Count inscription yet discovered is on Stela 2 at Chiapa de Corzo”; of Tres Zapotes Stela C, “This is the second oldest Long Count date yet discovered”; “La Mojarra Stela 1, the Tuxtla Statuette, Tres Zapotes Stela C and Chiapa Stela 2 are all inscribed in an Epi-Olmec, not Maya, style.”; and “The shell glyph was used to represent the zero concept.”; on the base-18 place: “the second digit from the right (and only that digit) rolls over to zero when it reaches 18.” and “Thus 0.0.1.0.0 does not represent 400 days, but rather only 360 days,” and its table gives 1 tun = 360 days, approximately 1 solar year.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.1", "url": "https://www.teclamairme.com/#r-3.1", "chapter": 3, "status": "confirmed", "claim": "The seven SI defining constants — among them the Planck constant h, the speed of light c, the elementary charge e, the Boltzmann constant k_{B}, and the Avogadro constant N_{A} — and the exact derived relations have the quoted status and values. Since 20 May 2019 the ampere is defined by fixing the exact numerical value of the elementary charge e. ε0 was exact before 20 May 2019; since then ε0 = e²/(2αhc), and it carries the relative uncertainty of α. F = N_A e, the charge of one mole of electrons, named after Michael Faraday. R = N_A k_B. The Planck length is ℓP = √(ℏG/c³), so ℓP² = ℏG/c³.", "sources": [ { "title": "BIPM, The International System of Units", "url": "https://www.bipm.org/en/measurement-units", "own_work": false, "note": "The SI defines its units through fixed constants from 20 May 2019.", "fetched": [] }, { "title": "NIST, SI historical resolutions", "url": "https://www.nist.gov/pml/special-publication-330/sp-330-appendix-1", "own_work": false, "note": "The 1983 meter definition fixes the speed of light; the 2018 resolution establishes the revised SI effective in 2019.", "fetched": [] }, { "title": "NIST 2022 CODATA complete listing", "url": "https://physics.nist.gov/cuu/Constants/Table/allascii.txt", "own_work": false, "note": "lists the exact SI constants and the quoted physical values.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Fundamental Physical Constants, front page", "url": "https://physics.nist.gov/cuu/Constants/", "own_work": false, "note": "the 2022 adjustment is the current one; the next regularly scheduled adjustment is 2026, for which data must be published or preprinted before 31 December 2026. Every CODATA-sourced value in this chapter is CODATA 2022; the cosmological, CMB, PDG, and electron-anomaly values are cited to their own sources at 3.2, 3.3, and 3.5.", "fetched": [ "2026-09-09" ] }, { "title": "2019 revision of the SI, Wikipedia", "url": "https://en.wikipedia.org/wiki/2019_revision_of_the_SI", "own_work": false, "note": "“Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (kB), and the Avogadro constant (NA), respectively.”", "fetched": [ "2026-09-22" ] }, { "title": "Vacuum permittivity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vacuum_permittivity", "own_work": false, "note": "“The elementary charge was redefined exactly in terms of the coulomb as from 20 May 2019, with the effect that the vacuum electric permittivity and the magnetic vacuum permeability no longer have exactly determined values in SI units.”; displays ε0 = 1/(μ0c²) = e²/(2αhc), “with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the values of each of ε0 and μ0 are therefore the same as that for the fine-structure constant”", "fetched": [ "2026-09-22" ] }, { "title": "Faraday constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Faraday_constant", "own_work": false, "note": "“It is named after the English scientist Michael Faraday. Since the 2019 revision of the SI, the Faraday constant has an exactly defined value, the product of the elementary charge (e, in coulombs) and the Avogadro constant (NA, in reciprocal moles)” and “One faraday is the amount of charge in one mole of electrons”", "fetched": [ "2026-09-22" ] }, { "title": "Gas constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gas_constant", "own_work": false, "note": "“The gas constant R is defined as the Avogadro constant NA multiplied by the Boltzmann constant k (or kB)”; “Since the 2019 revision of the SI, both N A and k are defined with exact numerical values when expressed in SI units. As a consequence, the SI value of the molar gas constant is exact.”", "fetched": [ "2026-09-22" ] }, { "title": "Planck units, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_units", "own_work": false, "note": "table of base units: “Planck length … length (L) … l_P = √(ħG/c³)”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.2", "url": "https://www.teclamairme.com/#r-3.2", "chapter": 3, "status": "confirmed", "claim": "The cosmological constant and density parameters have the displayed meanings and model dependence. In Planck units the measured cosmological constant is about 3 × 10⁻¹²². The Planck 2018 fit gives Λ ≈ 1.1 × 10^−52 m^−2; with the CODATA 2022 Planck length, ΛGℏ/c³ = Λ·ℓP² ≈ 1.1056 × 10^−52 × (1.616255 × 10^−35)² ≈ 2.9 × 10^−122 (arithmetic). Base ΛCDM assumes flat space (Ω_k = 0); curvature is fitted only as an extension, which shifts the other parameters. Planck’s total matter density includes the massive neutrinos. Ω_r, photons plus relativistic neutrinos, is about 9 × 10⁻⁵ today; radiation dominated until about fifty thousand years after the Big Bang. No accepted resolution of the cosmological constant problem is known; proposals include modified gravity and anthropic selection in the string theory landscape.", "sources": [ { "title": "Cosmological constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cosmological_constant", "own_work": false, "note": "confirms the 1917 term and vacuum-energy interpretation, and that “there is no known natural way to derive the tiny cosmological constant used in cosmology from particle physics.”", "fetched": [ "2026-09-09" ] }, { "title": "Planck (spacecraft), Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_(spacecraft)", "own_work": false, "note": "confirms the base flat LambdaCDM fit and its parameters.", "fetched": [ "2026-09-09" ] }, { "title": "Friedmann equations, Wikipedia", "url": "https://en.wikipedia.org/wiki/Friedmann_equations", "own_work": false, "note": "confirms the density balance and curvature term.", "fetched": [ "2026-09-09" ] }, { "title": "Cosmological constant problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cosmological_constant_problem", "own_work": false, "note": "gives the Planck-scale estimate as roughly 120 orders of magnitude too large, and attributes “probably the worst theoretical prediction in the history of physics” to Hobson, Efstathiou and Lasenby, General Relativity: An Introduction for Physicists (2006), p. 187. Under Proposed solutions: “Some proposals involve modifying gravity to diverge from general relativity.”; “Some propose an anthropic solution, and argue that we live in one region of a vast multiverse that has different regions with different vacuum energies.”; and “Anthropic arguments gradually gained credibility among many physicists after the discovery of dark energy and the development of the theoretical string theory landscape.”", "fetched": [ "2026-09-09" ] }, { "title": "H. Padmanabhan and T. Padmanabhan, CosMIn: The Solution to the Cosmological Constant Problem, arXiv:1302.3226 (2013)", "url": "https://arxiv.org/abs/1302.3226", "own_work": false, "note": "“observations [1, 2] suggest that ΛL_P² ≈ 3.4 × 10⁻¹²²”", "fetched": [ "2026-09-22" ] }, { "title": "Cosmological Constant Problem and H0 Tension in Void-dominated Cosmology, arXiv:2204.12180 (2022)", "url": "https://arxiv.org/abs/2204.12180", "own_work": false, "note": "quotes the Planck 2018 value (its ref. [39], Aghanim et al., A&A 641, A6): “The cosmological constant in the latest Planck data is reported as below [39], Λobs = 1.1056 × 10−52m−2.”", "fetched": [ "2026-09-22" ] }, { "title": "Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209", "url": "https://arxiv.org/abs/1807.06209", "own_work": false, "note": "“We find good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper)”; §7.3: “The base-ΛCDM model assumes that the spatial hypersurfaces are flat”; Fig. 29: “Constraints on a non-flat universe as a minimal extension to the base-ΛCDM model. Points show samples from the Planck TT,TE,EE+lowE chains coloured by the value of the Hubble parameter”", "fetched": [ "2026-09-22" ] }, { "title": "Lambda-CDM model, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lambda-CDM_model", "own_work": false, "note": "parameter table lists Ω_m as “Total matter density today (including massive neutrinos)”", "fetched": [ "2026-09-22" ] }, { "title": "K. Cahill, The eras of radiation, matter, and dark energy: new information from the Planck Collaboration, arXiv:1606.08865", "url": "https://arxiv.org/abs/1606.08865", "own_work": false, "note": "from Planck 2015 inputs: “The present ratio Ωr0 of the radiation density (2) to the critical density (6) is then” (9.0606 ± 0.1233) × 10⁻⁵; abstract: “The era of radiation ended, and the era of matter began, when the density of radiation dropped below that of matter. This happened 50,953 ± 2236 years after the time of infinite redshift”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.3", "url": "https://www.teclamairme.com/#r-3.3", "chapter": 3, "status": "confirmed", "claim": "The CMB and thermal constants have the quoted temperatures, laws, and identities, and Wien’s root is proven transcendental. Fixsen (2009) gives 2.72548 ± 0.00057 K as the determination from the measurements in the literature; the COBE FIRAS data alone, recalibrated with WMAP, give 2.7260 ± 0.0013 K. Wien’s constant b = 2.897771955… × 10⁻³ m K is exact; the 2.7255 K background peaks in the microwave, near 1 mm (b/T = 1.06 mm). The CMB is the most perfect black-body spectrum ever measured.", "sources": [ { "title": "ESA Planck CMB account", "url": "https://www.esa.int/Science_Exploration/Space_Science/Planck/Planck_reveals_first_stars_were_born_late", "own_work": false, "note": "confirms the CMB origin about 380,000 years after the beginning.", "fetched": [ "2026-09-09" ] }, { "title": "Cosmic microwave background, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cosmic_microwave_background", "own_work": false, "note": "(§ Discovery): supports 1964 — “On 20 May 1964 they made their first measurement clearly showing the presence of the microwave background”; and “the CMB is the most perfect black body spectrum ever measured.”", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Discovery of cosmic microwave background radiation", "url": "https://en.wikipedia.org/wiki/Discovery_of_cosmic_microwave_background_radiation", "own_work": false, "note": "Penzias and Wilson reported the 3.5 K excess in their note “A Measurement of Excess Antenna Temperature at 4080 Mc/s,” Astrophysical Journal 142, 419–421, July 1965 (doi:10.1086/148307).", "fetched": [ "2026-09-22" ] }, { "title": "Wien’s displacement law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Wien%27s_displacement_law", "own_work": false, "note": "confirms the peak equation and root.", "fetched": [ "2026-09-09" ] }, { "title": "Stefan-Boltzmann law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law", "own_work": false, "note": "confirms the Stefan-Boltzmann expression.", "fetched": [ "2026-09-09" ] }, { "title": "D. J. Fixsen, The Temperature of the Cosmic Microwave Background, Astrophysical Journal 707 (2009) 916–920", "url": "https://arxiv.org/abs/0911.1955", "own_work": false, "note": "the abstract: “The determination from the measurements from the literature is cosmic microwave background temperature of 2.72548 +/- 0.00057 K,” while “The FIRAS data are independently recalibrated using the WMAP data to obtain a CMB temperature of 2.7260 +/- 0.0013.” This is where the chapter’s 2.7255 kelvin comes from, and no later measurement supersedes it.", "fetched": [ "2026-09-09" ] }, { "title": "Lindemann–Weierstrass theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "e^{a} is transcendental for every nonzero algebraic a. Wien’s root satisfies x = 5(1 − e^{−x}), so if it were algebraic then e^{−x} = 1 − x/5 would be algebraic too. It is therefore transcendental, which is the only proven-transcendental claim in this chapter.", "fetched": [ "2026-09-09" ] }, { "title": "NIST 2022 CODATA complete listing", "url": "https://physics.nist.gov/cuu/Constants/Table/allascii.txt", "own_work": false, "note": "“Wien wavelength displacement law constant 2.897 771 955... e-3 (exact) m K”", "fetched": [ "2026-09-22" ] }, { "title": "Cosmic Microwave Background, OpenSpace documentation", "url": "https://docs.openspaceproject.com/latest/content/universe/cosmic-microwave-background/index.html", "own_work": false, "note": "“Today, the peak CMB light has redshifted 1,000 times, so it’s now in the microwave with a wavelength of 1 millimeter (1 micron × 1,000 = 1 mm).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.4", "url": "https://www.teclamairme.com/#r-3.4", "chapter": 3, "status": "confirmed", "claim": "The electromagnetic, quantum Hall, Josephson, and atomic identities use the displayed defining relations and CODATA values, and the Rydberg constant is among the most precisely measured constants rather than the single most precise. CODATA 2022 takes α chiefly from the electron’s magnetic-moment anomaly and from atom-interferometry measurements of h/m. R∞ = α²m_ec/(2h), inferred chiefly from hydrogen and deuterium spectroscopy.", "sources": [ { "title": "NIST 2022 CODATA complete listing", "url": "https://physics.nist.gov/cuu/Constants/Table/allascii.txt", "own_work": false, "note": "lists the gas, Faraday, Josephson, flux, atomic, and electron constants.", "fetched": [ "2026-09-09" ] }, { "title": "Quantum Hall effect, Wikipedia", "url": "https://en.wikipedia.org/wiki/Quantum_Hall_effect", "own_work": false, "note": "confirms the h/e^2 resistance relation.", "fetched": [ "2026-09-09" ] }, { "title": "Josephson effect, Wikipedia", "url": "https://en.wikipedia.org/wiki/Josephson_effect", "own_work": false, "note": "confirms the 2eV/h relation.", "fetched": [ "2026-09-09" ] }, { "title": "NIST CODATA 2022, Rydberg constant", "url": "https://physics.nist.gov/cgi-bin/cuu/Value?ryd", "own_work": false, "note": "10 973 731.568 157(12) m^{−1}, relative standard uncertainty 1.1 × 10^{−12}.", "fetched": [ "2026-09-09" ] }, { "title": "NIST CODATA 2022, electron g factor", "url": "https://physics.nist.gov/cgi-bin/cuu/Value?gem", "own_work": false, "note": "−2.002 319 304 360 92(36), relative standard uncertainty 1.8 × 10^{−13}, six times smaller than the Rydberg constant’s. So R_{∞} is among the most precisely measured constants and not the most precisely measured one.", "fetched": [ "2026-09-09" ] }, { "title": "Mohr et al., CODATA recommended values of the fundamental physical constants: 2022, arXiv:2409.03787", "url": "https://arxiv.org/abs/2409.03787", "own_work": false, "note": "carries the inputs for α: the electron anomaly gives “one of the three most accurate determinations of the fine-structure constant α in the 2022 adjustment,” and “the two input data h/m(87Rb) and h/m(133Cs) in the 2022 adjustment obtained using atom interferometry are also of great importance because they provide the two other accurate values of α”", "fetched": [ "2026-09-22" ] }, { "title": "Rydberg constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rydberg_constant", "own_work": false, "note": "“The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.5", "url": "https://www.teclamairme.com/#r-3.5", "chapter": 3, "status": "confirmed", "claim": "QED computes the electron anomaly; the weak mixing angle describes neutral electroweak field mixing; alpha_s is the running QCD coupling. The 2022 measurement of the electron magnetic moment was made by Gabrielse’s group at Northwestern University. Color confinement has no analytic proof in any non-abelian gauge theory. The CODATA 2022 value, 0.22305(23), is 1 − (m_W/m_Z)², which is the on-shell definition of sin²θ_W. Its value is not predicted by the Standard Model; the Weinberg angle is a free parameter.", "sources": [ { "title": "Anomalous magnetic dipole moment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment", "own_work": false, "note": "confirms QED agreement beyond ten significant figures.", "fetched": [ "2026-09-09" ] }, { "title": "Weinberg angle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Weinberg_angle", "own_work": false, "note": "confirms W3/B rotation and the on-shell value: “CODATA 2022 gives the value sin²θw = 1 − (mW/mZ)² = 0.22305(23)” (formula as typeset on the page); and “The specific value is not predicted by the Standard Model: The Weinberg angle θ_{w} is an open, free parameter.”", "fetched": [ "2026-09-09" ] }, { "title": "X. Fan, T. G. Myers, B. A. D. Sukra and G. Gabrielse, Measurement of the Electron Magnetic Moment, Physical Review Letters 130 (2023) 071801", "url": "https://arxiv.org/abs/2209.13084", "own_work": false, "note": "g/2 = 1.001 159 652 180 59(13), the Northwestern measurement the a_{e} card quotes, taken in 2022 and published in 2023.", "fetched": [ "2026-09-09" ] }, { "title": "Particle Data Group, Review of Particle Physics 2026, Quantum Chromodynamics review", "url": "https://pdg.lbl.gov/2026/reviews/rpp2026-rev-qcd.pdf", "own_work": false, "note": "section 9, revised August 2025, gives α_{s}(m_{Z}^{2}) = 0.1180 ± 0.0009 and its running. This is the current edition; of its world average it says that it “changed only marginally as compared to the values reported in the last edition of this Review,” so the chapter’s number stands.", "fetched": [ "2026-09-09" ] }, { "title": "Gabrielse group discovers new measurement of electron magnetic moment, Northwestern Physics and Astronomy", "url": "https://physics.northwestern.edu/about/news/2023/gabrielse-group-discovers-new-measurement-of-electron-magnetic-moment.html", "own_work": false, "note": "“A new measurement of the electron magnetic moment from Physics and Astronomy’s Board of Trustees Professor, Gerald Gabrielse and his research group at Northwestern University’s Center for Fundamental Physics has delivered the most precise measurement of a property of any elementary particle.”", "fetched": [ "2026-09-22" ] }, { "title": "Color confinement, Wikipedia", "url": "https://en.wikipedia.org/wiki/Color_confinement", "own_work": false, "note": "“There is not yet an analytic proof of color confinement in any non-abelian gauge theory.” and “It is sometimes believed that the sole origin of confinement is the very large value of the strong coupling near the Landau pole.” … “It is however incorrect”", "fetched": [ "2026-09-22" ] }, { "title": "Particle Data Group, Review of Particle Physics 2026, Electroweak Model and Constraints on New Physics review", "url": "https://pdg.lbl.gov/2026/reviews/rpp2026-rev-standard-model.pdf", "own_work": false, "note": "“the OS scheme promotes the tree-level formula sin²θW = 1 − MW²/MZ² to a definition of the renormalized sin²θW to all orders in perturbation theory”", "fetched": [ "2026-09-22" ] }, { "title": "Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, in The Millennium Prize Problems (Clay Mathematics Institute)", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "the strong force is the gauge theory of SU(3) with quarks: “The non-abelian gauge theory of the strong force is called Quantum Chromodynamics (QCD)”; the prize problem is the pure theory: “as of yet there is no effective approach to the gauge theory without fermions.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "" }, { "id": "r-3.6", "url": "https://www.teclamairme.com/#r-3.6", "chapter": 3, "status": "confirmed", "claim": "The six displayed laws express standard relativity, Maxwell, Noether, mass-energy, Bianchi, and charge-continuity identities. Luigi Bianchi published the identities named after him in 1902, rediscovering what Levi-Civita says Ricci had found in about 1889; the contracted form, which is the one that makes the divergence of the Einstein tensor vanish, had been published by Aurel Voss in 1880. Contracting the Bianchi identity with a pair of metric tensors gives the contracted form. No experiment has found evidence that magnetic monopoles exist. E = mc² has been called the world’s most famous equation, and it explains the source of the Sun’s energy.", "sources": [ { "title": "Maxwell’s equations, Wikipedia", "url": "https://en.wikipedia.org/wiki/Maxwell%27s_equations", "own_work": false, "note": "confirms div B=0.", "fetched": [ "2026-09-09" ] }, { "title": "Noether’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Noether%27s_theorem", "own_work": false, "note": "confirms symmetry-to-conservation within its hypotheses.", "fetched": [ "2026-09-09" ] }, { "title": "Mass-energy equivalence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence", "own_work": false, "note": "confirms E=mc^2 at rest.", "fetched": [ "2026-09-09" ] }, { "title": "Einstein tensor, Wikipedia", "url": "https://en.wikipedia.org/wiki/Einstein_tensor", "own_work": false, "note": "confirms the contracted Einstein-tensor identity.", "fetched": [ "2026-09-09" ] }, { "title": "Luigi Bianchi, Wikipedia", "url": "https://en.wikipedia.org/wiki/Luigi_Bianchi", "own_work": false, "note": "read as raw HTML: “In 1902, Bianchi rediscovered what are now called the Bianchi identities for the Riemann tensor … According to Tullio Levi-Civita, these identities had first been discovered by Ricci in about 1889, but Ricci apparently forgot all about the matter, which led to Bianchi’s rediscovery. However, the contracted Bianchi identities, which are sufficient for the proof that the divergence of the Einstein tensor always vanishes, had been published by Aurel Voss in 1880.” The 1902 paper is given as Bianchi, Sui simboli a quattro indici e sulla curvatura di Riemann, Rend. Acc. Naz. Lincei 11 (1902).", "fetched": [ "2026-09-17" ] }, { "title": "Continuity equation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Continuity_equation", "own_work": false, "note": "confirms charge conservation.", "fetched": [ "2026-09-09" ] }, { "title": "Contracted Bianchi identities, Wikipedia", "url": "https://en.wikipedia.org/wiki/Contracted_Bianchi_identities", "own_work": false, "note": "proof: “Contract both sides of the above equation with a pair of metric tensors,” arriving at ∇_m R^m_ℓ = ½ ∇_ℓ R", "fetched": [ "2026-09-22" ] }, { "title": "Magnetic monopole, Wikipedia", "url": "https://en.wikipedia.org/wiki/Magnetic_monopole", "own_work": false, "note": "“Magnetism in bar magnets and electromagnets is not caused by magnetic monopoles, and indeed, there is no known experimental or observational evidence that magnetic monopoles exist.”", "fetched": [ "2026-09-22" ] }, { "title": "Albert Einstein, Wikipedia", "url": "https://en.wikipedia.org/wiki/Albert_Einstein", "own_work": false, "note": "“His mass–energy equivalence formula E = mc², which arises from special relativity, has been called “the world’s most famous equation.””", "fetched": [ "2026-09-22" ] }, { "title": "Sun, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sun", "own_work": false, "note": "“Albert Einstein provided the essential clue to the source of the Sun’s energy output with his mass–energy equivalence relation E = mc².”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.7", "url": "https://www.teclamairme.com/#r-3.7", "chapter": 3, "status": "confirmed", "claim": "The Planck units, Hubble values, atomic key values, and Fermi coupling have the displayed definitions and values. α_G = Gm_e²/(ℏc) = (m_e/m_P)² ≈ 1.7518 × 10⁻⁴⁵, computed from the CODATA 2022 values. The Hubble tension is the most statistically significant tension in the ΛCDM model. The proton-to-electron mass ratio is a fundamental dimensionless constant that no theory explains; it is measured. The Planck mass, about a flea’s egg, is the one Planck base unit on a human scale, and (m_e/m_P)² is the gravitational coupling constant α_G. The universe is usually given a temperature of about 10³² K at the Planck time. The Planck time is the time light takes to cross one Planck length, t_P = ℓ_P/c. The classical electron radius is e²/(4πε₀m_ec²), the radius of a charged sphere whose field energy is m_ec², also called the Thomson scattering length, and equals α²a₀; the real electron is treated as a point charge. At the Planck temperature the wavelength of thermal radiation reaches the Planck length.", "sources": [ { "title": "NIST 2022 CODATA complete listing", "url": "https://physics.nist.gov/cuu/Constants/Table/allascii.txt", "own_work": false, "note": "lists the atomic, electron, magnetic, Fermi, and input constants.", "fetched": [ "2026-09-09" ] }, { "title": "Planck units, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_units", "own_work": false, "note": "confirms the Planck-unit definitions. These are the derived Planck length, time, and mass, built from ℏ, G, and c. The Planck constant h itself is a different thing — an SI defining constant, exact by definition — and is covered at 3.1. Of the Planck temperature: “At the Planck temperature, the wavelength of light emitted by thermal radiation reaches the Planck length. There are no known physical models able to describe temperatures greater than T_{P}.”", "fetched": [ "2026-09-09" ] }, { "title": "Planck (spacecraft), Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_(spacecraft)", "own_work": false, "note": "confirms the Planck Hubble and age values.", "fetched": [ "2026-09-09" ] }, { "title": "A. G. Riess et al., A Comprehensive Measurement of the Local Value of the Hubble Constant…, Astrophysical Journal Letters 934 (2022) L7", "url": "https://arxiv.org/abs/2112.04510", "own_work": false, "note": "H_{0} = 73.04 ± 1.04 km s^{−1} Mpc^{−1}, and the paper states a five-sigma difference with the value Planck and ΛCDM predict. This is the late-universe half of the Hubble tension.", "fetched": [ "2026-09-09" ] }, { "title": "Dimensionless physical constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dimensionless_physical_constant", "own_work": false, "note": "“A gravitational coupling constant is a constant characterizing the gravitational attraction between a given pair of elementary particles. The electron mass is typically used, and the associated constant typically denoted αG.” Displays αG = Gmₚ²/(ħc), mₚ the particle rest mass.", "fetched": [ "2026-09-22" ] }, { "title": "E. Di Valentino et al., In the Realm of the Hubble tension – a Review of Solutions, arXiv 2103.01183 (2021)", "url": "https://arxiv.org/abs/2103.01183", "own_work": false, "note": "carries the ranking of the tension: “The most statistically significant tension is the 4−6σ disagreement between predictions of the Hubble constant H0 by early time probes with ΛCDM model, and a number of late time, model-independent determinations of H0 from local measurements”", "fetched": [ "2026-09-22" ] }, { "title": "Planck units, OeisWiki", "url": "https://oeis.org/wiki/Planck_units", "own_work": false, "note": "carries the human scale and the flea’s egg: “Unlike all other Planck base units and most Planck derived units, the Planck mass has a scale more or less conceivable to humans. (It is traditionally said to be about the mass of a flea, but more accurately it is about the mass of a flea egg.)”", "fetched": [ "2026-09-22" ] }, { "title": "Ned Wright, Brief History of the Universe, UCLA", "url": "https://www.astro.ucla.edu/~wright/BBhistory.html", "own_work": false, "note": "carries the temperature at the Planck time (order of magnitude only): “The Planck time: 10⁻⁴³ seconds.” … “The temperature is T=10³² K.”", "fetched": [ "2026-09-22" ] }, { "title": "Planck time, Simple English Wikipedia", "url": "https://simple.wikipedia.org/wiki/Planck_time", "own_work": false, "note": "“One Planck time is the time it would take a photon travelling at the speed of light to cross a distance equal to one Planck length.”", "fetched": [ "2026-09-22" ] }, { "title": "Classical electron radius, Wikipedia", "url": "https://en.wikipedia.org/wiki/Classical_electron_radius", "own_work": false, "note": "carries the formula r_e = (1/4πε₀)e²/(m_ec²), the reading, the α-relations r_e = λ̄_e α = a₀α², and the point-like electron: “A classical charged sphere producing an electric field with energy equal to the electron’s rest mass energy would have a radius equal to the classical electron radius.”; “The classical electron radius is sometimes known as the Lorentz radius or the Thomson scattering length.”; “Consequently, electrons are considered point charges in modern theories.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-3.8", "url": "https://www.teclamairme.com/#r-3.8", "chapter": 3, "status": "confirmed", "claim": "The displayed product is approximately 1.3 times 10^-154 and finite positive.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Substitute the displayed CODATA-scale values; exact-one brackets cancel, the thermal bracket is pi^2/60, and the remaining positive factors multiply to approximately 1.3e-154.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-3.9", "url": "https://www.teclamairme.com/#r-3.9", "chapter": 3, "status": "judgement", "claim": "The Lambda wording is conventional shorthand, and the glossary supplies the dimensional distinction.", "sources": [ { "title": "Cosmological constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cosmological_constant", "own_work": false, "note": "discusses Lambda and vacuum energy.", "fetched": [ "2026-09-09" ] }, { "title": "Λ, the cosmological constant", "url": "#g-Lambda", "own_work": false, "note": "in this book — Λ has units of m^{−2}, and the vacuum-energy density that goes with it is built from Λ, c, and G.", "fetched": [] } ], "editorial_note": "The chapter calls Λ the energy of empty space. That is the ordinary popular description rather than the literal one: Λ is a curvature, measured in m−2, and the vacuum-energy density that corresponds to it has to be built from Λ, c, and G, with the factor depending on whether mass or energy density is meant. The prose keeps the familiar phrase and the glossary card carries the distinction." }, { "id": "r-3.10", "url": "https://www.teclamairme.com/#r-3.10", "chapter": 3, "status": "judgement", "claim": "The phrase describing 1.3 times 10^-154 as close to nothing is rhetorical rather than a literal minimum claim.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "The product evaluates to approximately 1.3e-154, but no fetched source can establish a global minimum among nonzero numbers.", "fetched": [] } ], "editorial_note": "The product is finite, positive, and about 1.3 × 10−154. There is no smallest positive real number, so “as close to nothing as a number can get” cannot be literally true; it is a remark about scale, not a claim about a minimum, and it is left as the author wrote it." }, { "id": "r-3.11", "url": "https://www.teclamairme.com/#r-3.11", "chapter": 3, "status": "confirmed", "claim": "Minkowski gave spacetime its invariant interval in 1908, and the interval between two events joined by a light ray is exactly zero.", "sources": [ { "title": "Minkowski space, Wikipedia", "url": "https://en.wikipedia.org/wiki/Minkowski_space", "own_work": false, "note": "dates Die Grundgleichungen fuer die elektromagnetischen Vorgaenge in bewegten Koerpern to 1908 and the Cologne Raum und Zeit lecture to 1908, and defines a null or lightlike vector by c^2t^2 = r^2, which is the interval being exactly zero along a light ray; and: “Minkowski spacetime is constructed so that the speed of light will be the same constant regardless of the reference frame in which it is measured.”", "fetched": [ "2026-09-10" ] }, { "title": "Hermann Minkowski, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hermann_Minkowski", "own_work": false, "note": "read as raw wikitext: the address “called “Space and Time” delivered at the 80th Assembly of German Natural Scientists and Physicians (21 September 1908),” in which Minkowski set out the four-dimensional geometry of spacetime; this page does not name the city (the Minkowski space line, now the Minkowski spacetime article, gives Cologne).", "fetched": [ "2026-09-10" ] } ], "editorial_note": "The chapter prose and the #dslight card both carry the year 1908 and both stood on 3.6, which sources div B = 0, Noether, E = mc², the Einstein tensor, and charge continuity but does not mention Minkowski at all. The date is correct; it simply had nothing behind it. Both the 1908 paper and the 1908 Cologne lecture are Minkowski’s; the lecture was published in 1909, which is why some sources give that year for Raum und Zeit." }, { "id": "r-3.12", "url": "https://www.teclamairme.com/#r-3.12", "chapter": 3, "status": "confirmed", "claim": "The hierarchy problem is the unexplained disparity between the strength of gravity and that of the other forces, of which the ratio of the electromagnetic to the gravitational coupling between two electrons is one statement. The ratio e²/(4πε₀Gm_e²) ≈ 4.17 × 10⁴² is one of the large-number coincidences behind Dirac’s hypothesis. Between two electrons the ratio e²/(4πε₀Gmₑ²) = α/α_G is about 4 × 10⁴²; proposed answers include supersymmetry, extra dimensions, and composite Higgs models.", "sources": [ { "title": "Hierarchy problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hierarchy_problem", "own_work": false, "note": "defines the hierarchy problem as the large discrepancy between aspects of the weak force and gravity, and gives the weakness of gravity relative to the other forces as its primary numerical example, alongside the Higgs-mass fine-tuning form of the same problem: “the question is why the Higgs boson is so much lighter than the Planck mass”; “one would expect that the large quantum contributions to the square of the Higgs boson mass would inevitably make the mass huge, comparable to the scale at which new physics appears unless there is an incredible fine-tuning cancellation between the quadratic radiative corrections and the bare mass.”", "fetched": [ "2026-09-10" ] }, { "title": "Dirac large numbers hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis", "own_work": false, "note": "“LNH was Dirac’s personal response to a set of large number “coincidences” that had intrigued other theorists of his time.” The page displays e²/(4πε₀ G mₑ²) ≈ 4.1666763·10⁴², which is α/α_G.", "fetched": [ "2026-09-22" ] }, { "title": "Composite Higgs models, Wikipedia", "url": "https://en.wikipedia.org/wiki/Composite_Higgs_models", "own_work": false, "note": "“Often referred to as “natural” composite Higgs models, CHMs are constructions that attempt to alleviate fine-tuning or “naturalness” problem of the Standard Model.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The chapter uses the broad sense: α/α_G is about 4 × 10⁴², so “about 10⁴²” is right, and it is the force-strength statement of the hierarchy problem rather than the Higgs-mass fine-tuning statement. The sentence stood on 3.7, which covers Planck units, the Hubble values, the atomic key values, and the Fermi coupling, and does not mention the hierarchy problem." }, { "id": "r-3.13", "url": "https://www.teclamairme.com/#r-3.13", "chapter": 3, "status": "confirmed", "claim": "The chapter epigraph is Feynman’s, on the fine-structure constant, from QED: The Strange Theory of Light and Matter of 1985.", "sources": [ { "title": "Fine-structure constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fine-structure_constant", "own_work": false, "note": "read as raw HTML: it quotes Feynman at length — “It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics” — which is the epigraph and the source of the “greatest damn mysteries” line the #alpha card also uses.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker." }, { "id": "r-3.14", "url": "https://www.teclamairme.com/#r-3.14", "chapter": 3, "status": "confirmed", "claim": "The Standard Model is the quantum field theory whose defining internal symmetry is the local gauge group SU(3) × SU(2) × U(1), whose three factors give rise to the three non-gravitational interactions. Writing the most general Lagrangian with that symmetry leaves 19 parameters whose numerical values must be established by experiment; explaining neutrino mass is believed to require a further 7 or 8.", "sources": [ { "title": "Standard Model, Wikipedia", "url": "https://en.wikipedia.org/wiki/Standard_Model", "own_work": false, "note": "read as raw HTML: “The local SU(3) × SU(2) × U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model… the three factors of the gauge symmetry give rise to the three fundamental interactions”; “the dynamics depends on 19 parameters, whose numerical values are established by experiment”; and “explaining neutrino mass will require an additional 7 or 8 constants.” The numbered parameter table runs to 19, ending at the Higgs mass.", "fetched": [ "2026-09-17" ] }, { "title": "Review of Particle Physics, Electroweak model and constraints on new physics, Particle Data Group", "url": "https://pdg.lbl.gov/2024/reviews/rpp2024-rev-standard-model.pdf", "own_work": false, "note": "the field’s own reference review of the model and of the measured inputs the chapter’s three couplings are quoted from.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-3.15", "url": "https://www.teclamairme.com/#r-3.15", "chapter": 3, "status": "confirmed", "claim": "ΛCDM is the standard cosmological model, with a cosmological constant Λ for dark energy and cold dark matter. The Planck collaboration’s version is fitted with six parameters — baryon density, dark matter density, the angular size of the sound horizon, the amplitude and the spectral index of the primordial fluctuations, and the reionization optical depth — six being the smallest number that gives an acceptable fit; the Hubble constant and the dark-energy density are then calculated from them rather than fitted, and the total density is fixed at 1 and the dark-energy equation of state at −1.", "sources": [ { "title": "Lambda-CDM model, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lambda-CDM_model", "own_work": false, "note": "read as raw HTML: “The Planck collaboration version of the ΛCDM model is based on six parameters… Six is the smallest number of parameters needed to give an acceptable fit to the observations; other possible parameters are fixed at ‘natural’ values, e.g. total density parameter = 1.00, dark energy equation of state = −1… From these six parameters, the other model values, such as the Hubble constant and the dark energy density, can be calculated.”", "fetched": [ "2026-09-17" ] }, { "title": "Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209 (A&A 641 (2020) A6)", "url": "https://arxiv.org/abs/1807.06209", "own_work": false, "note": "the same paper, free on arXiv: “good consistency with the standard spatially-flat 6-parameter ΛCDM cosmology,” with H₀ = 67.4 ± 0.5 km/s/Mpc inferred from the fit rather than fitted.", "fetched": [ "2026-09-22" ] }, { "title": "Planck 2018 results. VI, readable HTML (ar5iv)", "url": "https://ar5iv.labs.arxiv.org/html/1807.06209", "own_work": false, "note": "section 2.1 names the six fitted base-ΛCDM parameters: the baryon density ω_b, the cold dark matter density ω_c, θ_MC (“an approximation to the observed angular size of the sound horizon at recombination”), the reionization optical depth τ, the amplitude A_s and the spectral index n_s of the primordial perturbations — one amplitude.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "This is the model the chapter means by “taken consistently from the same cosmological model” in the Ω bracket. It is why the four density parameters may be added and why the individual values should not be mixed between fits." }, { "id": "r-3.16", "url": "https://www.teclamairme.com/#r-3.16", "chapter": 3, "status": "confirmed", "claim": "The 26th General Conference on Weights and Measures voted unanimously on 16 November 2018 to redefine four SI base units, and the new definitions took effect on 20 May 2019, the 144th anniversary of the Metre Convention. The kilogram, ampere, kelvin, and mole are now defined by fixing exact numerical values for the Planck constant h, the elementary charge e, the Boltzmann constant k_B, and the Avogadro constant N_A; the second, meter, and candela had already been defined by physical constants.", "sources": [ { "title": "2019 revision of the SI, Wikipedia", "url": "https://en.wikipedia.org/wiki/2019_revision_of_the_SI", "own_work": false, "note": "read as raw HTML: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e), the Boltzmann constant (k_B), and the Avogadro constant (N_A), respectively. The second, metre, and candela had previously been redefined using physical constants,” and “The vote, which was held on 16 November 2018 at the 26th CGPM, was unanimous.”", "fetched": [ "2026-09-17" ] }, { "title": "Measurement units, Bureau International des Poids et Mesures", "url": "https://www.bipm.org/en/measurement-units", "own_work": false, "note": "the defining body’s own page for the seven defining constants of the SI.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-3.17", "url": "https://www.teclamairme.com/#r-3.17", "chapter": 3, "status": "confirmed", "claim": "A dimensionless quantity, or quantity of dimension one, has no unit of measurement, so its numerical value is the same in every system of units; the fine-structure constant, about 1/137, and the proton-to-electron mass ratio, about 1836, are the standard examples, and CODATA lists both as pure numbers.", "sources": [ { "title": "Dimensionless quantity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dimensionless_quantity", "own_work": false, "note": "read as raw wikitext: “Dimensionless quantities, or quantities of dimension one, are quantities with no associated units of measurement,” and of a ratio, “its value remains independent of the specific units of volume used”; its list of dimensionless quantities in physics and engineering includes the fine-structure constant, “α ≈ 1/137,” and “β (or μ) ≈ 1836, the proton-to-electron mass ratio.”", "fetched": [ "2026-09-18" ] }, { "title": "NIST CODATA 2022, proton-electron mass ratio", "url": "https://physics.nist.gov/cgi-bin/cuu/Value?mpsme", "own_work": false, "note": "read as raw HTML: numerical value 1836.152 673 426(32), given with no unit.", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026. The #dimensionless card was the only one of 723 glossary cards that cited nothing at all; a sweep with citecheck.py found it." }, { "id": "r-3.18", "url": "https://www.teclamairme.com/#r-3.18", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Bohr’s model of the hydrogen atom, whose innermost orbit is the Bohr radius, dates from 1913; Dirac’s equation of 1928 gives the electron a g-factor of exactly 2; Schwinger first computed the leading QED correction a_e = alpha/2pi in 1948; The Planck 2018 fit (final data release), combined with BAO, gives t0 = 13.787 +/- 0.020 billion years; Arnold Sommerfeld introduced the fine-structure constant in 1916; Ștefan Procopiu obtained the electron’s magnetic moment in 1913, and Bohr obtained it the same year from his atom model, and Pauli named the Bohr magneton in 1920; Pauli gave the Bohr magneton its name in 1920; Ole Rømer made the first estimate of the speed of light in 1676 from the moons of Jupiter; Arthur Compton explained the scattering of X-rays by electrons in 1923, with a shift proportional to h/(m_e c). The Bohr radius is about 0.53 Å, half an ångström. The age follows from H₀ and the density parameters through the Friedmann equation, and the oldest stars’ ages are compatible with the Planck 2018 value. The electron’s magnetic moment is 1.00115965218046(18) Bohr magnetons in magnitude, that is (1 + a_e) of one. Below the Compton wavelength an electron cannot be localized without enough energy to make a pair, and quantum field theory becomes essential. α is Sommerfeld’s ratio of the electron’s speed in the first Bohr orbit to c, and Feynman called its unexplained value one of the greatest damn mysteries of physics. The Bohr radius is approximately the most probable electron–nucleus distance in ground-state hydrogen.", "sources": [ { "title": "Bohr radius, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bohr_radius", "own_work": false, "note": "read raw: “In the Bohr model for atomic structure, put forward by Niels Bohr in 1913, electrons orbit a central nucleus under electrostatic attraction.”; “The Bohr radius (a_{0}) is a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state.”", "fetched": [ "2026-09-18" ] }, { "title": "Dirac equation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dirac_equation", "own_work": false, "note": "read raw: “In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928..”", "fetched": [ "2026-09-18" ] }, { "title": "Anomalous magnetic dipole moment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment", "own_work": false, "note": "read raw: “where \\alpha is the fine-structure constant. This result was first found by Julian Schwinger in 1948.”", "fetched": [ "2026-09-18" ] }, { "title": "Planck (spacecraft), Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_(spacecraft)", "own_work": false, "note": "read raw: “Results from the final analysis of Planck data were released in July 2018..”", "fetched": [ "2026-09-18" ] }, { "title": "Fine-structure constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fine-structure_constant", "own_work": false, "note": "read raw: “The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom.”; it gives α as “The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom … to the speed of light in vacuum, c,” “Sommerfeld’s original physical interpretation”; and quotes Feynman: “Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans.”", "fetched": [ "2026-09-18" ] }, { "title": "Bohr magneton, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bohr_magneton", "own_work": false, "note": "read raw: “The Romanian physicist Ștefan Procopiu had obtained the expression for the magnetic moment of the electron in 1913.”; “In 1920, Wolfgang Pauli gave the Bohr magneton its name in an article where he contrasted it with the magneton of the experimentalists which he called the Weiss magneton.”; “In the summer of 1913, the values for the natural units of atomic angular momentum and magnetic moment were obtained by the Danish physicist Niels Bohr as a consequence of his atom model.”", "fetched": [ "2026-09-18" ] }, { "title": "Speed of light, Wikipedia", "url": "https://en.wikipedia.org/wiki/Speed_of_light", "own_work": false, "note": "read raw: “Ole Rømer used an astronomical measurement to make the first quantitative estimate of the speed of light in the year 1676..”", "fetched": [ "2026-09-18" ] }, { "title": "Compton wavelength, Wikipedia", "url": "https://en.wikipedia.org/wiki/Compton_wavelength", "own_work": false, "note": "read raw: “It was introduced by Arthur Compton in 1923 in his explanation of the scattering of photons by electrons (a process known as Compton scattering)..”", "fetched": [ "2026-09-18" ] }, { "title": "Angstrom, Wikipedia", "url": "https://en.wikipedia.org/wiki/Angstrom", "own_work": false, "note": "carries the size in ångströms: “the atomic unit of length, the bohr—about 0.53 Å”", "fetched": [ "2026-09-22" ] }, { "title": "Age of the universe, Wikipedia", "url": "https://en.wikipedia.org/wiki/Age_of_the_universe", "own_work": false, "note": "carries the method and the stellar check: “With accurate measurements of these parameters, the age of the universe can be determined by using the Friedmann equation.”; “Both values are compatible with the Planck 2018 number.” (the Methuselah star, 14.27±0.80 billion years, and BD +17°3248, 13.8±4 billion years)", "fetched": [ "2026-09-22" ] }, { "title": "Electron magnetic moment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Electron_magnetic_moment", "own_work": false, "note": "carries the ratio: “In units of the Bohr magneton (μB), it is −1.00115965218046(18), which has a relative uncertainty of 1.8×10−13.”", "fetched": [ "2026-09-22" ] }, { "title": "John Baez, Length Scales in Physics, UC Riverside", "url": "https://math.ucr.edu/home/baez/lengths.html", "own_work": false, "note": "carries both readings: “Trying to localize an electron to within less than its Compton wavelength makes its momentum so uncertain that it can have an energy large enough to make an extra electron-positron pair!” and “the length scale at which relativistic quantum field theory becomes crucial for its accurate description”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.19", "url": "https://www.teclamairme.com/#r-3.19", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Maxwell’s equations of 1865 (the ‘Dynamical Theory’ paper), from which charge continuity follows; Coulomb measured/published the inverse-square law of electric force with a torsion balance in 1785; The four Maxwell equations were assembled in 1861–65; Millikan and Fletcher’s oil-drop experiment was begun in 1909 (final form published 1913); Einstein’s field equations date from November 1915; In a paper received in September 1905 and published that November, Einstein showed that a body’s inertia depends on its energy content (E = mc²); Faraday’s laws of electrolysis of 1834; Enrico Fermi’s 1934 theory of beta decay; In 1961 Deaver and Fairbank, and Doll and Näbauer, independently found flux quantization in steps of h/2e. Heaviside recast Maxwell’s equations in modern vector form in 1884. Faraday’s first law: the mass deposited at an electrode is proportional to the charge passed. Fermi’s theory was a four-fermion contact interaction; G_F is measured most precisely from the muon lifetime and is set by the mass of the heavy W boson. Deaver and Fairbank worked at Stanford; Doll and Näbauer in Herrsching. Mass–energy equivalence accounts for the energy released by nuclear fission in both nuclear weapons and nuclear power.", "sources": [ { "title": "A Dynamical Theory of the Electromagnetic Field, Wikipedia", "url": "https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field", "own_work": false, "note": "read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..”", "fetched": [ "2026-09-18" ] }, { "title": "Coulomb’s law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Coulomb%27s_law", "own_work": false, "note": "read raw: “Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his application of a torsion balance to determine that the magnitude of the electric force between two point charges..”", "fetched": [ "2026-09-18" ] }, { "title": "Maxwell’s equations, Wikipedia", "url": "https://en.wikipedia.org/wiki/Maxwell%27s_equations", "own_work": false, "note": "read raw: “Maxwell’s equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law..”", "fetched": [ "2026-09-18" ] }, { "title": "Oil drop experiment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Oil_drop_experiment", "own_work": false, "note": "read raw: “The oil drop experiment was performed by Robert A. Millikan and Harvey Fletcher in 1909 to measure the elementary electric charge (the charge of the electron)..”", "fetched": [ "2026-09-18" ] }, { "title": "Einstein field equations, Wikipedia", "url": "https://en.wikipedia.org/wiki/Einstein_field_equations", "own_work": false, "note": "read raw: “The equations were published by Albert Einstein in 1915 in the form of a tensor equation.”", "fetched": [ "2026-09-18" ] }, { "title": "Mass–energy equivalence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence", "own_work": false, "note": "read raw: “Einstein did not write the exact formula E = mc² in his 1905 Annus Mirabilis paper “Does the Inertia of an object Depend Upon Its Energy Content?”; rather, the paper states that if a body gives off the energy L by emitting light, its mass diminishes by L/c².”; and “a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power.”", "fetched": [ "2026-09-18" ] }, { "title": "M. Faraday, Experimental researches in electricity — Seventh Series, Phil. Trans. R. Soc. 124 (1834) 77–122, Crossref record", "url": "https://api.crossref.org/works/10.1098/rstl.1834.0008", "own_work": false, "note": "read raw: “the publisher’s Crossref record gives the year of publication as 1834.”", "fetched": [ "2026-09-18" ] }, { "title": "E. Fermi, Tentativo di una teoria dei raggi β, Il Nuovo Cimento 11 (1934) 1–19, Crossref record", "url": "https://api.crossref.org/works/10.1007/BF02959820", "own_work": false, "note": "read raw: “the publisher’s Crossref record gives the year of publication as 1934.”", "fetched": [ "2026-09-18" ] }, { "title": "Magnetic flux quantum, Wikipedia", "url": "https://en.wikipedia.org/wiki/Magnetic_flux_quantum", "own_work": false, "note": "read raw: “The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank, and independently by R. Doll and M. Näbauer, in 1961..”", "fetched": [ "2026-09-18" ] }, { "title": "Oliver Heaviside, Wikipedia", "url": "https://en.wikipedia.org/wiki/Oliver_Heaviside", "own_work": false, "note": "“In 1884 he recast Maxwell’s mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell’s equations.”", "fetched": [ "2026-09-22" ] }, { "title": "Annus mirabilis papers, Wikipedia", "url": "https://en.wikipedia.org/wiki/Annus_mirabilis_papers", "own_work": false, "note": "“On 21 November Annalen der Physik published a fourth paper (received September 27): “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?” (“Does the Inertia of a Body Depend Upon Its Energy Content?”)”", "fetched": [ "2026-09-22" ] }, { "title": "Faraday’s laws of electrolysis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Faraday%27s_laws_of_electrolysis", "own_work": false, "note": "“Michael Faraday reported that the mass (m) of a substance deposited or liberated at an electrode during electrolysis is directly proportional to the charge (Q, for which the SI unit is the ampere-second or coulomb) passed through the electrolyte.”", "fetched": [ "2026-09-22" ] }, { "title": "Fermi’s interaction, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermi%27s_interaction", "own_work": false, "note": "carries the contact interaction, the muon lifetime, and the heavy boson: “The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram).”; “In the original theory, Fermi assumed that the form of interaction is a contact coupling of two vector currents.”; “The most precise experimental determination of the Fermi constant comes from measurements of the muon lifetime, which is inversely proportional to the square of GF (when neglecting the muon mass against the mass of the W boson).”; “MW is the mass of the W boson, which mediates the decay in question.”", "fetched": [ "2026-09-22" ] }, { "title": "Wikipedia (German), Robert Doll", "url": "https://de.wikipedia.org/wiki/Robert_Doll", "own_work": false, "note": "places Doll at the Herrsching outstation of the low-temperature commission, where he and Martin Näbauer showed flux quantization in 1961: “Ab 1953 war er wissenschaftlicher Mitarbeiter an der Herrschinger Außenstelle der Kommission für Tieftemperaturforschung.”", "fetched": [ "2026-09-24" ] }, { "title": "Wikipedia, William M. Fairbank", "url": "https://en.wikipedia.org/wiki/William_M._Fairbank", "own_work": false, "note": "the Stanford half of the 1961 experiments: “At Stanford, one of his most famous discoveries has been the flux quantization in superconducting tin.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.20", "url": "https://www.teclamairme.com/#r-3.20", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Φ0 = h/(2e) exact since 2019; Newton’s Principia of 1687 stated the inverse-square law but never wrote G as a number; Cavendish’s torsion-balance experiment of 1798 weighed the Earth; CODATA 2018 gives G = 6.67430(15) x 10^-11 m^3 kg^-1 s^-2 (same as CODATA 2022); Clapeyron’s ideal-gas law (1834); Max Planck introduced h in 1900 to explain black-body radiation; Einstein took the quantum literally in 1905; Hubble’s 1929 law v = H0 d; Planck satellite’s 2018 fit gives H0 = 67.4 ± 0.5 km/s/Mpc. PV = nRT combines the laws of Boyle, Charles, and Avogadro. Planck regarded dividing energy into increments as a mathematical artifice. Since h is fixed, a Kibble balance measures the kilogram by fixing it.", "sources": [ { "title": "Magnetic flux quantum, Wikipedia", "url": "https://en.wikipedia.org/wiki/Magnetic_flux_quantum", "own_work": false, "note": "read raw: “As a result of the 2019 revision of the SI, the Planck constant h has a fixed value.”", "fetched": [ "2026-09-18" ] }, { "title": "Philosophiae Naturalis Principia Mathematica, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Philosophi%C3%A6_Naturalis_Principia_Mathematica", "own_work": false, "note": "read raw: “authorized (imprimatur) by Samuel Pepys, then-president of the Royal Society on 5 July 1686 and first published in 1687..”", "fetched": [ "2026-09-18" ] }, { "title": "Gravitational constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gravitational_constant", "own_work": false, "note": "read raw: “The first direct measurement of gravitational attraction between two bodies in the laboratory was performed in 1798, seventy-one years after Newton’s death, by Henry Cavendish..”", "fetched": [ "2026-09-18" ] }, { "title": "Fundamental Physical Constants — Complete Listing, 2018 CODATA adjustment, NIST", "url": "https://physics.nist.gov/cuu/Constants/ArchiveASCII/allascii_2018.txt", "own_work": false, "note": "read raw: “2018 CODATA adjustment ... Newtonian constant of gravitation 6.674 30 e-11 0.000 15 e-11 m^3 kg^-1 s^-2.”", "fetched": [ "2026-09-18" ] }, { "title": "Émile Clapeyron, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=%C3%89mile_Clapeyron", "own_work": false, "note": "read raw: “In 1834, he made his first contribution to the creation of modern thermodynamics by publishing a report entitled Mémoire sur la puissance motrice de la chaleur (Memoir on the Motive Power of Heat).”", "fetched": [ "2026-09-18" ] }, { "title": "Planck constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_constant", "own_work": false, "note": "read raw: “The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation.”; “Technologies of mass metrology such as the Kibble balance measure the kilogram by fixing the Planck constant.”", "fetched": [ "2026-09-18" ] }, { "title": "Photoelectric effect, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Photoelectric_effect", "own_work": false, "note": "read raw: “In 1905, Albert Einstein proposed a theory of the photoelectric effect using a concept that light consists of tiny packets of energy known as photons or light quanta..”", "fetched": [ "2026-09-18" ] }, { "title": "Hubble’s law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hubble%27s_law", "own_work": false, "note": "read raw: “The discovery of Hubble’s law is attributed to work published by Edwin Hubble in 1929,.”", "fetched": [ "2026-09-18" ] }, { "title": "Planck (spacecraft), Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_(spacecraft)", "own_work": false, "note": "read raw: “Results from the final analysis of Planck data were released in July 2018. ... Planck measured the Hubble constant to be 67.4 ± 0.5 km/s/Mpc, increasing the tension with direct local measurements..”", "fetched": [ "2026-09-18" ] }, { "title": "Ideal gas law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ideal_gas_law", "own_work": false, "note": "“It was first stated by Benoît Paul Émile Clapeyron and independently of him, Dmitry Mendeleev in 1834 as a combination of the empirical Boyle’s law, Charles’s law, Avogadro’s law, and Gay-Lussac’s law.” and “The equation of state given here (PV = nRT) applies only to an ideal gas”", "fetched": [ "2026-09-22" ] }, { "title": "Planck’s law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck%27s_law", "own_work": false, "note": "“While Planck originally regarded the hypothesis of dividing energy into increments as a mathematical artifice, introduced merely to get the correct answer, other physicists including Albert Einstein built on his work”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.21", "url": "https://www.teclamairme.com/#r-3.21", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Dirac wrote plain h for h/2π from 1926 until 1930, when he introduced the symbol ℏ in The Principles of Quantum Mechanics; Dirac’s 1930 Principles of Quantum Mechanics; K_J = 2e/h exact since 2019; Boltzmann’s 1877 relation S = k log W; Planck introduced k in the same 1900 paper as h; 1998 discovery of accelerated expansion; accelerated expansion recognized by the 2011 Nobel Prize; γ appeared in Lorentz’s transformations of 1904; Einstein’s special relativity of 1905; Karl Schwarzschild solved the field equations for a point mass in 1916, within months of their publication, and died on 11 May 1916 in Potsdam, having been invalided home from the Russian front that March. γ is 1 at rest and about 7000 for LHC protons. Brian Josephson predicted the effect in 1962. The Josephson voltage standard gives the standard representation of the volt. The Schwarzschild radius r_s = 2GM/c² is an event horizon, about 3 km for the Sun and 9 mm for the Earth, and equals twice the reduced Compton wavelength of one Planck mass.", "sources": [ { "title": "Planck constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_constant", "own_work": false, "note": "read raw: “Then, in 1926, in their seminal papers, Schrödinger and Dirac again introduced special symbols for it: K in the case of Schrödinger, and h in the case of Dirac. Dirac continued to use h in this way until 1930, when he introduced the symbol ℏ in his book The Principles of Quantum Mechanics..”", "fetched": [ "2026-09-18" ] }, { "title": "The Principles of Quantum Mechanics, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=The_Principles_of_Quantum_Mechanics", "own_work": false, "note": "read raw: “The Principles of Quantum Mechanics is an influential monograph written by Paul Dirac and first published by Oxford University Press in 1930..”", "fetched": [ "2026-09-18" ] }, { "title": "2019 revision of the SI, Wikipedia", "url": "https://en.wikipedia.org/wiki/2019_revision_of_the_SI", "own_work": false, "note": "read raw: “Effective 20 May 2019, the 144th anniversary of the Metre Convention, the kilogram, ampere, kelvin, and mole are defined by setting exact numerical values, when expressed in SI units, for the Planck constant (h), the elementary electric charge (e).”", "fetched": [ "2026-09-18" ] }, { "title": "Boltzmann constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Boltzmann_constant", "own_work": false, "note": "read raw: “Although Boltzmann first linked entropy and probability in 1877, the relation was never expressed with a specific constant until Max Planck first introduced k”; “the relation was never expressed with a specific constant until Max Planck first introduced k ... in his derivation of the law of black-body radiation in 1900–1901..”", "fetched": [ "2026-09-18" ] }, { "title": "Cosmological constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cosmological_constant", "own_work": false, "note": "read raw: “In 1998, two teams of astrophysicists, the Supernova Cosmology Project and the High-Z Supernova Search Team, carried out measurements on distant supernovae which showed that the speed of galaxies’ recession in relation to the Milky Way increases over time.”; “For this work, Perlmutter, Schmidt, and Riess jointly received the Nobel Prize in Physics in 2011..”", "fetched": [ "2026-09-18" ] }, { "title": "History of Lorentz transformations, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=History_of_Lorentz_transformations", "own_work": false, "note": "read raw: “They were completed by Larmor (1897, 1900) and Lorentz (1899, 1904), and brought into their modern form by Poincaré (1905) who gave the transformation the name of Lorentz..”", "fetched": [ "2026-09-18" ] }, { "title": "Lorentz transformation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lorentz_transformation", "own_work": false, "note": "read raw: “All of the work up to 1905 assumed that the constant speed of light was a consequence of an aether and hypothesized the transformation. In that year Albert Einstein published what is now called special relativity.”", "fetched": [ "2026-09-18" ] }, { "title": "Lorentz factor, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lorentz_factor", "own_work": false, "note": "carries the value at rest: “Its initial value is 1 (when v = 0)”", "fetched": [ "2026-09-22" ] }, { "title": "Large Hadron Collider, Wikipedia", "url": "https://en.wikipedia.org/wiki/Large_Hadron_Collider", "own_work": false, "note": "carries the LHC value at 6.5 TeV per proton: “At this energy, the protons have a Lorentz factor of about 6,930”", "fetched": [ "2026-09-22" ] }, { "title": "Schwarzschild radius, Wikipedia", "url": "https://en.wikipedia.org/wiki/Schwarzschild_radius", "own_work": false, "note": "carries the solution, formula, horizon, and values: “In 1916, Karl Schwarzschild obtained an exact solution to the Einstein field equations for the gravitational field outside a non-rotating, spherically symmetric body”; “The Schwarzschild radius (2GM/c²) of a given mass M equals twice its reduced Compton wavelength when M equals one Planck mass”; “The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body”; “the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi), whereas Earth’s is approximately 9 mm (0.35 in)”", "fetched": [ "2026-09-22" ] }, { "title": "Karl Schwarzschild, Wikipedia", "url": "https://en.wikipedia.org/wiki/Karl_Schwarzschild", "own_work": false, "note": "carries the timing: “an exact solution which he first set down in a letter to Einstein of 22 December 1915”; “In March 1916, Schwarzschild left military service because of his illness”; “Schwarzschild died of immune complications related to his illness on 11 May 1916, at the age of 42.” Its infobox gives the place of death as Potsdam; its “returned to Göttingen” is his leaving military service in March 1916.", "fetched": [ "2026-09-22" ] }, { "title": "Karl Schwarzschild, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Schwarzschild/", "own_work": false, "note": "“Died 11 May 1916 Potsdam, Germany”; “after being invalided home in March 1916, he died two months later.”", "fetched": [ "2026-09-22" ] }, { "title": "Josephson effect, Wikipedia", "url": "https://en.wikipedia.org/wiki/Josephson_effect", "own_work": false, "note": "read as raw wikitext: “The effect is named after the British physicist Brian Josephson, who predicted in 1962 the mathematical relationships for the current and voltage across the weak link.”; and “The Josephson voltage standard takes the caesium standard definition of frequency and gives the standard representation of a volt.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.22", "url": "https://www.teclamairme.com/#r-3.22", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: J. J. Thomson identified the electron in 1897 and measured its charge-to-mass ratio; the mole was adopted as an SI base unit in 1971, then defined by the atoms in 0.012 kilograms of carbon-12; Maxwell’s 1865 result c^2 = 1/(μ0 ε0); from 1948 to 2019 the ampere’s definition fixed μ0 at exactly 4π × 10^-7 H/m; since 2019 μ0 = 2αh/(e^2 c), with the uncertainty of α; Avogadro’s 1811 hypothesis (equal volumes, equal numbers of molecules); Jean Perrin coined the name Avogadro’s number in 1909; Emmy Noether’s work on the invariance theorem began in 1915, when she was helping Felix Klein and David Hilbert with general relativity; Emmy Noether’s Invariante Variationsprobleme was presented for her by Felix Klein and published in 1918. Klein presented it on 26 July 1918; Noether presumably did not because she was not a member of the society. An electron and a positron annihilating with negligible momentum give photons whose energies total 1.022 MeV, twice m_ec².", "sources": [ { "title": "Electron, Wikipedia", "url": "https://en.wikipedia.org/wiki/Electron", "own_work": false, "note": "read raw: “In 1897, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson, performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.”; and of annihilation: “If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons whose energies total 1.022 MeV.”", "fetched": [ "2026-09-18" ] }, { "title": "Mole (unit), Wikipedia", "url": "https://en.wikipedia.org/wiki/Mole_(unit)", "own_work": false, "note": "read raw: “The mole was made the seventh SI base unit in 1971 by the 14th CGPM.”; and of the definition that replaced oxygen-16 “during the 1960s”: “The International Bureau of Weights and Measures defined the mole as “the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon-12.””", "fetched": [ "2026-09-18" ] }, { "title": "A Dynamical Theory of the Electromagnetic Field, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=A_Dynamical_Theory_of_the_Electromagnetic_Field", "own_work": false, "note": "read raw: “A Dynamical Theory of the Electromagnetic Field” is a paper by James Clerk Maxwell on electromagnetism, published in 1865..”", "fetched": [ "2026-09-18" ] }, { "title": "Impedance of free space, Wikipedia", "url": "https://en.wikipedia.org/wiki/Impedance_of_free_space", "own_work": false, "note": "read raw: “Between 1948 and 2019, the SI unit the ampere was defined by choosing the numerical value of μ0 to be exactly 4π × 10^-7 H/m..”", "fetched": [ "2026-09-18" ] }, { "title": "Vacuum permeability, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vacuum_permeability", "own_work": false, "note": "read raw: “Since the revision of the SI in 2019 (when the values of e and h were fixed as defined quantities), μ0 is an experimentally determined constant with its value proportional to the dimensionless fine-structure constant.”", "fetched": [ "2026-09-18" ] }, { "title": "Avogadro constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Avogadro_constant", "own_work": false, "note": "read raw: “The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas.”; “The name Avogadro’s number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 32 grams of oxygen gas.”; “Perrin himself determined the Avogadro number, which he called “Avogadro’s constant” (constante d’Avogadro), by several different experimental methods.”", "fetched": [ "2026-09-18" ] }, { "title": "Noether’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Noether%27s_theorem", "own_work": false, "note": "read raw: “Emmy Noether’s work on the invariance theorem began in 1915 when she was helping Felix Klein and David Hilbert with their work related to Albert Einstein’s theory of general relativity”; “This is the first of two theorems (see Noether’s second theorem) published by the mathematician Emmy Noether in 1918..”", "fetched": [ "2026-09-18" ] }, { "title": "Emmy Noether, Wikipedia", "url": "https://en.wikipedia.org/wiki/Emmy_Noether", "own_work": false, "note": "“The paper, Invariante Variationsprobleme, was presented by Felix Klein on 26 July 1918 at a meeting of the Royal Society of Sciences at Göttingen. Noether presumably did not present it herself because she was not a member of the society.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.23", "url": "https://www.teclamairme.com/#r-3.23", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Max Planck proposed his natural units in 1899, using h where the modern definitions use ℏ; Klaus von Klitzing discovered the quantized Hall resistance in 1980; the quantum Hall effect won the 1985 Nobel Prize; R_K = h/e^2 exact since 2019 because h and e are; Johannes Rydberg’s 1888 formula for spectral lines; Bohr’s 1913 atom explained the constant; Jožef Stefan found the fourth-power law in 1879, and his student Boltzmann derived it in 1884; Planck’s 1900 law gave σ its value; the Weinberg angle of the 1967 electroweak theory. Bohr’s model gives R∞ = m_e e⁴/(8ε₀²h³c).", "sources": [ { "title": "Planck units, Wikipedia", "url": "https://en.wikipedia.org/wiki/Planck_units", "own_work": false, "note": "read raw: “In 1899, one year before the advent of quantum theory, Max Planck introduced what later became known as the Planck constant. At the end of the paper, he proposed the base units that were later named in his honor.”; and “At the Planck scale, the predictions of the Standard Model, quantum field theory and general relativity are not expected to apply, and quantum effects of gravity are expected to dominate.”", "fetched": [ "2026-09-18" ] }, { "title": "Quantum Hall effect, Wikipedia", "url": "https://en.wikipedia.org/wiki/Quantum_Hall_effect", "own_work": false, "note": "read raw: “In 1980, Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based MOSFET samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall resistance was exactly quantized.”; “For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics.”; “Later, the 2019 revision of the SI fixed exact values of h and e, resulting in an exact R_K = h/e^2.”", "fetched": [ "2026-09-18" ] }, { "title": "Rydberg formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rydberg_formula", "own_work": false, "note": "read raw: “It was first empirically stated in 1888 by the Swedish physicist Johannes Rydberg, then theoretically by Niels Bohr in 1913, who used a primitive form of quantum mechanics..”", "fetched": [ "2026-09-18" ] }, { "title": "Josef Stefan, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Stefan_Josef/", "own_work": false, "note": "read raw: “Stefan showed empirically, in 1879, that total radiation from a blackbody is proportional to the fourth power of its absolute temperature.”; “Boltzmann, who was one of Stefan’s students, showed in 1884 that this Stefan-Boltzmann law could be demonstrated mathematically.”", "fetched": [ "2026-09-18" ] }, { "title": "Stefan–Boltzmann law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law", "own_work": false, "note": "read raw: “The law, including the theoretical prediction of the Stefan–Boltzmann constant as a function of the speed of light, the Boltzmann constant and the Planck constant, is a direct consequence of Planck’s law as formulated in 1900..”", "fetched": [ "2026-09-18" ] }, { "title": "Steven Weinberg, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Steven_Weinberg", "own_work": false, "note": "read raw: “In 1967 he was a visiting professor at MIT. It was in that year at MIT that Weinberg proposed his model of unification of electromagnetism and nuclear weak forces.”", "fetched": [ "2026-09-18" ] }, { "title": "Rydberg constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rydberg_constant", "own_work": false, "note": "“The Rydberg constant was one of the most precisely determined physical constants, with a relative standard uncertainty of 1.1×10⁻¹².” and “the Rydberg constant is inferred from measurements of atomic transition frequencies in three different atoms (hydrogen, deuterium, and antiprotonic helium).” Displays R∞ = α²mₑc/(2h) and R∞ = mₑe⁴/(8ε₀²h³c).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.24", "url": "https://www.teclamairme.com/#r-3.24", "chapter": 3, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Wilhelm Wien found the displacement law in 1893; Z0 was exact before 2019 and is now derived from alpha. Since 2019 μ0, and so Z0 = μ0c, is fixed by the measured fine-structure constant.", "sources": [ { "title": "Wien’s displacement law, Wikipedia", "url": "https://en.wikipedia.org/wiki/Wien%27s_displacement_law", "own_work": false, "note": "read raw: “The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument..”", "fetched": [ "2026-09-18" ] }, { "title": "Impedance of free space, Wikipedia", "url": "https://en.wikipedia.org/wiki/Impedance_of_free_space", "own_work": false, "note": "read raw: “This chain of dependencies changed when the ampere was redefined on 20 May 2019..”", "fetched": [ "2026-09-18" ] }, { "title": "Vacuum permeability, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vacuum_permeability", "own_work": false, "note": "“In the 2019 revision of the SI, the ampere is defined exactly in terms of the elementary charge and the second, and the value of μ0 is now determined experimentally (based on the measured value of the fine-structure constant)”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-3.26", "url": "https://www.teclamairme.com/#r-3.26", "chapter": 3, "status": "confirmed", "claim": "Quantum electrodynamics is the quantum field theory of the electromagnetic interaction, developed by Feynman, Schwinger, and Tomonaga in the late 1940s, and its prediction for the electron’s anomalous magnetic moment agrees with measurement to more than ten significant figures. Schwinger first computed the anomalous moment in 1948, and Feynman, Schwinger, and Tomonaga shared the 1965 Nobel Prize.", "sources": [ { "title": "Quantum electrodynamics, Wikipedia", "url": "https://en.wikipedia.org/wiki/Quantum_electrodynamics", "own_work": false, "note": "describes the theory’s development and its precision agreement for the electron’s magnetic moment.", "fetched": [ "2026-09-22" ] }, { "title": "Anomalous magnetic dipole moment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_moment", "own_work": false, "note": "carries Schwinger’s 1948 one-loop result and the agreement: “found by Julian Schwinger in 1948” and theory and experiment “agree to more than 10 significant figures,” “making the magnetic moment of the electron one of the most accurately verified predictions in the history of physics”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.1", "url": "https://www.teclamairme.com/#r-4.1", "chapter": 4, "status": "confirmed", "claim": "The chapter’s elementary identities and classifications of integers, rationals, and algebraic numbers are correct, and whether any factorial other than 4!, 5!, and 7! is one less than a square is Brocard’s problem, which is open. 42 is the Catalan number C₅, which counts the ways to bracket six factors. 0.999… and 1 are exactly the same number: the 10x − x argument shows it informally, proofs by series and limits show it rigorously, and the misconception is old. φ is defined as the positive root of x² − x − 1, so φ² − φ − 1 = 0. The silver ratio 1 + √2 ≈ 2.414 is the positive root of x² − 2x − 1 and the limit of ratios of consecutive Pell numbers; it is constructible. Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, and base sixty survives in the 60 minutes of an hour, the 60 seconds of a minute, and the 360 degrees of a circle. 4 = 2² and 9 = 3² are the first two square numbers after 1. The decimal expansion of 1/3 repeats the digit 3 forever. 0! = 1 is the empty product and the one arrangement of nothing; it keeps n! = n·(n−1)! valid at n = 1, makes the binomial coefficients and the power series of e^x come out right, and matches Γ(1) = 1.", "sources": [ { "title": "DLMF, §24.2 Bernoulli and Euler numbers", "url": "https://dlmf.nist.gov/24.2", "own_work": false, "note": "confirms the Bernoulli-number table and generating-function conventions.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Algebraic Number", "url": "https://mathworld.wolfram.com/AlgebraicNumber.html", "own_work": false, "note": "confirms the algebraic-number definition and the listed algebraic constants.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Brocard’s problem", "url": "https://en.wikipedia.org/wiki/Brocard%27s_problem", "own_work": false, "note": "n! + 1 = m² is known only for n = 4, 5, and 7; whether there are others is open, no further solution has been found below n = 10¹⁵, and Erdős conjectured there are none.", "fetched": [ "2026-09-09" ] }, { "title": "Catalan number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Catalan_number", "own_work": false, "note": "“The first Catalan numbers for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...”; C_n is “the number of different ways n + 1 factors can be completely parenthesized”", "fetched": [ "2026-09-22" ] }, { "title": "0.999..., Wikipedia", "url": "https://en.wikipedia.org/wiki/0.999...", "own_work": false, "note": "“Despite common misconceptions, 0.999... is not “almost exactly 1” or “very, very nearly but not quite 1”; rather, “0.999...” and “1” represent exactly the same number.”; Byers’s argument displayed: x = 0.999…, 10x = 9.999…, 10x = 9 + x, 9x = 9, x = 1; “Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits.”; “the 1895 Arithmetic for Schools says, “when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small.””", "fetched": [ "2026-09-22" ] }, { "title": "Golden ratio, Wikipedia", "url": "https://en.wikipedia.org/wiki/Golden_ratio", "own_work": false, "note": "the defining polynomial: “Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is x^2 − x − 1.”; “φ is defined as the positive solution.”", "fetched": [ "2026-09-22" ] }, { "title": "Silver ratio, Wikipedia", "url": "https://en.wikipedia.org/wiki/Silver_ratio", "own_work": false, "note": "polynomial, value, and Pell limit: “It follows that the silver ratio is the positive solution of quadratic equation σ^2 − 2σ − 1 = 0. The quadratic formula gives the two solutions 1 ± √2, the decimal expansion of the positive root begins with 2.414213562373095...”; of the Pell numbers: “The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,...”; “The limit ratio of consecutive terms is the silver mean.”", "fetched": [ "2026-09-22" ] }, { "title": "Constructible number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_number", "own_work": false, "note": "constructibility of numbers built from square roots: “Equivalently, r is constructible if and only if there is a closed-form expression for r using only integers and the operations for addition, subtraction, multiplication, division, and square roots.”", "fetched": [ "2026-09-22" ] }, { "title": "Sexagesimal, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sexagesimal", "own_work": false, "note": "carries the divisors, the hour, and the degree: “The number 60, a superior highly composite number, has twelve divisors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified.”; “One hour of time is divided into 60 minutes, and one minute is divided into 60 seconds.”; “Similarly, the practical unit of angular measure is the degree, of which there are 360 (six sixties) in a circle.”", "fetched": [ "2026-09-22" ] }, { "title": "Square number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Square_number", "own_work": false, "note": "lists the small squares: “The squares (sequence A000290 in the OEIS) smaller than 60² = 3600 are:” … “2² = 4” … “3² = 9”", "fetched": [ "2026-09-22" ] }, { "title": "Repeating decimal, Wikipedia", "url": "https://en.wikipedia.org/wiki/Repeating_decimal", "own_work": false, "note": "“the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit “3” forever, i.e. 0.333....”", "fetched": [ "2026-09-22" ] }, { "title": "Factorial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Factorial", "own_work": false, "note": "§ Factorial of zero: “the definition of n! as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product, a product of no factors, is equal to the multiplicative identity.”; “There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.”; “the number of ways to choose all n elements from a set of n is (n choose n) = n!/(n!0!) = 1, a binomial coefficient identity that would only be valid with 0!=1.”; “With 0!=1, the recurrence relation for the factorial remains valid at n=1.”; “Setting 0!=1 allows for the compact expression of many formulae, such as the exponential function, as a power series”; “This choice matches the gamma function 0! = Γ(0+1) = 1”; and “there are n! different ways of arranging n distinct objects into a sequence.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.2", "url": "https://www.teclamairme.com/#r-4.2", "chapter": 4, "status": "confirmed", "claim": "1729 is the smallest number expressible as two positive cubes in two distinct ways, and subtracting the two representations gives the displayed zero.", "sources": [ { "title": "Wolfram MathWorld, Taxicab Number", "url": "https://mathworld.wolfram.com/TaxicabNumber.html", "own_work": false, "note": "lists Taxicab(2)=1729 and its two cube representations.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-4.3", "url": "https://www.teclamairme.com/#r-4.3", "chapter": 4, "status": "confirmed", "claim": "The named integer curiosities, perfect and amicable numbers, Kaprekar’s constant, the cyclic number 142857, the Answer of Douglas Adams’s Deep Thought, and the displayed arithmetic values have the stated definitions. Belphegor’s prime, 1000000000000066600000000000001, is a 31-digit palindromic prime named by Clifford A. Pickover after the demon Belphegor, for its 666 and two runs of thirteen zeros. Deep Thought’s words are quoted from the novel, from “The Answer to the Great Question . . .” to “Forty-two,” said Deep Thought, with infinite majesty and calm. Keeping leading zeros, any four-digit number with at least two distinct digits reaches 6174 within seven iterations; if they are dropped, some reach 0 and up to eight are needed.", "sources": [ { "title": "Wolfram MathWorld, Kaprekar Routine", "url": "https://mathworld.wolfram.com/KaprekarRoutine.html", "own_work": false, "note": "confirms 6174 and the four-digit routine.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Perfect Number", "url": "https://mathworld.wolfram.com/PerfectNumber.html", "own_work": false, "note": "confirms the proper-divisor definition and the first even perfect numbers; the Euclid form is perfect “as stated in Proposition IX.36 of Euclid’s Elements,” and “In a posthumous 1849 paper, Euler provided the first proof that Euclid’s construction gives all possible even perfect numbers.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Amicable Numbers", "url": "https://en.wikipedia.org/wiki/Amicable_numbers", "own_work": false, "note": "confirms 220 and 284 and their divisor sums.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Cyclic Number", "url": "https://mathworld.wolfram.com/CyclicNumber.html", "own_work": false, "note": "142857 is the cyclic number generated by 1/7, a cyclic number being an (n−1)-digit integer whose products with 1, 2, …, n−1 are its own digits reordered; 7 is the first full reptend prime, and multiplying a cyclic number by its generator gives a string of nines.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Phrases from The Hitchhiker’s Guide to the Galaxy", "url": "https://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker%27s_Guide_to_the_Galaxy", "own_work": false, "note": "Deep Thought’s Answer is 42, first broadcast in Fit the Fourth on 29 March 1978 and published in the novel of 1979; Adams wrote on 3 November 1993 that “It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one.”", "fetched": [ "2026-09-09" ] }, { "title": "Belphegor’s prime, Wikipedia", "url": "https://en.wikipedia.org/wiki/Belphegor%27s_prime", "own_work": false, "note": "“Belphegor’s prime is the palindromic prime number 1000000000000066600000000000001”; “The name “Belphegor’s prime” was coined by author Clifford A. Pickover in 2012.”; “Belphegor is one of the Seven Princes of Hell”; “This number is surrounded on either side by thirteen zeroes and is 31 digits in length (thirteen reversed)”", "fetched": [ "2026-09-22" ] }, { "title": "Quote Origin: The Answer to Life, the Universe and Everything Is Forty-Two, Quote Investigator", "url": "https://quoteinvestigator.com/2022/10/29/answer-universe/", "own_work": false, "note": "quotes the novel’s passage: ““All right,” said Deep Thought. “The Answer to the Great Question . . .”” … ““Of Life, the Universe and Everything . . .” said Deep Thought.” … ““Is . . .” said Deep Thought, and paused.” … ““Forty-two,” said Deep Thought, with infinite majesty and calm.”", "fetched": [ "2026-09-22" ] }, { "title": "Kaprekar’s routine, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kaprekar%27s_routine", "own_work": false, "note": "“Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations.”; “When leading zeroes are retained, only repdigits lead to the trivial Kaprekar’s constant.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.4", "url": "https://www.teclamairme.com/#r-4.4", "chapter": 4, "status": "confirmed", "claim": "The large-number values and historical attributions in the integer brace, including Eddington’s number, Shannon’s chess-game estimate, googol/googolplex, and Skewes’ bound, are as stated. Leonhard Euler proved 2³¹ − 1 prime in 1772 and it held the record for the largest known prime until 1867, when Fortuné Landry found (2⁵⁹ − 1)/179951 = 3,203,431,780,337. 2³¹ − 1 = 2,147,483,647; Euler’s proof used trial division; it is the largest value of a signed 32-bit integer, hexadecimal 7FFFFFFF, and as seconds from 1970 it ends at 19 January 2038. Moser’s number is 2 in a megagon in Steinhaus–Moser notation, Moser’s extension of Steinhaus’s polygons; it exceeds Skewes’ number and is far below Graham’s. Archimedes’ Sand Reckoner, third century BC, used Aristarchus’ universe and his orders and periods to bound the grains of sand filling it by 10^63.", "sources": [ { "title": "Wikipedia, Large Numbers", "url": "https://en.wikipedia.org/wiki/Large_numbers", "own_work": false, "note": "gives the values of googol, googolplex, and the first Skewes’s number (approximately 10^10^10^34), and the Shannon number for chess, estimated at 10^120; the historical attributions are on this entry’s other lines.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Shannon Number", "url": "https://en.wikipedia.org/wiki/Shannon_number", "own_work": false, "note": "confirms Shannon’s 10^120 lower-bound estimate.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Skewes’ Number", "url": "https://en.wikipedia.org/wiki/Skewes%27_number", "own_work": false, "note": "confirms the historical bound and the modern upper bound.", "fetched": [ "2026-09-09" ] }, { "title": "Largest known prime number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Largest_known_prime_number", "own_work": false, "note": "read as raw HTML. Its “Largest known prime by year” table gives, consecutively, “M₁₇ | 6 | 1588 | Pietro Cataldi,” “M₁₉ | 6 | 1588 | Pietro Cataldi,” “M₃₁ | 10 | 1772 | Leonhard Euler,” “M₅₉/179951 | 13 | 1867 | Fortuné Landry” and “M₁₂₇ | 39 | 1876 | Édouard Lucas,” which is the 95-year gap the #m31 card names.", "fetched": [ "2026-09-17" ] }, { "title": "2,147,483,647, Wikipedia", "url": "https://en.wikipedia.org/wiki/2,147,483,647", "own_work": false, "note": "“By 1772, Leonhard Euler had proven that 2,147,483,647 is a prime.”; “Euler used trial division, improving on Pietro Cataldi’s method, so that at most 372 divisions were needed.”; “In 1867, the number 3,203,431,780,337 was proven to be prime.”; “In computing, this number is the largest value that a signed 32-bit integer field can hold.”; “hexadecimal 7FFFFFFF”; “The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch).”; “This means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem.”; “On 1 January 2022, a bug was reported for Microsoft Exchange systems where email delivery would fail. An internal malware scanner (enabled by default since 2013) used the date and time as a signed 32-bit integer.”", "fetched": [ "2026-09-22" ] }, { "title": "Steinhaus–Moser notation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation", "own_work": false, "note": "“It is an extension (devised by Leo Moser) of Hugo Steinhaus’s polygon notation.”; “a number n in a triangle means n^n”; “a number n in a square is equivalent to “the number n inside n triangles, which are all nested.””; “Moser’s number is the number represented by “2 in a megagon.””; “Therefore, Moser’s number, although incomprehensibly large, is vanishingly small compared to Graham’s number”", "fetched": [ "2026-09-22" ] }, { "title": "big numbers, Susan Stepney (University of York)", "url": "https://www-users.york.ac.uk/~ss44/cyc/b/big.htm", "own_work": false, "note": "gives Skewes’ number as e^e^e^79, approximately 10^10^10^34, “< 4^^5, occurs in theories on the distribution of primes”; with Wikipedia’s mega ≈ (256↑)^256 257 this puts Moser’s number far above it", "fetched": [ "2026-09-22" ] }, { "title": "The Sand Reckoner, Wikipedia", "url": "https://en.wikipedia.org/wiki/The_Sand_Reckoner", "own_work": false, "note": "“The Sand Reckoner (Greek: Ψαμμίτης, Psammites) is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the universe.”; “To do this, he used the heliocentric model of Aristarchus of Samos.”; “He then constructed the orders of the second period by taking multiples of this unit”; “Multiplying 10^21 by 10^42 yields 10^63, the number of grains of sand in the Aristarchian Universe.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.5", "url": "https://www.teclamairme.com/#r-4.5", "chapter": 4, "status": "confirmed", "claim": "The notation and comparisons involving Knuth arrows, Conway chains, hyperoperations, the fast-growing hierarchy, TREE(3), SSCG(3), Graham’s number, and Loader’s number are standard as qualified in the cards. Loader’s number is said on a community wiki (the Googology Wiki line below) to be far larger than TREE(3), a comparison this book does not rest on, and the Busy Beaver function grows faster than any computable function. The last digits of Graham’s number are known; its leading digits are not. 3↑↑4 has about 3.6 trillion digits: more than the 41,024,320 digits of the largest known prime (September 2026), fewer than the googolplex, 10^(10^100). Skewes’ 1933 bound is below 10^(10^(10^34)), far below a tower of 7,625,597,484,987 threes. SSCG(k) is Harvey Friedman’s function: the longest sequence of simple subcubic graphs, the i-th on at most i + k vertices, with no earlier one homeomorphically embeddable in a later one; the Robertson–Seymour theorem makes every such sequence finite; SSCG(0) = 2, SSCG(1) = 5, SSCG(2) = 3·2^(3·2^95) − 8, and SSCG(3) is greater than TREE(3).", "sources": [ { "title": "Wikipedia, Knuth’s up-arrow notation", "url": "https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation", "own_work": false, "note": "confirms the recursive arrow definitions and 3↑↑3 value.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Conway chained arrow notation", "url": "https://en.wikipedia.org/wiki/Conway_chained_arrow_notation", "own_work": false, "note": "confirms the chain notation and its growth comparisons.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Fast-growing hierarchy", "url": "https://en.wikipedia.org/wiki/Fast-growing_hierarchy", "own_work": false, "note": "confirms the ordinal-indexed hierarchy and its use for growth comparison; it gives “f_{ω+1}(64) = f_{ω}^{64}(64) > Graham’s number” (sub- and superscripts as typeset on the page).", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Graham’s number", "url": "https://en.wikipedia.org/wiki/Graham%27s_number", "own_work": false, "note": "the number Gardner published in 1977 is not the bound in Graham and Rothschild’s 1971 paper, which is smaller; the best known bounds on the original problem are 13 (Barkley, 2008) below and, above, the 2014 bound 2↑↑(2↑↑(3 + 2↑↑8)) of Lavrov, Lee and Mackey, sharpened again in 2019; the last ten digits are …2464195387; Martin Gardner described it in Scientific American “in November 1977,” writing that Graham had established “a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.”; “At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof.”", "fetched": [ "2026-09-09" ] }, { "title": "Loader’s number, Googology Wiki", "url": "https://googology.miraheze.org/wiki/Loader's_number", "own_work": false, "note": "community wiki, not primary: “The final output of D^5(99) is much larger than TREE(3), SCG(13), and, say, BH(100).”", "fetched": [ "2026-09-22" ] }, { "title": "Busy beaver, Wikipedia", "url": "https://en.wikipedia.org/wiki/Busy_beaver", "own_work": false, "note": "“This is a noncomputable function, because it grows asymptotically faster than any computable function.”", "fetched": [ "2026-09-22" ] }, { "title": "Largest known prime number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Largest_known_prime_number", "own_work": false, "note": "perishable: “The largest known prime number as of September 2026 is 2^136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system.”", "fetched": [ "2026-09-22" ] }, { "title": "Googolplex, Wikipedia", "url": "https://en.wikipedia.org/wiki/Googolplex", "own_work": false, "note": "“A googolplex is the large number 10^(10^100), that is, 10 raised to the power of a googol.”", "fetched": [ "2026-09-22" ] }, { "title": "Skewes’s number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Skewes%27s_number", "own_work": false, "note": "“Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < li(x), below e^e^e^79 < 10^10^10^34.” The comparison with 3↑↑↑3 is arithmetic.", "fetched": [ "2026-09-22" ] }, { "title": "Friedman’s SSCG function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Friedman%27s_SSCG_function", "own_work": false, "note": "“Friedman’s SSCG function is a mathematical function defined by Harvey Friedman.”; definition (displayed): “There is a sequence G1,…,Gn of simple subcubic graphs such that each Gi has at most i+k vertices and for no i < j is Gi homeomorphically embeddable into Gj.”; “The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-quasi-ordered by homeomorphic embeddability, implying such a sequence cannot be infinite.”; “The first few terms of the sequence are SSCG(0) = 2, SSCG(1) = 5, and SSCG(2) = 3·2^(3·2^95) − 8” (displayed formulas); “It has been shown that the next term, SSCG(3), is greater than TREE(3).”", "fetched": [ "2026-09-22" ] }, { "title": "Marco Ripà, Graham’s number stable digits: An exact solution (Notes on Number Theory and Discrete Mathematics 31(3), 2025, 607–616; arXiv 2024)", "url": "https://arxiv.org/abs/2411.00015", "own_work": false, "note": "read in the arXiv HTML full text: the most significant digit of G is 1 in binary and in ternary, but “the problem of expressing the most significant digit of G in the decimal number system is still open.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.6", "url": "https://www.teclamairme.com/#r-4.6", "chapter": 4, "status": "judgement", "claim": "65537 is the fifth known Fermat prime and the largest known; the regular 65537-gon can be drawn with compass and straightedge, by the sufficiency Gauss proved and the necessity Wantzel proved in 1837; that no sixth Fermat prime exists is a heuristic expectation and not a theorem. 65537 is commonly used as the public exponent in RSA.", "sources": [ { "title": "Wolfram MathWorld, 65537-gon", "url": "https://mathworld.wolfram.com/65537-gon.html", "own_work": false, "note": "confirms constructibility from 65537 being a Fermat prime and attributes the result to Gauss.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Fermat number", "url": "https://en.wikipedia.org/wiki/Fermat_number", "own_work": false, "note": "confirms 65537 is the largest known Fermat prime.", "fetched": [ "2026-09-09" ] }, { "title": "Boklan and Conway, Expect at most one billionth of a new Fermat prime!", "url": "https://arxiv.org/abs/1605.01371", "own_work": false, "note": "gives the heuristic evidence for no further Fermat primes.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Constructible polygon", "url": "https://en.wikipedia.org/wiki/Constructible_polygon", "own_work": false, "note": "Gauss proved the Fermat-prime condition sufficient and stated its necessity without proof; Wantzel gave the full proof of necessity in 1837, which is why the result is the Gauss–Wantzel theorem.", "fetched": [ "2026-09-09" ] }, { "title": "65,537, Wikipedia", "url": "https://en.wikipedia.org/wiki/65,537", "own_work": false, "note": "“65537 is commonly used as a public exponent in the RSA cryptosystem.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The page hedges to “almost certainly the last” rather than asserting it. What is proved is that 65537 is the largest Fermat prime anyone has found; Boklan and Conway put the chance of another at under one in a billion, which is evidence and not a proof. On the constructibility, Gauss’s celebrated result of 1796 is the 17-gon; the general criterion is his Disquisitiones Arithmeticae of 1801, and the converse half is Wantzel’s." }, { "id": "r-4.7", "url": "https://www.teclamairme.com/#r-4.7", "chapter": 4, "status": "confirmed", "claim": "The rational-number identities, continued-fraction facts, Bernoulli values, and harmonic-number statement are correct. The first harmonic numbers are 1, 3/2, 11/6, and 25/12, so H₄ is the first whose denominator (12) is not a factorial (1, 2, and 6 are). Hₙ − ln n decreases towards Euler’s γ ≈ 0.5772, so Hₙ grows like ln n + γ. 355/113, the Milü found by Zu Chongzhi in the fifth century, is accurate to six decimal places, and the next better rational approximation of π, by size of denominator, is 52163/16604.Counting the initial 3 as the first convergent, 22/7 and 355/113 are the second and fourth convergents of π = [3;7, 15, 1,…]; both π and 355/113 round to 3.141593 at six decimal places.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "direct high-precision evaluation of π gives initial partial quotients 3, 7, 15, 1. Their exact truncations are 3, 22/7, 333/106, and 355/113. π ≈ 3.141592653589793 and 355/113 ≈ 3.141592920353982, both rounding to 3.141593 at six places.", "fetched": [ "2026-09-30" ] }, { "title": "Wolfram MathWorld, Continued Fraction", "url": "https://mathworld.wolfram.com/ContinuedFraction.html", "own_work": false, "note": "the simple-continued-fraction notation (convergents are only linked from it, under See also; the π approximations are on the Milü line).", "fetched": [ "2026-09-09" ] }, { "title": "DLMF, §24.2 Bernoulli and Euler numbers", "url": "https://dlmf.nist.gov/24.2", "own_work": false, "note": "confirms B0, B1 convention, B2, and B4.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Harmonic Number", "url": "https://en.wikipedia.org/wiki/Harmonic_number", "own_work": false, "note": "confirms H4=25/12 and the nonintegrality theorem for Hn, n>1; on growth: “The values of the sequence H_n − ln n decrease monotonically towards the limit” … “where γ ≈ 0.5772156649 is the Euler–Mascheroni constant.”", "fetched": [ "2026-09-09" ] }, { "title": "Milü, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mil%C3%BC", "own_work": false, "note": "“355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places.”; “The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, though it is still only correct to six decimal places.”; the Milü “is the name given to an approximation of π ( pi ) found by the Chinese mathematician and astronomer Zu Chongzhi during the 5th century.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.8", "url": "https://www.teclamairme.com/#r-4.8", "chapter": 4, "status": "confirmed", "claim": "The constructibility and algebraic descriptions of φ, √2, the silver ratio, cos(2π/17), ∛2, cos(2π/7), the plastic, tribonacci, Lehmer, and Conway constants are correct as stated. cos(2π/17) ≈ 0.93247 has degree 8; Gauss’s diary records the 17-gon’s constructibility on 30 March 1796, the first progress in regular polygon construction since the Greeks; Gauss, born on 30 April 1777, was eighteen at that diary entry; that he asked for it on his gravestone is a story, probably apocryphal. Lehmer’s number ≈ 1.176280818, of degree 10, is the smallest known Salem number; it is the spectral radius of the Coxeter element of E₁₀ and the Mahler measure of the Alexander polynomial of the (−2, 3, 7)-pretzel knot.", "sources": [ { "title": "Wolfram MathWorld, Constructible Polygon", "url": "https://mathworld.wolfram.com/ConstructiblePolygon.html", "own_work": false, "note": "confirms the Fermat-prime criterion and the 17-gon/heptagon distinction.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Plastic Constant", "url": "https://mathworld.wolfram.com/PlasticConstant.html", "own_work": false, "note": "confirms x^3=x+1 and the value 1.32471795… .", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Conway’s Constant", "url": "https://mathworld.wolfram.com/ConwaysConstant.html", "own_work": false, "note": "confirms the look-and-say growth rate, value, and degree-71 polynomial.", "fetched": [ "2026-09-09" ] }, { "title": "Constructible number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_number", "own_work": false, "note": "the power-of-two criterion and the √2 example: “For instance, the square root of 2 is constructible, because it can be described by the formulas √2 or √(1+1).”; “It follows from this decomposition that the degree of the field extension [Q(γ):Q] is 2^r, where r counts the number of quadratic extension steps.”; “Consequently, if a complex number γ is constructible, then the above characterization implies that [Q(γ):Q] is a power of two.”; Gauss’s argument “was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular n-gon.”", "fetched": [ "2026-09-22" ] }, { "title": "Trigonometry Angles--Pi/17, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/TrigonometryAnglesPi17.html", "own_work": false, "note": "the value, equation (12): “cos((2pi)/(17)) = 1/(16)[delta+sqrt(2)(alpha+epsilon^*)] approx 0.93247”", "fetched": [ "2026-09-22" ] }, { "title": "Minimal polynomial of 2cos(2pi/n), Wikipedia", "url": "https://en.wikipedia.org/wiki/Minimal_polynomial_of_2cos(2pi/n)", "own_work": false, "note": "the degree, 8 for n = 17: “it follows that the degree of Ψn(x) is 1 for n=1,2 and φ(n)/2 for n≥3.”", "fetched": [ "2026-09-22" ] }, { "title": "Gauss’s diary, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gauss%27s_diary", "own_work": false, "note": "the date: “Entry 1, dated 1796, March 30, states “Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc.,” which records Gauss’s discovery of the construction of a heptadecagon by ruler and compass.”", "fetched": [ "2026-09-22" ] }, { "title": "Heptadecagon, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heptadecagon", "own_work": false, "note": "the first since the Greeks: “This proof represented the first progress in regular polygon construction in over 2000 years.”", "fetched": [ "2026-09-22" ] }, { "title": "Constructing the Heptadecagon, MathPages", "url": "https://www.mathpages.com/home/kmath487.htm", "own_work": false, "note": "the gravestone as a story: “Gauss was clearly fond of this discovery, and there’s a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes’ tombstone. The story is probably apochryphal”", "fetched": [ "2026-09-22" ] }, { "title": "Heptagon, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heptagon", "own_work": false, "note": "gives 2cos(2π/7) ≈ 1.247 and: “The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible cubic x³ + x² − 2x − 1. Consequently, this polynomial is the minimal polynomial of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.”", "fetched": [ "2026-09-22" ] }, { "title": "Lehmer’s conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lehmer%27s_conjecture", "own_work": false, "note": "the value to nine places: “The smallest known Mahler measure (greater than 1) is for “Lehmer’s polynomial,”” x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1, “for which the Mahler measure is the Salem number” 1.176280818…", "fetched": [ "2026-09-22" ] }, { "title": "Salem number, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Salem_number", "own_work": false, "note": "degree and polynomial: “The smallest known Salem number is the number σ1 = 1.1762808… of degree 10 known as Lehmer’s number. The minimum polynomial of σ1 is Lehmer’s polynomial: x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1.”", "fetched": [ "2026-09-22" ] }, { "title": "Eriko Hironaka, What is... Lehmer’s number?, Notices of the AMS 56 (2009), author’s copy", "url": "https://www.math.fsu.edu/~hironaka/papers/Whatis.pdf", "own_work": false, "note": "E10 and knot theory: “Thus, Lehmer’s number is the geometric dilatation of φ, and the spectral radius of the Coxeter element of E10.”; “In particular, Lehmer’s number λ is the homological dilatation of the (-2, 3, 7)-pretzel knot (shown in Fig.1, center), and is the Mahler measure of its Alexander polynomial.”", "fetched": [ "2026-09-22" ] }, { "title": "Algebraic Number, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/AlgebraicNumber.html", "own_work": false, "note": "degrees of algebraic numbers: “An example of a complex algebraic number is i, and an example of a real algebraic number is sqrt(2), both of which are of degree 2.”; its table of degrees lists “tribonacci constant 3.”", "fetched": [ "2026-09-22" ] }, { "title": "Carl Friedrich Gauss, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/", "own_work": false, "note": "the birth date: “Born 30 April 1777 Brunswick, Duchy of Brunswick (now Germany).” (Wikipedia’s Gauss article has him “only 19 years old” when he proved the construction; the diary entry of 30 March 1796 is a month before his nineteenth birthday.)", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-4.9", "url": "https://www.teclamairme.com/#r-4.9", "chapter": 4, "status": "confirmed", "claim": "The classical transcendence claims for e, π, logarithms, trigonometric values, 2^√2, e^π, log2(3), i^i, and √2^√2 follow from the cited transcendence theorems. The Dottie number 0.7390851…, the unique real fixed point of cosine in radians, is transcendental by Lindemann–Weierstrass; it is named for a professor of French who found it by pressing cos repeatedly. Gelfond proved e^π transcendental in 1929; most combinations of e and π, π^e among them, are not known to be transcendental. Hilbert posed his seventh problem in 1900 and gave 2^√2 as one of its two explicit examples. All values of i^i are e^(−π/2)·e^(−2kπ), all real, the principal one e^(−π/2). Lindemann’s proof that π is transcendental in 1882, extending the work of Hermite, is what showed squaring the circle impossible. ln 2 ≈ 0.6931 is the sum of the alternating harmonic series and is transcendental by Lindemann–Weierstrass. log2(3) ≈ 1.58496 is the information in one trit, in bits, and it is irrational because a positive power of 2 is even and a power of 3 is odd. The universal parabolic constant P = ln(1 + √2) + √2 ≈ 2.2955, the ratio for any parabola of the latus-rectum segment’s arc length to the focal parameter, is transcendental by Lindemann–Weierstrass. A constructible number is algebraic, so π’s transcendence means the circle cannot be squared. e^(π√163) is transcendental by Gelfond–Schneider, like every e^(π√n) with n a positive integer. A non-constant algebraic function of a transcendental number is transcendental, so 2/π is transcendental because π is.", "sources": [ { "title": "Wikipedia, Lindemann–Weierstrass theorem", "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "states the theorem — if α₁, …, αₙ are algebraic and linearly independent over ℚ then e^α₁, …, e^αₙ are algebraically independent over ℚ — records that Lindemann proved the e^α case in 1882 and Weierstrass the general statement in 1885, and derives the transcendence of e and of π, and adds: “A slight variant on the same proof will show that if α is a non-zero algebraic number then sin(α), cos(α), tan(α) and their hyperbolic counterparts are also transcendental.” (The logarithms are on the Natural logarithm of 2 line.)", "fetched": [ "2026-09-09", "2026-09-10" ] }, { "title": "Wikipedia, Gelfond–Schneider theorem", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem", "own_work": false, "note": "“If a and b are algebraic numbers with a ∉ {0,1} and b irrational, then any value of a^b is a transcendental number”; “originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider”; and it “answers affirmatively Hilbert’s seventh problem.”", "fetched": [ "2026-09-09", "2026-09-10" ] }, { "title": "Dottie number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dottie_number", "own_work": false, "note": "value, fixed point, transcendence, and name: “D = 0.739085133215160641655312087673...”; “It is the single real-valued fixed point of the cosine function”; “It is also a transcendental number because of the Lindemann–Weierstrass theorem.”; “It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.”", "fetched": [ "2026-09-22" ] }, { "title": "Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf", "own_work": false, "note": "Gelfond’s 1929 proof for e^π: “The transcendence of e^π had already been proved in 1929 by A.O. Gel’fond.”; and on e^(π√163) (Remark 14), with τ = (1 + i√163)/2: “the class number of the imaginary quadratic field Q(τ) is 1, we have j(τ) = −(640 320)³ and |j(τ) − 1/q − 744| < 10^−12”.", "fetched": [ "2026-09-22" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "e–π combinations mostly open: “Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational,” listing π^e among them.", "fetched": [ "2026-09-22" ] }, { "title": "Hilbert’s seventh problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem", "own_work": false, "note": "the year of the problem: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900.”", "fetched": [ "2026-09-22" ] }, { "title": "Gelfond–Schneider constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant", "own_work": false, "note": "2^√2 as Hilbert’s test case: “In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2^√2.”; also “which was proved to be a transcendental number by Rodion Kuzmin in 1930.”", "fetched": [ "2026-09-22" ] }, { "title": "Exponentiation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Exponentiation", "own_work": false, "note": "the branches of i^i, i^i = e^(−π/2) e^(−2kπ): “So, all values of i^i are real, the principal one being e^(−π/2) ≈ 0.2079.”", "fetched": [ "2026-09-22" ] }, { "title": "Constructible number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_number", "own_work": false, "note": "on squaring the circle: “It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number.”", "fetched": [ "2026-09-22" ] }, { "title": "Natural logarithm of 2, Wikipedia", "url": "https://en.wikipedia.org/wiki/Natural_logarithm_of_2", "own_work": false, "note": "value ln 2 ≈ 0.693 147 180 559…, the series 1 − 1/2 + 1/3 − …, and transcendence: “This is the well-known “alternating harmonic series.””; “By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.”", "fetched": [ "2026-09-22" ] }, { "title": "Ternary numeral system, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ternary_numeral_system", "own_work": false, "note": "the trit in bits: “One trit is equivalent to log2 3 (about 1.58496) bits of information.”", "fetched": [ "2026-09-22" ] }, { "title": "Irrational number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Irrational_number", "own_work": false, "note": "why log2 3 is irrational: “The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2).”", "fetched": [ "2026-09-22" ] }, { "title": "Universal parabolic constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Universal_parabolic_constant", "own_work": false, "note": "value, definition and transcendence: “It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter.”; “P = ln(1+√2)+√2 = 2.29558714939…”; “Hence P is transcendental.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.10", "url": "https://www.teclamairme.com/#r-4.10", "chapter": 4, "status": "confirmed", "claim": "Champernowne’s and the Thue–Morse constants are transcendental, so are the Komornik–Loreti and paperfolding constants, and the digit-constructed constants are described by their stated sequences. The paperfolding constant 0.85073618… has the regular paperfolding sequence 1101100111001001… as its binary digits; the sequence is automatic and not ultimately periodic, and the constant is transcendental. The Thue–Morse sequence starts from 0 and repeatedly appends the complement of what has been written so far.", "sources": [ { "title": "OEIS A066716, Champernowne-related constant record", "url": "https://oeis.org/A066716", "own_work": false, "note": "records Mahler’s transcendence result for the base-b Champernowne constant.", "fetched": [ "2026-09-09" ] }, { "title": "OEIS A010060, Thue–Morse sequence", "url": "https://oeis.org/A010060/internal", "own_work": false, "note": "confirms the binary sequence and the transcendence statement for its associated constant.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Komornik–Loreti constant", "url": "https://en.wikipedia.org/wiki/Komornik%E2%80%93Loreti_constant", "own_work": false, "note": "q = 1.787231650…, the smallest base in which 1 has a unique expansion, defined by Vilmos Komornik and Paola Loreti in 1998; Jean-Paul Allouche and Michel Cosnard proved it transcendental in the American Mathematical Monthly 107 (2000), 448–449; its digits are the Thue–Morse sequence: “The Komornik–Loreti constant is the value q such that” 1 = Σ t_{k}/q^{k}, “where t_{k} is the Thue–Morse sequence.”", "fetched": [ "2026-09-09" ] }, { "title": "Adamczewski and Bugeaud, On the complexity of algebraic numbers I: expansions in integer bases", "url": "https://arxiv.org/abs/math/0511674", "own_work": false, "note": "the abstract states that irrational automatic and morphic numbers are transcendental, which is the theorem that settles the paperfolding constant; published as Annals of Mathematics 165 (2007), 547–565.", "fetched": [ "2026-09-09" ] }, { "title": "Regular paperfolding sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Regular_paperfolding_sequence", "own_work": false, "note": "value, digits, not periodic: “This number is known as the paperfolding constant and has the value,” 0.85073618820186...; “The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence”; “A paperfolding sequence is not ultimately periodic.”", "fetched": [ "2026-09-22" ] }, { "title": "Automatic sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Automatic_sequence", "own_work": false, "note": "“Both the Baum–Sweet sequence (OEIS: A086747) and the regular paperfolding sequence (OEIS: A014577) are automatic.”", "fetched": [ "2026-09-22" ] }, { "title": "Borwein and Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv (2008)", "url": "https://arxiv.org/abs/0806.1694", "own_work": false, "note": "the same binary digits, proved transcendental: “define the Gaussian Liouville number by γ:=0.110110011100100111011...”; “we prove that the Gaussian Liouville number and its relatives are transcendental.”", "fetched": [ "2026-09-22" ] }, { "title": "Thue–Morse sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence", "own_work": false, "note": "the construction: “the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.11", "url": "https://www.teclamairme.com/#r-4.11", "chapter": 4, "status": "confirmed", "claim": "The lemniscate constant, Gauss’s constant and Cahen’s constant have the stated definitions, values and theorem attributions. Siegel (1932) and Schneider both proved it through periods of elliptic functions; Schneider’s paper is dated 1937 by Wikipedia and 1936 by Waldschmidt, so the card gives no year for it. Corrections welcome.", "sources": [ { "title": "Wikipedia, Lemniscate constant", "url": "https://en.wikipedia.org/wiki/Lemniscate_constant", "own_work": false, "note": "ϖ ≈ 2.622 was proved transcendental by Carl Ludwig Siegel in 1932 and again by Theodor Schneider in 1937; Gauss’s constant is G = ϖ/π ≈ 0.8346268 = 1/M(1,√2), and Chudnovsky proved π and ϖ algebraically independent.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Cahen’s constant", "url": "https://en.wikipedia.org/wiki/Cahen%27s_constant", "own_work": false, "note": "C ≈ 0.643410546288, the alternating sum of 1/(s_i − 1) over Sylvester’s sequence s_i (1/1 − 1/2 + 1/6 − 1/42 + …); Eugène Cahen introduced it and proved it irrational in 1891, and Davison and Shallit proved it transcendental in 1991 from its continued fraction.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Conway’s Constant", "url": "https://mathworld.wolfram.com/ConwaysConstant.html", "own_work": false, "note": "confirms the look-and-say constant used in the algebraic brace.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Particular values of the gamma function", "url": "https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function", "own_work": false, "note": "credits G. V. Chudnovsky with proving Γ(1/3) and Γ(1/4) transcendental, and Yuri Nesterenko (1996) with the algebraic independence of Γ(1/4), π and e^π; Chudnovsky’s own independence result (π and ϖ, 1975) is on the Lemniscate constant line.", "fetched": [ "2026-09-09" ] }, { "title": "Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf", "own_work": false, "note": "Siegel and Schneider on periods: “The first transcendence result on periods of elliptic functions was proved by C.L. Siegel [199] as early as 1932.”; “Theorem 18 (Schneider, 1936) Assume that the invariants g2 and g3 of ℘ are algebraic. Then for any non-zero period ω of ℘, the numbers ω and η(ω) are transcendental.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.12", "url": "https://www.teclamairme.com/#r-4.12", "chapter": 4, "status": "judgement", "claim": "Γ(1/4) and Γ(1/3) are transcendental, because Chudnovsky proved that Γ(1/4) and π are algebraically independent; the result was first published in the 1970s and collected in his 1984 monograph. The lemniscate constant ϖ was proved transcendental by Siegel in 1932 and Gauss’s constant by Schneider in 1941, before Chudnovsky; Γ(1/2) = √π is transcendental with π.", "sources": [ { "title": "Wikipedia, Particular values of the gamma function", "url": "https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function", "own_work": false, "note": "confirms Chudnovsky’s authorship of the algebraic-independence result but gives no 1984/1976 date.", "fetched": [ "2026-09-09" ] }, { "title": "Michel Waldschmidt, Elliptic Functions and Transcendence (survey, PDF)", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf", "own_work": false, "note": "“In the 1970’s G.V. Chudnovsky proved strong results of algebraic independence,” including that π and Γ(1/4), and π and Γ(1/3), are algebraically independent; it cites “Chudnovsky’s lecture at the Helsinki ICM in 1978” (G. V. Chudnovsky, Algebraic independence of values of exponential and elliptic functions, Proc. ICM Helsinki 1978, vol. 1, 339–350); on Γ(1/5): “at least two of the three numbers π, Γ(1/5) and Γ(2/5) are algebraically independent,” and among the “Further open problems,” “the algebraic independence of three numbers among π, Γ(1/5), Γ(2/5), e^(π√5).”", "fetched": [ "2026-09-22" ] }, { "title": "Michel Waldschmidt, Les travaux de G. V. Čudnovskiĭ sur les nombres transcendants, Séminaire Bourbaki 488 (1975/76), Numdam", "url": "http://www.numdam.org/item/SB_1975-1976__18__274_0/", "own_work": false, "note": "the contemporary account of the result: “Γ(1/4) et π sont algébriquement indépendants,” and likewise Γ(1/3) and π.", "fetched": [ "2026-09-22" ] }, { "title": "Arshay Sheth, Real geometric transcendence for the Gamma function", "url": "https://arxiv.org/html/2605.15117", "own_work": false, "note": "“Γ(1/3) and Γ(1/4) are known to be transcendental by the work of Chudnovsky”, citing Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698–701, 767.", "fetched": [ "2026-09-09" ] }, { "title": "Lemniscate constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lemniscate_constant", "own_work": false, "note": "earlier proofs: “The lemniscate constant ϖ and Todd’s first lemniscate constant A were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941.”", "fetched": [ "2026-09-22" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "under “Numbers proven to be transcendental:,” “Values of the gamma function of rational numbers that are of the form Γ(n/2), Γ(n/3), Γ(n/4) or Γ(n/6).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Three dates are in circulation for one result, so the cards give none of them as a bare year. The first publication is the Doklady paper of 1976, which is what the transcendence literature cites; Wikipedia’s article on the lemniscate constant dates the same theorem to 1975; and the American Mathematical Society monograph that collects it, Contributions to the Theory of Transcendental Numbers, is 1984. “In the 1970s, collected in 1984” is true under all three, and that is what the cards say. Ruled, 10 September 2026: the first publication is the Doklady paper of 1976. Chudnovsky’s own 1980 paper cites it by title, issue and pages, and Waldschmidt, writing on precisely this theorem, dates it to 1976; Wikipedia’s 1975 is an undated secondary assertion carrying no reference. The cards keep the decade wording, which needs no year, and this entry carries 1976 where a date is wanted. Reference 17.24 carries the same ruling." }, { "id": "r-4.13", "url": "https://www.teclamairme.com/#r-4.13", "chapter": 4, "status": "confirmed", "claim": "The transcendental numerator is exactly zero because sin π=0, regardless of the status of its other displayed factors; its approximate nonzero parenthetical value is an own computation. π is the smallest positive number at which the sine function equals zero.", "sources": [ { "title": "Wikipedia, Lindemann–Weierstrass theorem", "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "confirms π is transcendental and the standard exponential consequences.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "The numerator contains sin(π) as a multiplicative factor, and sin(π)=0; evaluating the remaining displayed parenthetical with standard high-precision arithmetic gives approximately 2.4, but it cannot affect the product.", "fetched": [] }, { "title": "Pi, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pi", "own_work": false, "note": "sin π = 0 with π the first positive zero: “π is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The rider deliberately mixes algebraic, proven-transcendental and status-open ingredients; no claim that the rider itself is transcendental is made." }, { "id": "r-4.14", "url": "https://www.teclamairme.com/#r-4.14", "chapter": 4, "status": "confirmed", "claim": "The displayed formula has four zero numerators and nonzero denominators, so its chapter factor is zero; the denominator positivity check is an own computation from the displayed terms.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "1729−1729=0; 3·(1/3)−1=0; 0.999…−1=0; sin(π)=0. Every denominator is a sum/product of positive displayed values, hence nonzero, so the product is 0·0·0·0=0.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-4.15", "url": "https://www.teclamairme.com/#r-4.15", "chapter": 4, "status": "judgement", "claim": "The Laplace limit is about 0.6627434 and is the unique real x with x·e^√(1+x²)/(1+√(1+x²)) = 1; it is transcendental, since if it were algebraic then s = √(1+x²) would be a nonzero algebraic number and e^s = (1+s)/x would be algebraic, which Hermite–Lindemann forbids. Francesco Carlini had the limit before Laplace computed 0.66195 in 1827, and Cauchy sharpened it to 0.66274 in 1829.", "sources": [ { "title": "Wolfram MathWorld, Laplace Limit", "url": "https://mathworld.wolfram.com/LaplaceLimit.html", "own_work": false, "note": "gives the defining equation and the value 0.6627434…, and says nothing about the number’s arithmetic.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Laplace limit", "url": "https://en.wikipedia.org/wiki/Laplace_limit", "own_work": false, "note": "gives the same equation and 0.66274 34193 49181 58097…, and the history: Carlini had the limit five years before Laplace, who calculated 0.66195 in 1827, and Cauchy gave 0.66274 in 1829. It too says nothing about irrationality or transcendence.", "fetched": [ "2026-09-09" ] }, { "title": "Michel Waldschmidt, Introduction to Diophantine methods: irrationality and transcendence", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/IntroductionDiophantineMethods.pdf", "own_work": false, "note": "Theorem 2.1, Hermite–Lindemann: “let β be a non-zero algebraic number. Then e^β is transcendental.” This is the premise the deduction uses.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "At x = 0.66274341934918158097474209710925, evaluating x·exp(√(1+x²))/(1+√(1+x²)) to fifty digits gives 1 exactly at that precision; s = √(1+x²) = 1.19967864…, which is not zero, so the theorem applies.", "fetched": [] } ], "editorial_note": "No reference work states this number’s arithmetic nature. MathWorld and Wikipedia both give the equation and the value and stop there, and Wikipedia’s list of transcendental numbers does not name it. The status printed here is therefore a deduction from Hermite–Lindemann rather than a result quoted from a source, which is why the card sets the whole argument out in three lines for the reader to check. It is short, it uses nothing but the defining equation and a nineteenth-century theorem, and transcendence carries irrationality with it." }, { "id": "r-4.16", "url": "https://www.teclamairme.com/#r-4.16", "chapter": 4, "status": "confirmed", "claim": "Carmichael’s 561 passes the Fermat congruence for all bases coprime to it, while the usual b^(n−1)=1 test is only stated for coprime bases. The priority runs Václav Šimerka, who found the first seven in 1885 in a Czech journal that went unnoticed; A. Korselt, who gave the criterion in 1899 without an example; Robert Carmichael, who published 561 in 1910 and gave the numbers their name; and Alford, Granville and Pomerance, who proved in 1994 that there are infinitely many.", "sources": [ { "title": "Wikipedia, Carmichael number", "url": "https://en.wikipedia.org/wiki/Carmichael_number", "own_work": false, "note": "gives the coprime-base formulation and the equivalent b^n≡b formulation. Re-read raw on 17 September 2026 for the priority: “The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885” and his work “remained unnoticed”; “Theorem (A. Korselt 1899)” and “he did not give any examples”; “In 1910, Carmichael himself also published the smallest such number, 561, and the numbers were later named after him”; “In 1994 W. R. (Red) Alford, Andrew Granville and Carl Pomerance ... show that there really do exist infinitely many Carmichael numbers”.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-4.17", "url": "https://www.teclamairme.com/#r-4.17", "chapter": 4, "status": "confirmed", "claim": "Cohn proved that the only square Fibonacci numbers are 0, 1 and F12=144, so 144 is the only one after 1. The sequence first appears, as a count of rabbits, in Leonardo of Pisa’s Liber Abaci of 1202.", "sources": [ { "title": "Wolfram MathWorld, Fibonacci Number", "url": "https://mathworld.wolfram.com/FibonacciNumber.html", "own_work": false, "note": "cites Cohn (1964) and states that the only square Fibonacci numbers are 1 and F12=144 under the convention beginning 1,1.", "fetched": [ "2026-09-09" ] }, { "title": "Fibonacci sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fibonacci_sequence", "own_work": false, "note": "carries the 1202 date and the rabbits: “The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci, where it is used to calculate the growth of rabbit populations.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.18", "url": "https://www.teclamairme.com/#r-4.18", "chapter": 4, "status": "confirmed", "claim": "The largest known prime is M136279841, found by Luke Durant’s GIMPS search on 12 October 2024, with 41,024,320 digits.", "sources": [ { "title": "GIMPS, M136279841 press release", "url": "https://www.mersenne.org/primes/?press=M136279841", "own_work": false, "note": "confirms the discoverer, date, exponent and digit count, and “It is only the 52nd known Mersenne prime ever discovered.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-4.19", "url": "https://www.teclamairme.com/#r-4.19", "chapter": 4, "status": "confirmed", "claim": "Hilbert’s 1900 seventh-problem examples were 2^√2 and e^π; √2^√2 is a related number covered by Gelfond–Schneider. √2^√2 is also the number in Dov Jarden’s non-constructive proof that an irrational number raised to an irrational power can be rational.", "sources": [ { "title": "Wikipedia, Hilbert’s seventh problem", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem", "own_work": false, "note": "names Hilbert’s examples 2^√2 and e^π.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Gelfond–Schneider theorem", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem", "own_work": false, "note": "confirms the theorem used for √2^√2.", "fetched": [ "2026-09-09" ] }, { "title": "Irrational number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Irrational_number", "own_work": false, "note": "the classic proof: “Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a^b is rational:”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.20", "url": "https://www.teclamairme.com/#r-4.20", "chapter": 4, "status": "confirmed", "claim": "A real number is constructible with compass and straightedge exactly when it lies at the top of a finite tower of quadratic extensions of ℚ, so its degree over ℚ is a power of 2; Wantzel proved this in 1837 and deduced the impossibility of doubling the cube and of trisecting the general angle. A degree that is a power of 2 is necessary but not sufficient for constructibility. Non-constructible numbers are still computable: every real algebraic number, and π, can be approximated to any precision by an algorithm.", "sources": [ { "title": "J. S. Milne, Fields and Galois Theory", "url": "https://www.jmilne.org/math/Books/FT0.pdf", "own_work": false, "note": "Read 12 September 2026: Theorem 1.37 and Corollaries 1.38–1.39 give the successive-square-root criterion, algebraicity, and the impossibility of constructing ∛2; Remark 3.27 explains why power-of-two degree alone is insufficient.", "fetched": [] }, { "title": "Constructible number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_number", "own_work": false, "note": "(§ Algebraic properties, § History, § Impossible constructions): a real is constructible exactly when it “lies in a field at the top of a finite tower of real quadratic extensions,” with [ℚ(γ):ℚ] = 2ʳ; “Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using only compass and straightedge”; and the two minimal polynomials, x³ − 2 for ∛2 and 8x³ − 6x − 1 for cos(π/9), both of degree 3; and “However, this condition is not sufficient - there exist field extensions whose degree is a power of two, but which cannot be factored into a sequence of quadratic extensions.”", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor, Pierre Wantzel", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Wantzel/", "own_work": false, "note": "the biography for the 1837 paper in which the criterion and the two impossibility results appear.", "fetched": [ "2026-09-10" ] }, { "title": "Computable number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Computable_number", "own_work": false, "note": "“The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.21", "url": "https://www.teclamairme.com/#r-4.21", "chapter": 4, "status": "confirmed", "claim": "Joseph Liouville proved in 1844 that transcendental numbers exist and gave the first explicit decimal example, Liouville’s constant, in 1851; Georg Cantor proved in 1874 that the real algebraic numbers are countable while the reals are not, so almost all real numbers are transcendental. Cantor’s 1874 proof is constructive: applied to a list of the real algebraic numbers, it produces a transcendental one.", "sources": [ { "title": "Transcendental number, Wikipedia (§ History)", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "“Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “In 1874 Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable”; and “almost all real and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers and complex numbers are both uncountable.”", "fetched": [ "2026-09-10" ] }, { "title": "Cantor’s first set theory article, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article", "own_work": false, "note": "Cantor’s paper of 1874, Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, proves that the set of real algebraic numbers can be put in one-to-one correspondence with the positive integers and that an interval of reals cannot, which together give a transcendental number in every interval.", "fetched": [ "2026-09-10" ] }, { "title": "Joseph Liouville, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Liouville/", "own_work": false, "note": "he proved “the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "The chapter’s opening history sentence previously carried 4.9, which states only that the classical transcendence claims follow from the Lindemann-Weierstrass and Gelfond-Schneider theorems; neither Liouville’s priority nor Cantor’s counting argument is in it, and Cantor’s 1874 carried no footnote at all. This entry carries both." }, { "id": "r-4.22", "url": "https://www.teclamairme.com/#r-4.22", "chapter": 4, "status": "confirmed", "claim": "In the workbench aside: 8 and 9 are the only consecutive perfect powers, conjectured by Catalan in a note published in 1844 and proved by Mihăilescu in 2002; and 4900 is, apart from 1, the only perfect square that is also a square pyramidal number.", "sources": [ { "title": "Wolfram MathWorld, Catalan’s Conjecture", "url": "https://mathworld.wolfram.com/CatalansConjecture.html", "own_work": false, "note": "“The conjecture made by Belgian mathematician Eugene Charles Catalan in 1844” that 8 and 9 are the only consecutive powers, with Mihailescu’s proof sent to several mathematicians on 18 April 2002; the original note is cited as Catalan, J. reine angew. Math. 27, 192, 1844.", "fetched": [ "2026-09-10" ] }, { "title": "Catalan’s conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Catalan%27s_conjecture", "own_work": false, "note": "“conjectured by the mathematician Eugène Charles Catalan in 1842 and proven in 2002 by Preda Mihăilescu,” with the same primary note cited as J. Reine Angew. Math. 27 (1844), 192, and the proof published in 2004.", "fetched": [ "2026-09-10" ] }, { "title": "Square pyramidal number, Wikipedia (§ Cannonball problem)", "url": "https://en.wikipedia.org/wiki/Square_pyramidal_number", "own_work": false, "note": "“Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number”; the first complete proof that no others exist is G. N. Watson’s, 1918.", "fetched": [ "2026-09-10" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Enumerating the square pyramidal numbers n(n+1)(2n+1)/6 for n up to 200 and testing each for squareness gives exactly n = 0, 1 and 24, that is 0, 1 and 4900, so 4900 is the only one after 1.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Two comparable sources date the conjecture differently — MathWorld says 1844, Wikipedia’s lead says 1842 — while both cite the same primary note, Crelle 27 (1844), 192. The page keeps 1844, the year of the publication both sources point at; no tilde is needed because the disagreement is over an undated secondary assertion, not over two datable events. The aside’s superlative for 4900 is corrected to “the only square after 1,” the same form this chapter’s fib144 card already uses for 144." }, { "id": "r-4.23", "url": "https://www.teclamairme.com/#r-4.23", "chapter": 4, "status": "confirmed", "claim": "The numbers left on the workbench have the stated definitions and values: 153 is the sum of the cubes of its digits; 2520 is the least common multiple of 1 through 10; the chessboard holds 2⁶⁴ − 1 grains; the centillion is 10³⁰³ on the short scale and 10⁶⁰⁰ on the long, and is the highest -illion name the major English dictionaries include; Theodorus proved √3 irrational; the supergolden ratio is the real root of x³ = x² + 1; 2^(1/12) is the equal-tempered semitone; τ = 2π; the golden angle is 2π/φ² and separates the florets of a sunflower; the magic angle is arccos(1/√3), the tilt of magic-angle spinning NMR; nonzero values of arccos at algebraic arguments are transcendental, and so is any nonzero algebraic multiple of π. Turning by the golden angle keeps successive elements off any common radial line; Vogel’s sunflower model of 1979 places floret n at n × 137.508°. √3, a square root of a positive integer, is constructible.", "sources": [ { "title": "Narcissistic number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Narcissistic_number", "own_work": false, "note": "gives the base-10 example 153 = 1³ + 5³ + 3³.", "fetched": [ "2026-09-16" ] }, { "title": "2520 (number), Wikipedia", "url": "https://en.wikipedia.org/wiki/2520_(number)", "own_work": false, "note": "the smallest number divisible by all integers from one to ten, their least common multiple.", "fetched": [ "2026-09-16" ] }, { "title": "Wheat and chessboard problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem", "own_work": false, "note": "the total number of grains is 2^64 − 1, or 18,446,744,073,709,551,615.", "fetched": [ "2026-09-16" ] }, { "title": "Names of large numbers, Wikipedia", "url": "https://en.wikipedia.org/wiki/Names_of_large_numbers", "own_work": false, "note": "lists centillion as 10^303 (short scale) and 10^600 (long scale), and says centillion appears to be the highest name ending in -illion included in the dictionaries it surveys.", "fetched": [ "2026-09-16" ] }, { "title": "Theodorus of Cyrene, Wikipedia", "url": "https://en.wikipedia.org/wiki/Theodorus_of_Cyrene", "own_work": false, "note": "quotes Plato’s Theaetetus: Theodorus showed the square roots of the non-square numbers from 3 up to 17 irrational; his method is unknown.", "fetched": [ "2026-09-16" ] }, { "title": "Square root of 3, Wikipedia", "url": "https://en.wikipedia.org/wiki/Square_root_of_3", "own_work": false, "note": "value 1.7320508075688772…, also called Theodorus’s constant.", "fetched": [ "2026-09-16" ] }, { "title": "Supergolden ratio, Wikipedia", "url": "https://en.wikipedia.org/wiki/Supergolden_ratio", "own_work": false, "note": "the unique real solution of x³ = x² + 1, 1.465571231876768….", "fetched": [ "2026-09-16" ] }, { "title": "Twelfth root of two, Wikipedia", "url": "https://en.wikipedia.org/wiki/Twelfth_root_of_two", "own_work": false, "note": "1.0594630943592952646; in the equal-tempered chromatic scale each note is 2^(1/12) times the frequency of the one below.", "fetched": [ "2026-09-16" ] }, { "title": "Turn (angle), Wikipedia", "url": "https://en.wikipedia.org/wiki/Turn_(angle)", "own_work": false, "note": "one turn is 2π radians, proposed to be written τ, citing Michael Hartl’s Tau Manifesto (2010); 1 turn = τ rad ≈ 6.283185307 rad.", "fetched": [ "2026-09-16" ] }, { "title": "Golden angle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Golden_angle", "own_work": false, "note": "≈ 137.50776°, 360(2 − φ) degrees or 2π/φ² radians; the angle separating the florets on a sunflower; Vogel (1979) is in its reading list, and his model is on the Fermat’s spiral line.", "fetched": [ "2026-09-16" ] }, { "title": "Magic Angle, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/MagicAngle.html", "own_work": false, "note": "θ_m = cos^(-1)(1/√3) = 54.7356103… degrees.", "fetched": [ "2026-09-16" ] }, { "title": "Magic angle spinning, Wikipedia", "url": "https://en.wikipedia.org/wiki/Magic_angle_spinning", "own_work": false, "note": "the sample is spun at the magic angle, about 54.74°, to the magnetic field in solid-state NMR.", "fetched": [ "2026-09-16" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "lists nonzero values of arcsin and arccos at algebraic arguments as transcendental by Lindemann–Weierstrass, and says a non-constant algebraic function of π, such as 5π, is transcendental.", "fetched": [ "2026-09-16" ] }, { "title": "Phyllotaxis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Phyllotaxis", "own_work": false, "note": "why successive elements never line up: “Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge.”", "fetched": [ "2026-09-22" ] }, { "title": "Fermat’s spiral, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_spiral", "own_work": false, "note": "Vogel’s model: “The full model proposed by H. Vogel in 1979 is r = c√n, θ = n × 137.508°, where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor.” and “The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.”", "fetched": [ "2026-09-22" ] }, { "title": "Constructible number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_number", "own_work": false, "note": "carries √3’s constructibility as a square root of a positive number: “The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.24", "url": "https://www.teclamairme.com/#r-4.24", "chapter": 4, "status": "confirmed", "claim": "The chapter’s small-integer history is as stated: Christian Kramp introduced the notation n! in 1808; Plato in the Laws proposes 5040 as the number of citizens of a polis because it divides so many ways, being divisible by every number from 1 to 12 except 11, and it has 60 divisors in all; the name primorial was coined by Harvey Dubner, in Factorial and primorial primes, Journal of Recreational Mathematics 19 (1987), 197-203; and Euclid gives the first known definition of the golden ratio, as a line cut in extreme and mean ratio. 30 is the largest number all of whose totatives other than 1 are prime (Schatunowsky’s theorem of 1893).", "sources": [ { "title": "Factorial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Factorial", "own_work": false, "note": "read as raw HTML: “The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808.”", "fetched": [ "2026-09-17" ] }, { "title": "5040 (number), Wikipedia", "url": "https://en.wikipedia.org/wiki/5040_(number)", "own_work": false, "note": "read as raw HTML: “Plato mentions in his dialogue Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis,” and “this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11.” The page gives the reason Plato gives, which is divisibility rather than the divisor count; 5040 = 2⁴·3²·5·7 has 5·3·2·2 = 60 divisors, which is the book’s own arithmetic.", "fetched": [ "2026-09-17" ] }, { "title": "Primorial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Primorial", "own_work": false, "note": "read as raw HTML: “The name ‘primorial’, coined by Harvey Dubner, draws an analogy to primes similar to the way the name ‘factorial’ relates to factors,” and its reference list gives Dubner, Harvey (1987), “Factorial and primorial primes,” J. Recr. Math. 19, 197-203.", "fetched": [ "2026-09-17" ] }, { "title": "Golden ratio, Wikipedia", "url": "https://en.wikipedia.org/wiki/Golden_ratio", "own_work": false, "note": "read as raw HTML: “The golden ratio was called the extreme and mean ratio by Euclid,” and Euclid’s Elements (c. 300 BC) “contains its first known definition,” “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”", "fetched": [ "2026-09-17" ] }, { "title": "Hala King and Victor Pambuccian, Generalized Schatunowsky theorem in a weak arithmetic, arXiv:2506.08256", "url": "https://arxiv.org/abs/2506.08256", "own_work": false, "note": "abstract: “Schatunowsky’s 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 17 September 2026. Six cards in this chapter hung datable history on r-4.1, whose claim line covers the chapter’s elementary identities and type classifications and whose three sources are the DLMF Bernoulli tables, MathWorld on algebraic numbers and Brocard’s problem. None of that history was in it. Two of the six, #sixty and #twenty, were re-pointed at this chapter’s own r-2.3 and r-2.4; #sqrt2’s Pythagorean tradition already stood on r-8.9; the remaining three, #fact7, #primorial5 and #phi, are what this entry carries. Note what the Plato source does and does not say: the reason he gives is that 5040 divides by every number from 1 to 12 but 11, not that it has 60 divisors, so the card now says both and attributes only the first to him." }, { "id": "r-4.25", "url": "https://www.teclamairme.com/#r-4.25", "chapter": 4, "status": "confirmed", "claim": "The chapter epigraph is Kronecker’s, said at the 1886 Berlin meeting of natural scientists and reported by Heinrich Weber in the Jahresbericht der Deutschen Mathematiker-Vereinigung of 1893.", "sources": [ { "title": "Leopold Kronecker, Wikiquote", "url": "https://en.wikiquote.org/wiki/Leopold_Kronecker", "own_work": false, "note": "read as raw HTML: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God made the integers, all the rest is the work of man. In a lecture to the Berliner Naturforschung-Versammlung, 1886; quoted in the obituary ‘Leopold Kronecker’, by H[einrich] Weber, in Jahresbericht der Deutsche Mathematiker Vereinigung, 1893, page 19,” which is the page’s footer line word for word.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is one of the four secondhand epigraphs where the book asserts provenance rather than quoting a printed original, so the reporter is named in the claim line as well as in the footer." }, { "id": "r-4.26", "url": "https://www.teclamairme.com/#r-4.26", "chapter": 4, "status": "confirmed", "claim": "A simple continued fraction writes a real number as a₀ + 1/(a₁ + 1/(a₂ + ⋯)) with integer partial quotients; the finite truncations are its convergents. The convergents give a systematic sequence of rational approximations. The universal equal-denominator superlative is not asserted: for example, 8/5 = [1;1, 1, 2] has initial convergent 1/1, but 2/1 is closer with the same denominator. The expansion terminates exactly for rationals, so an infinite expansion is a proof of irrationality.", "sources": [ { "title": "Simple continued fraction, Wikipedia", "url": "https://en.wikipedia.org/wiki/Simple_continued_fraction", "own_work": false, "note": "The earlier annotation used this page for the procedure and described a best-approximation superlative too broadly. That universal equal-denominator assertion is no longer used in this entry: the direct rational counterexample in its claim and the NIST definitions below control the book’s wording.", "fetched": [] }, { "title": "Continued Fraction, MathWorld", "url": "https://mathworld.wolfram.com/ContinuedFraction.html", "own_work": false, "note": "the same simple-continued-fraction notation (convergents are only linked from it, under See also).", "fetched": [ "2026-09-17" ] }, { "title": "NIST DLMF, §1.12 Continued Fractions", "url": "https://dlmf.nist.gov/1.12", "own_work": false, "note": "Section 1.12(ii), equations 1.12.3–4, defines the finite truncations as convergents. Its indexing begins at zero; the book counts the initial integer truncation as first. The equal-denominator counterexample in the claim is direct rational arithmetic.", "fetched": [ "2026-09-30" ] } ], "editorial_note": "" }, { "id": "r-4.27", "url": "https://www.teclamairme.com/#r-4.27", "chapter": 4, "status": "confirmed", "claim": "A Salem number is a real algebraic integer greater than 1 whose other conjugates all lie in the closed unit disc with at least one on the unit circle; the minimal polynomial is therefore reciprocal, of even degree at least 4, and every Salem number is a unit. The smallest known Salem number is the largest real root of Lehmer’s degree-10 polynomial x¹⁰ + x⁹ − x⁷ − x⁶ − x⁵ − x⁴ − x³ + x + 1, about 1.1762808, which is also the smallest known Mahler measure above 1; whether anything smaller exists is open. A Pisot number is a real algebraic integer greater than 1 all of whose other conjugates lie strictly inside the unit circle.", "sources": [ { "title": "Salem number, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Salem_number", "own_work": false, "note": "read as raw HTML: “An algebraic integer θ > 1 such that all other Galois conjugates of θ lie inside the closed unit disc |z| ≤ 1, with at least one lying on the boundary”; reciprocity, even degree ≥ 4; “The smallest known Salem number is the number σ₁ = 1.1762808… of degree 10 known as Lehmer’s number… This is also the smallest known value > 1 of the Mahler measure.”", "fetched": [ "2026-09-17" ] }, { "title": "Salem number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Salem_number", "own_work": false, "note": "read as raw HTML: the same definition, names them after Raphaël Salem, gives Lehmer’s polynomial explicitly with root about 1.17628, states that every Salem number is a unit and a Perron number, and that it is conjectured to be the smallest.", "fetched": [ "2026-09-17" ] }, { "title": "Pisot–Vijayaraghavan number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number", "own_work": false, "note": "the definition: “In mathematics, a Pisot–Vijayaraghavan number (or Pisot number or PV number) is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.28", "url": "https://www.teclamairme.com/#r-4.28", "chapter": 4, "status": "confirmed", "claim": "The class number of an imaginary quadratic field ℚ(√−d) is 1 exactly when its ring of integers has unique factorization; by the Stark–Heegner theorem there are exactly nine such d — 1, 2, 3, 7, 11, 19, 43, 67 and 163. Gauss conjectured the list; Heegner proved it in 1952 up to a minor gap, and Baker and Stark proved it independently in 1966. When the class number is 1 the j-invariant at the corresponding point is an integer, which is why e^(π√163) misses an integer so narrowly.", "sources": [ { "title": "Heegner number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heegner_number", "own_work": false, "note": "read as raw HTML: “Heegner numbers are square-free positive integers d such that the imaginary quadratic field ℚ(√−d) has class number 1. Equivalently, the ring of algebraic integers… has unique factorization”; “there are precisely nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, and 163… conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966”; and “if the imaginary quadratic extension ℚ(τ) has class number 1… the j-invariant is an integer.”", "fetched": [ "2026-09-17" ] }, { "title": "Class Number, MathWorld", "url": "https://mathworld.wolfram.com/ClassNumber.html", "own_work": false, "note": "the class number as the order of the ideal class group, and its being 1 exactly for unique factorization.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-4.29", "url": "https://www.teclamairme.com/#r-4.29", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: the sixteen-year-old Nicolò Paganini found the amicable pair 1184, 1210, published in the Atti of the Turin Academy for 1866–67 (MathWorld dates the find 1866, Wikipedia 1867); Jacob Bernoulli published the Bernoulli numbers in the Ars Conjectandi of 1713 (Seki a year earlier); Pierre Wantzel proved in 1837 that doubling the cube with compass and straightedge is impossible; Conway chained-arrow notation comes from The Book of Numbers by Conway and Guy, 1996; David Champernowne, an undergraduate, showed in 1933 that the constant is normal in base 10; Kurt Mahler proved Champernowne’s constant transcendental in 1937; Conway’s paper on the look-and-say sequence, The Weird and Wonderful Chemistry of Audioactive Decay, appeared in Eureka 46 in January 1986 and was reprinted by Springer in 1987; Wantzel proved in 1837 that the polygons outside Gauss’s criterion (the heptagon first) cannot be constructed; e first appears in (Jacob) Bernoulli’s 1683 study of compound interest. By Kummer’s theorem the Bernoulli numerators decide which primes are regular, and for a regular prime p Fermat’s equation of degree p has no solution in nonzero integers.", "sources": [ { "title": "Amicable Pair, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/AmicablePair.html", "own_work": false, "note": "read raw: “In 1866, 16-year old B. Nicolò I. Paganini found the small amicable pair (1184, 1210) which had eluded his more illustrious predecessors (Paganini 1866-1867; Dickson 2005, p. 47).”; “In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), although these results were actually rediscoveries of numbers known to Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended to 60.”", "fetched": [ "2026-09-18" ] }, { "title": "Bernoulli number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bernoulli_number", "own_work": false, "note": "read raw: “Bernoulli’s result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.”; “The Bernoulli numbers are related to Fermat’s Last Theorem (FLT) by Kummer’s theorem, which says: If the odd prime p does not divide any of the numerators of the Bernoulli numbers B₂, B₄, ..., Bₚ₋₃ then xᵖ + yᵖ + zᵖ = 0 has no solutions in nonzero integers. Prime numbers with this property are called regular primes.” (sub- and superscripts as typeset on the page)", "fetched": [ "2026-09-18" ] }, { "title": "Doubling the cube, Wikipedia", "url": "https://en.wikipedia.org/wiki/Doubling_the_cube", "own_work": false, "note": "read raw: “The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837..”", "fetched": [ "2026-09-18" ] }, { "title": "Conway chained arrow notation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Conway_chained_arrow_notation", "own_work": false, "note": "read raw: “John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62.”", "fetched": [ "2026-09-18" ] }, { "title": "Champernowne constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Champernowne_constant", "own_work": false, "note": "read raw: “It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.”; “K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428..”", "fetched": [ "2026-09-18" ] }, { "title": "Look-and-say sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Look-and-say_sequence", "own_work": false, "note": "read raw: “title=The Weird and Wonderful Chemistry of Audioactive Decay |journal=Eureka |date=January 1986 |volume=46 |pages=5–16.”", "fetched": [ "2026-09-18" ] }, { "title": "Constructible polygon, Wikipedia", "url": "https://en.wikipedia.org/wiki/Constructible_polygon", "own_work": false, "note": "read raw: “A full proof of necessity was given by Pierre Wantzel in 1837..”", "fetched": [ "2026-09-18" ] }, { "title": "e (mathematical constant), Wikipedia", "url": "https://en.wikipedia.org/wiki/E_(mathematical_constant)", "own_work": false, "note": "read raw: “Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest.”", "fetched": [ "2026-09-18" ] }, { "title": "Amicable numbers, Wikipedia", "url": "https://en.wikipedia.org/wiki/Amicable_numbers", "own_work": false, "note": "dates the find a year later than MathWorld: “The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.30", "url": "https://www.teclamairme.com/#r-4.30", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: the letter e appears in a letter Euler wrote to Goldbach in 1731 (MacTutor calls this its first appearance; Wikipedia has Euler using the letter from 1727 or 1728, in an unpublished paper on explosive forces in cannons); Charles Hermite proved e transcendental in 1873, the first number proved transcendental without having been specifically constructed for the purpose; Eddington’s claim was made in his Tarner lectures of 1938, published in 1939; Fermat conjectured in 1640 that all numbers 2^(2^n)+1 are prime; Euler factored F5 = 641 x 6700417 in 1732; Johann Gustav Hermes spent ten years on the 65537-gon construction, finished in 1894; the fast-growing hierarchy of Martin Löb and Stanley Wainer, 1970; Gauss found in 1799 that the reciprocal of the arithmetic-geometric mean of 1 and sqrt 2 matches 2/π times the lemniscate integral; Theodor Schneider proved Gauss’s constant transcendental in 1941. In the same passage Eddington claims the same number of electrons as protons.", "sources": [ { "title": "The number e, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/e/", "own_work": false, "note": "read raw: “Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731..”", "fetched": [ "2026-09-18" ] }, { "title": "e (mathematical constant), Wikipedia", "url": "https://en.wikipedia.org/wiki/E_(mathematical_constant)", "own_work": false, "note": "read raw: “It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873..”", "fetched": [ "2026-09-18" ] }, { "title": "Eddington number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Eddington_number", "own_work": false, "note": "read raw: “During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that:.”", "fetched": [ "2026-09-18" ] }, { "title": "Some historical notes on proving Fermat numbers and cofactors to be composite, 64ordle.au", "url": "https://64ordle.au/fermat/historical/", "own_work": false, "note": "read raw: “Thus in the second half of 1640 we find Fermat writing at least three letters to Frénicle directly, and at least once more to Mersenne..”", "fetched": [ "2026-09-18" ] }, { "title": "Fermat number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat_number", "own_work": false, "note": "read raw: “Fermat’s conjecture was refuted by Leonhard Euler in 1732.”", "fetched": [ "2026-09-18" ] }, { "title": "65537-gon, Wikipedia", "url": "https://en.wikipedia.org/wiki/65537-gon", "own_work": false, "note": "read raw: “the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.”; “Although it was known to Carl Friedrich Gauss by 1801 that the regular 65537-gon was constructible.”", "fetched": [ "2026-09-18" ] }, { "title": "Hierarchies of number-theoretic functions. I (Löb and Wainer), Archiv für mathematische Logik und Grundlagenforschung, Springer", "url": "https://link.springer.com/article/10.1007/BF01967649", "own_work": false, "note": "read raw: “Arch math Logik 13, 39–51 (1970). https://doi.org/10.1007/BF01967649.”", "fetched": [ "2026-09-18" ] }, { "title": "Elliptic functions: introduction course, lecture 2 (the AGM), Vladimir G. Tkachev, Linköping University", "url": "https://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf", "own_work": false, "note": "read raw: “By May 30th, 1799, Gauss had observed, purely computationally, that” 1/M(1, √2) and (2/π)∫₀¹ dt/√(1 − t⁴) “agreed to at least eleven (!) decimal places.”", "fetched": [ "2026-09-18" ] }, { "title": "Lemniscate constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lemniscate_constant", "own_work": false, "note": "read raw: “Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.31", "url": "https://www.teclamairme.com/#r-4.31", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Hilbert set his problems (the seventh among them) in 1900; Kasner’s nine-year-old nephew Milton Sirotta named the googol in 1920; Kasner published the name in Mathematics and the Imagination in 1940; googolplex was named at the same time in 1920 as the googol; Leopold Theisinger proved in 1915 that H_n is an integer only for n = 1; Wilhelm Ackermann’s three-argument function of 1928 is an early version of the hyperoperations; Reuben Goodstein defined the sequence and coined the Greek names (tetration, pentation...) in 1947; D. R. Kaprekar found the 6174 routine in 1949; Knuth introduced up-arrow notation in 1976, in Science. Kasner’s nephew was asked to name the number, and named the googolplex at the same time. The hyperoperation sequence as a whole is a version of Ackermann’s original three-argument function. A googol exceeds the roughly 10^80 elementary particles in the observable universe.", "sources": [ { "title": "Hilbert’s seventh problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem", "own_work": false, "note": "read raw: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900..”", "fetched": [ "2026-09-18" ] }, { "title": "Googol, Wikipedia", "url": "https://en.wikipedia.org/wiki/Googol", "own_work": false, "note": "read raw: “The term was coined in 1920 by nine-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.”; “Kasner popularized the concept in his 1940 book Mathematics and the Imagination.”; “Carl Sagan pointed out that the total number of elementary particles in the observable universe is around 10^80 (the Eddington number)” (exponent as typeset on the page).", "fetched": [ "2026-09-18" ] }, { "title": "Googolplex, Wikipedia", "url": "https://en.wikipedia.org/wiki/Googolplex", "own_work": false, "note": "read raw: “In 1920, Edward Kasner’s nine-year-old nephew, Milton Sirotta, coined the term googol.”", "fetched": [ "2026-09-18" ] }, { "title": "Bemerkung über die harmonische Reihe (L. Theisinger), Monatshefte für Mathematik und Physik, Springer", "url": "https://link.springer.com/article/10.1007/BF01999444", "own_work": false, "note": "read raw: “Monatsh. f. Mathematik und Physik 26, 132–134 (1915). https://doi.org/10.1007/BF01999444.”", "fetched": [ "2026-09-18" ] }, { "title": "Hyperoperation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hyperoperation", "own_work": false, "note": "read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function”; “In his 1947 paper, Reuben Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation.”", "fetched": [ "2026-09-18" ] }, { "title": "Kaprekar Routine, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/KaprekarRoutine.html", "own_work": false, "note": "read raw: “The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers.”", "fetched": [ "2026-09-18" ] }, { "title": "Knuth’s up-arrow notation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation", "own_work": false, "note": "read raw: “Knuth’s up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976..”", "fetched": [ "2026-09-18" ] }, { "title": "Edward Kasner, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Kasner/", "own_work": false, "note": "quoting Mathematics and the Imagination: “The name “googol” was invented by a child (Dr Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it.”; “At the same time that he suggested “googol” he gave a name for a still larger number: “Googolplex.””", "fetched": [ "2026-09-22" ] }, { "title": "Yang, Li, Feng and Jiang, On the integrality of the first and second elementary symmetric functions, AIMS Mathematics 2(4) (2017)", "url": "https://www.aimspress.com/fileOther/PDF/Math/Math-02-00682.pdf", "own_work": false, "note": "the first name and year: “The first published proof went back to 1915 by Leopold Theisinger.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.32", "url": "https://www.teclamairme.com/#r-4.32", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Ackermann’s function of 1928 (preceding Goodstein’s naming of tetration, pentation, hexation in 1947); Lehmer’s degree-10 polynomial and its root 1.17628 found by Derrick Lehmer in 1933; Gauss’s diary records his results on the lemniscate in 1798, and the relation between the arithmetic-geometric mean of 1 and √2 and π/ϖ in May 1799; Gauss proved in 1799 that M(1, √2) = π/ϖ; Liouville announced in 1844 that numbers approximated too well by rationals must be transcendental; Liouville gave the constant Σ10^(−n!) in his full paper of 1851, the first decimal constant proven transcendental; the Bignum Bakeoff, which Ralph Loader’s program won, was a contest of December 2001; Jinkōki first printed in 1627 with many errors; Jinkōki corrected in the 1631 edition. Ralph Loader’s program won the Bignum Bakeoff, whose entries were C programs of at most 512 characters; it diagonalizes over the calculus of constructions and returns D^5(99).", "sources": [ { "title": "Hyperoperation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hyperoperation", "own_work": false, "note": "read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function \\phi(a, b, n) which gradually evolved into a 2-argument function known as the Ackermann function..”", "fetched": [ "2026-09-18" ] }, { "title": "Lehmer’s Mahler Measure Problem, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/LehmersMahlerMeasureProblem.html", "own_work": false, "note": "read raw: “Lehmer’s Mahler measure problem is an unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials..”", "fetched": [ "2026-09-18" ] }, { "title": "B. Sury, The Arithmetico-geometric Mean of Gauss, Resonance, August 2000", "url": "https://www.ias.ac.in/article/fulltext/reso/005/08/0072-0083", "own_work": false, "note": "read raw: “Gauss’s 92nd entry made in 1798 in his famous mathematical diary reads “I have obtained most elegant results concerning the lemniscate, which surpasses all expectation.”", "fetched": [ "2026-09-18" ] }, { "title": "Arithmetic–geometric mean, Wikipedia", "url": "https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean", "own_work": false, "note": "read raw: “In 1799, Gauss proved that M(1,√2) = π/ϖ where ϖ is the lemniscate constant..”", "fetched": [ "2026-09-18" ] }, { "title": "Joseph Liouville, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Liouville/", "own_work": false, "note": "read raw: “led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions”; “In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number 0 . 1100010000000000000000010000 ....”", "fetched": [ "2026-09-18" ] }, { "title": "David Moews, BIGNUM BAKEOFF contest rules (the organiser’s own file, dated 10 Dec 2001)", "url": "http://djm.cc/bignum-rules-posted.txt", "own_work": false, "note": "read raw: “Entries must be e-mailed to dmoews@xraysgi.ims.uconn.edu and received before 23:59:59 PST, December 31, 2001..”", "fetched": [ "2026-09-18" ] }, { "title": "Japanese numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Japanese_numerals", "own_work": false, "note": "read raw: “The initial edition was published in 1627 and had many errors, most of which were fixed in the 1631 edition. In 1634, there was yet another edition which again changed a few values.”; “Variation is due to the Jinkōki (塵劫記), Japan’s oldest mathematics text.” Its table of large numbers runs in steps of 10⁴ up to 極 (10⁴⁸) and then gives each higher name two values, from “10^52 (or 10^56)” for 恒河沙 to “10^68 (or 10^88)” for 無量大数.", "fetched": [ "2026-09-18" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "the first decimal examples: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”", "fetched": [ "2026-09-22" ] }, { "title": "Liouville number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Liouville_number", "own_work": false, "note": "the 1844 approximation bound: “In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound.”", "fetched": [ "2026-09-22" ] }, { "title": "BIGNUM BAKEOFF contest recap, David Moews", "url": "http://djm.cc/bignum-results.txt", "own_work": false, "note": "“The aim of this contest was to write a C program of 512 characters or less (excluding whitespace) that returned as large a number as possible from main()”; “The winner is Ralph Loader”; “The final and winning entry, {loader.c}, diagonalizes over the Huet-Coquand ‘calculus of constructions’.”; “The program’s return value is D@@5(99).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.33", "url": "https://www.teclamairme.com/#r-4.33", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Jinkōki changed again in the 1634 edition; the folded-strip pattern was described by Martin Gardner in 1967, and many of its properties were first published by Chandler Davis and Donald Knuth in 1970; Davis and Knuth’s paper on the dragon curve is of 1970; Sylvester called an odd perfect number “little short of a miracle” in 1888; Lambert proved π irrational in 1761; 22/7 is Archimedes’ upper bound for π from about 250 BC, from 96-sided polygons; Gérard Cordonnier found the plastic ratio in 1924; Hans van der Laan, who came to it in 1928, first called it the fundamental ratio and used the name plastic number from the 1950s; Hans van der Laan discovered his system of proportions in 1928; e^(π√163) as an almost-integer first noticed by Charles Hermite in 1859. Van der Laan’s plastic number describes three-dimensional growth, 1 + a = a³, where the golden section is 1 + a = a².", "sources": [ { "title": "Japanese numerals, Wikipedia", "url": "https://en.wikipedia.org/wiki/Japanese_numerals", "own_work": false, "note": "read raw: “In 1634, there was yet another edition which again changed a few values. The above variation is due to inconsistencies in the latter two editions..”", "fetched": [ "2026-09-18" ] }, { "title": "Dragon curve, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dragon_curve", "own_work": false, "note": "read raw: “It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth.”; “Chandler Davis (1970), Donald J. Knuth (1970).”", "fetched": [ "2026-09-18" ] }, { "title": "Perfect number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Perfect_number", "own_work": false, "note": "read raw: “In 1888, Sylvester stated: ... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.”; “The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.”", "fetched": [ "2026-09-18" ] }, { "title": "Irrational number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Irrational_number", "own_work": false, "note": "read raw: “Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e^n is irrational if n is rational (unless n = 0)..”", "fetched": [ "2026-09-18" ] }, { "title": "Pi, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pi", "own_work": false, "note": "read raw: “The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion..”", "fetched": [ "2026-09-18" ] }, { "title": "Plastic ratio, Wikipedia", "url": "https://en.wikipedia.org/wiki/Plastic_ratio", "own_work": false, "note": "read raw: “French high school student Gérard Cordonnier discovered the ratio for himself in 1924. In his correspondence with Hans van der Laan a few years later, he called it the radiant number. Van der Laan initially referred to it as the fundamental ratio, using the plastic number from the 1950s onward..”", "fetched": [ "2026-09-18" ] }, { "title": "arXiv:1805.07178, Appendix C", "url": "https://arxiv.org/abs/1805.07178", "own_work": false, "note": "read in the paper’s PDF, Appendix C (the abstract page does not carry it): “In 1928, shortly after abandoning his architectural studies and becoming a novice monk of the Benedictine Order, Hans van der Laan discovered a new, unique system of architectural proportions..”", "fetched": [ "2026-09-18" ] }, { "title": "Heegner number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heegner_number", "own_work": false, "note": "read raw: “This number was discovered in 1859 by the mathematician Charles Hermite.”; it gives e^(π√163) “≈ 640 320³ + 744” and “In a 1975 April Fools’ Day article in Scientific American magazine, ‘Mathematical Games’ columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.”", "fetched": [ "2026-09-18" ] }, { "title": "The Plastic Number ratio, Dom Hans van der Laan foundation", "url": "https://domhansvanderlaan.nl/theory-practice/theory/the-plastic-number-ratio/", "own_work": false, "note": "the three-dimensional contrast with the golden section: “This growth occurs three-dimensionally. Moreover: the Plastic Number represents the smallest proportional growth of three-dimensional objects.”; “This is different than the golden section, which defines the Fibonacci series, and is defined as 1 + a = a2.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.34", "url": "https://www.teclamairme.com/#r-4.34", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Ramanujan gave almost-integer examples such as e^(π√58) in a paper of 1913–14 (the volume’s dates; 1913 is its first year); Claude Shannon’s 1950 paper Programming a Computer for Playing Chess gives the 10^120 estimate; Skewes proved in 1933, assuming RH, a crossing below e^(e^(e^79)) < 10^(10^(10^34)); In 1955 Skewes gave an unconditional bound of about 10^(10^(10^964)); Bays and Hudson’s 2000 crossing near 1.398×10^316, sharpened by Saouter and Demichel in 2010 to below about 1.397×10^316; No crossing below 10^19 (Büthe 2015); The 1729 taxi story is Hardy’s own, from his 1940 book on Ramanujan, where he went to see Ramanujan ill at Putney; Kurt Mahler proved the Prouhet–Thue–Morse constant transcendental in 1929; Kruskal’s tree theorem of 1960. G. H. Hardy described Skewes’ number as the largest number that has ever served any definite purpose in mathematics.", "sources": [ { "title": "Ramanujan Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/RamanujanConstant.html", "own_work": false, "note": "read raw: “Although Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such ), he did not actually mention the particular near-identity given above.”; “The name “Ramanujan’s constant” was coined by Simon Plouffe and derives from an April Fool’s joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American.”", "fetched": [ "2026-09-18" ] }, { "title": "Shannon number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Shannon_number", "own_work": false, "note": "read raw: “His influential 1950 paper on it, “Programming a Computer for Playing Chess”; the estimate is “based on an average of about 10^3 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves” (exponent as typeset on the page).", "fetched": [ "2026-09-18" ] }, { "title": "Skewes’s number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Skewes%27s_number", "own_work": false, "note": "read raw: “Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) ‹ li(x), below e^{e^{e^{79}}}‹10^{10^{10^{34}}}.”; “Without assuming the Riemann hypothesis, Skewes (1955) later proved that there exists a value of x below e^{e^{e^{e^{7.705}}}}‹10^{10^{10^{964}}}.”; “|2010|| 2.2 || Saouter and Demichel || [exp(727.95132478), exp(727.95134682)] (without RH)”; “by Platt & Trudgian (2014) to 1.39\\times 10^{17}, and by Büthe (2015) to 10^{19}..”", "fetched": [ "2026-09-18" ] }, { "title": "1729 (number), Wikipedia", "url": "https://en.wikipedia.org/wiki/1729_(number)", "own_work": false, "note": "read raw: “{{cite book |title = Ramanujan | url = https://archive.org/details/pli.kerala.rare.37877 | last = Hardy | first = G. H. | author-link = G. H. Hardy |location = New York | publisher = Cambridge University Press | year = 1940”; the page quotes it (p. 12): “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 1729 and remarked that the number seemed to me rather a dull one.”", "fetched": [ "2026-09-18" ] }, { "title": "Prouhet–Thue–Morse constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant", "own_work": false, "note": "read raw: “The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929..”", "fetched": [ "2026-09-18" ] }, { "title": "Kruskal’s tree theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem", "own_work": false, "note": "read raw: “The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963)..”", "fetched": [ "2026-09-18" ] }, { "title": "Stanley Skewes and the Skewes Number, H. P. Williams (LSE)", "url": "https://personal.lse.ac.uk/williahp/pubs/Stanley%20Skewes%20and%20the%20Skewes%20Number.html", "own_work": false, "note": "“the famous mathematician G.H.Hardy described the Skewes number as “the largest number that has ever served any definite purpose in mathematics.””", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.35", "url": "https://www.teclamairme.com/#r-4.35", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Harvey Friedman introduced the TREE function in the 2000s; the tribonacci sequence appears in an 1892 puzzle about sheep; The word ‘tribonacci’ coined in 1963 by Mark Feinberg, aged fourteen, in the Fibonacci Quarterly; Knuth introduced up-arrow notation in ‘Mathematics and Computer Science: Coping with Finiteness’, Science, 1976; Euler proved 2^31 - 1 prime in 1772; cos(2π/17): the number Gauss found in 1796; Wantzel proved in 1837 that the cube root of 2 cannot be constructed with compass and straightedge, which settles the doubling of the cube; Liouville’s constant Σ 10^(−n!) was constructed in 1851 so that it could be proved transcendental, one of the first decimal examples of a transcendental number; Hermite showed in 1873 that e is transcendental, and Lindemann did the same for π in 1882. Friedman defined TREE[k] in a Foundations of Mathematics posting of 28 March 2006.", "sources": [ { "title": "Kruskal’s tree theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem", "own_work": false, "note": "read raw: “By incorporating labels, Friedman defined a far faster-growing function.{{Cite web |last=Friedman |first=Harvey |date=28 March 2006 |title=273:Sigma01/optimal/size |url=http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html.”", "fetched": [ "2026-09-18" ] }, { "title": "Generalizations of Fibonacci numbers, Wikipedia", "url": "https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers", "own_work": false, "note": "read raw: “In 1892, the sequence of numbers appeared in the solution of a recreational problem, concerning a farmer and the raising of sheep, that was posed by the American mathematician Artemas Martin (1835–1918).”; “The moniker tribonacci appeared much later, not until 1963, and is due to Mark Feinberg, at the time a fourteen-year-old high-school student, who introduced the term in an article in the Fibonacci Quarterly.”; “The most notable historical occurrence of the sequence is connected to Charles Darwin (1809–1882) and his seminal book On the Origin of Species, where the procreation and population growth of elephants is considered as an illustrative example.”", "fetched": [ "2026-09-18" ] }, { "title": "Knuth’s up-arrow notation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation", "own_work": false, "note": "read raw: “introduced by Donald Knuth in 1976.{{cite journal | last =Knuth | first = Donald E.| year=1976|title=Mathematics and Computer Science: Coping with Finiteness |journal=Science | volume=194|issue=4271| pages=1235–1242.”", "fetched": [ "2026-09-18" ] }, { "title": "2,147,483,647, Wikipedia", "url": "https://en.wikipedia.org/wiki/2,147,483,647", "own_work": false, "note": "read raw: “The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772..”", "fetched": [ "2026-09-18" ] }, { "title": "Heptadecagon, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heptadecagon", "own_work": false, "note": "read raw: “the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796..”", "fetched": [ "2026-09-18" ] }, { "title": "Doubling the cube, Wikipedia", "url": "https://en.wikipedia.org/wiki/Doubling_the_cube", "own_work": false, "note": "read raw: “But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass..”", "fetched": [ "2026-09-18" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "read raw: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..”", "fetched": [ "2026-09-18" ] }, { "title": "[FOM] 273:Sigma01/optimal/size, Harvey Friedman, 28 March 2006 (archived)", "url": "https://web.archive.org/web/20240918084441/https://cs.nyu.edu/pipermail/fom/2006-March/010279.html", "own_work": false, "note": "posted “Tue Mar 28 12:57:25 EST 2006”: “Let TREE[k] be the length n of the longest sequence of trees T_1,...,T_n with vertices labeled from {1,...,k}, where each T_i has at most i vertices, and no T_i is inf preserving and label preserving embeddable into a later T_j.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.36", "url": "https://www.teclamairme.com/#r-4.36", "chapter": 4, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Γ(1/4) was proved transcendental by Gregory Chudnovsky around 1976; Graham’s number was published by Martin Gardner in 1977.", "sources": [ { "title": "Real geometric transcendence for the Gamma function (Arshay Sheth), arXiv", "url": "https://arxiv.org/pdf/2605.15117v1", "own_work": false, "note": "read raw: “Gregory Volfovich Chudnovsky, Algebraic independence of constants connected with the exponential and elliptic functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698.”", "fetched": [ "2026-09-18" ] }, { "title": "Graham’s number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Graham%27s_number", "own_work": false, "note": "read raw: “This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-4.37", "url": "https://www.teclamairme.com/#r-4.37", "chapter": 4, "status": "confirmed", "claim": "Kruskal’s tree theorem of 1960 says that any infinite sequence of finite trees contains one tree that embeds into a later one, and Harvey Friedman’s finite miniaturized version of the theorem defines the function whose value TREE(3) the book uses.", "sources": [ { "title": "Kruskal’s tree theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem", "own_work": false, "note": "states the theorem and its role in defining Friedman’s TREE function.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.38", "url": "https://www.teclamairme.com/#r-4.38", "chapter": 4, "status": "confirmed", "claim": "Hilbert posed the transcendence of a^b, for algebraic a not equal to 0 or 1 and algebraic irrational b, as the seventh of his 1900 list of problems, naming 2^√2 and e^π as his own examples, and Gelfond and Schneider answered it affirmatively and independently in 1934. The list had twenty-three problems.", "sources": [ { "title": "Hilbert’s seventh problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem", "own_work": false, "note": "states the problem, Hilbert’s own examples, and the 1934 resolution.", "fetched": [ "2026-09-22" ] }, { "title": "Hilbert’s problems, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_problems", "own_work": false, "note": "carries the count of the list: “Hilbert’s problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900.”", "fetched": [ "2026-09-22" ] }, { "title": "Gelfond–Schneider theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem", "own_work": false, "note": "carries e^π as (−1)^(−i) and the cases the theorem covers: “(−1)^(−i) = 23.14069263…,” with 2^√2, √2^√2 and i^i = e^(−π/2) listed as consequences.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.41", "url": "https://www.teclamairme.com/#r-4.41", "chapter": 4, "status": "confirmed", "claim": "The Bernoulli numbers are the rational coefficients Bn in the expansion x/(e^x − 1) = sum Bn x^n/n!, published by Jacob Bernoulli in the Ars Conjectandi of 1713, and they give closed forms for sums of integer powers and for the Riemann zeta function at even integers. Seki Takakazu reached the same numbers independently in Japan in 1712.", "sources": [ { "title": "Bernoulli number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bernoulli_number", "own_work": false, "note": "gives the defining generating function and the applications to power sums and zeta.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.44", "url": "https://www.teclamairme.com/#r-4.44", "chapter": 4, "status": "confirmed", "claim": "The harmonic numbers Hn are the partial sums 1 + 1/2 + ... + 1/n of the harmonic series; Leopold Theisinger proved in 1915 that Hn is an integer only for n = 1, and every harmonic number is rational. The sequence begins 1, 3/2, 11/6, 25/12. Hₙ grows roughly like ln n, so the harmonic series diverges, slowly.", "sources": [ { "title": "Harmonic number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Harmonic_number", "own_work": false, "note": "defines the harmonic numbers and states that H_n is an integer only for n = 1, “a result often attributed to Theisinger”; it gives no year (1915 is on the arXiv line). On growth: “The harmonic numbers roughly approximate the natural logarithm function” … “and thus the associated harmonic series grows without limit, albeit slowly.”", "fetched": [ "2026-09-22" ] }, { "title": "About the Non-Integer Property of Hyperharmonic Numbers, arXiv:0811.0043", "url": "https://arxiv.org/abs/0811.0043", "own_work": false, "note": "names Theisinger and the year: “It was proven in 1915 by Leopold Theisinger that the H_n harmonic numbers are never integers.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.45", "url": "https://www.teclamairme.com/#r-4.45", "chapter": 4, "status": "confirmed", "claim": "Lehmer’s problem, posed by Derrick Lehmer in 1933, asks whether there is an algebraic integer with Mahler measure strictly between 1 and the value of Lehmer’s number, about 1.1762808, the smallest Mahler measure above 1 ever found, and no smaller example is known nearly a century later. Lehmer’s number is the largest root of his degree-10 polynomial.", "sources": [ { "title": "Mahler measure, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mahler_measure", "own_work": false, "note": "states Lehmer’s conjecture, that there is a constant μ > 1 such that an irreducible integer polynomial p has either M(p) = 1 or M(p) > μ; Lehmer’s number and its smallest-known status are on the Lehmer’s conjecture and arXiv lines.", "fetched": [ "2026-09-22" ] }, { "title": "Lehmer’s conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lehmer%27s_conjecture", "own_work": false, "note": "gives Lehmer’s 1933 paper, his degree-10 polynomial and its measure, and the open status: “P(x) = x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1”; “M(P(x)) = 1.176280818…”; “Lehmer’s conjecture is not yet completely proved and is still a question of much interest.”", "fetched": [ "2026-09-22" ] }, { "title": "E. Hironaka, Lehmer’s Problem, McKay’s Correspondence, and 2,3,7, arXiv math/0204040", "url": "https://arxiv.org/html/math/0204040v1", "own_work": false, "note": "carries the search record: “Boyd and Mossinghoff have done searches up to degree 40, but so far no one has found a monic noncyclotomic integer polynomial with smaller Mahler measure.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.49", "url": "https://www.teclamairme.com/#r-4.49", "chapter": 4, "status": "confirmed", "claim": "The look-and-say sequence starts at 1 and each later term describes the digit runs of the one before, giving 1, 11, 21, 1211, 111221, ...; John Conway analyzed it in a 1986 paper and showed the length of the n-th term grows like Conway’s constant to the n. Conway’s constant is about 1.3036.", "sources": [ { "title": "Look-and-say sequence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Look-and-say_sequence", "own_work": false, "note": "defines the sequence and states Conway’s 1986 growth-rate result.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-4.54", "url": "https://www.teclamairme.com/#r-4.54", "chapter": 4, "status": "confirmed", "claim": "States that the transcendence of Γ(1/5) is open, unlike Γ(1/3) and Γ(1/4). Γ(1/2) = √π and Γ(1/6) are also known to be transcendental.", "sources": [ { "title": "Open Diophantine Problems, M. Waldschmidt, arXiv math/0312440", "url": "https://arxiv.org/pdf/math/0312440", "own_work": false, "note": "states Γ(1/5) is open and Γ(1/2) transcendental: “there is no proof so far that Γ(1/5) is transcendental”; “Lindemann’s result on the transcendence of π which implies that Γ(1/2) = √π is also transcendental”", "fetched": [ "2026-09-22" ] }, { "title": "Gamma function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gamma_function", "own_work": false, "note": "gives the known transcendental values: “It has been proved that Γ(n + r) is a transcendental number and algebraically independent of π for any integer n and each of the fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, 5/6.”", "fetched": [ "2026-09-22" ] }, { "title": "Lemniscate constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lemniscate_constant", "own_work": false, "note": "ties Γ(1/4) to the lemniscate constant: “ϖ = Γ(1/4)² / (2√(2π))”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.1", "url": "https://www.teclamairme.com/#r-5.1", "chapter": 5, "status": "confirmed", "claim": "Legendre’s correction was about 1.08366 in 1808 and de la Vallée Poussin proved the best constant was exactly 1 in 1899.", "sources": [ { "title": "Legendre’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Legendre%27s_constant", "own_work": false, "note": "“Legendre proposed in 1808 the formula y = x/log(x) − 1.08366”; Chebyshev showed in 1849 that the limit, if it exists, must be 1, and de la Vallée Poussin proved in 1899 that it is.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-5.2", "url": "https://www.teclamairme.com/#r-5.2", "chapter": 5, "status": "confirmed", "claim": "The Erdős–Borwein constant is irrational and its transcendence is open.", "sources": [ { "title": "Erdős–Borwein constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant", "own_work": false, "note": "the value 1.606695152415291763…, Paul Erdős’s irrationality proof of 1948 and Peter Borwein’s second proof of 1992. Nothing is asserted about transcendence either way, which is the sense in which the card calls it open.", "fetched": [ "2026-09-09" ] }, { "title": "List of mathematical constants, Wikipedia", "url": "https://en.wikipedia.org/wiki/List_of_mathematical_constants", "own_work": false, "note": "the Erdős–Borwein constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic), 𝒫 and 𝓔𝓟: not rational, algebraic status unknown.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.3", "url": "https://www.teclamairme.com/#r-5.3", "chapter": 5, "status": "confirmed", "claim": "The reciprocal Fibonacci constant is about 3.3599 and irrational, with transcendence open.", "sources": [ { "title": "Reciprocal Fibonacci constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant", "own_work": false, "note": "the value 3.359885666243177553…, and Richard André-Jeannin’s 1989 proof of irrationality, settling a conjecture of Erdős, Graham and Carlitz. The transcendence results there are for the Fibonacci zeta function at even arguments, not for ψ itself.", "fetched": [ "2026-09-09" ] }, { "title": "List of mathematical constants, Wikipedia", "url": "https://en.wikipedia.org/wiki/List_of_mathematical_constants", "own_work": false, "note": "the Reciprocal Fibonacci constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic) and 𝒫: irrational, algebraic status unknown.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.4", "url": "https://www.teclamairme.com/#r-5.4", "chapter": 5, "status": "confirmed", "claim": "Copeland–Erdős is the concatenation of the primes, normal in base 10 and therefore irrational. Its transcendence has not been proved.", "sources": [ { "title": "Copeland–Erdős constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant", "own_work": false, "note": "“In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős” (1946, per the page’s reference), and “It follows that the constant is irrational; this can also be proven with Dirichlet’s theorem on arithmetic progressions or Bertrand’s postulate.”", "fetched": [ "2026-09-09" ] }, { "title": "Copeland–Erdős Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/Copeland-ErdosConstant.html", "own_work": false, "note": "the decimal 0.23571113171923… and the 1946 normality result; no transcendence claim is made.", "fetched": [ "2026-09-09" ] }, { "title": "Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015", "url": "https://arxiv.org/abs/1003.4015", "own_work": false, "note": "“Arthur Herbert Copeland and Paul Erdős have shown that CCE is normal, but apparently it is not proved that CCE is transcendental.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.5", "url": "https://www.teclamairme.com/#r-5.5", "chapter": 5, "status": "confirmed", "claim": "Apéry proved ζ(3) irrational in 1978 and its transcendence is open. No closed form for ζ(3) is known, and Apéry’s proof, greeted with disbelief, used only methods available for 200 years.", "sources": [ { "title": "Apéry’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant", "own_work": false, "note": "the value 1.202056903159594285…, “Roger Apéry, who proved in 1978 that it is an irrational number,” and “Apéry’s constant has not yet been proved transcendental.”", "fetched": [ "2026-09-09" ] }, { "title": "Roger Apéry, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Apery/", "own_work": false, "note": "“In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was greeted with doubt and disbelief.” and “The problem had long been an open question, yet Apéry’s proof only used methods which had been available for 200 years.”", "fetched": [ "2026-09-22" ] }, { "title": "Computing ζ(3), John D. Cook", "url": "https://www.johndcook.com/blog/2021/08/28/computing-zeta-3/", "own_work": false, "note": "“There are closed-form expressions for ζ(s) when s is an even integer but so far not nobody has found one for s = 3.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.6", "url": "https://www.teclamairme.com/#r-5.6", "chapter": 5, "status": "confirmed", "claim": "Euler’s constant is about 0.5772 and its irrationality is unknown; at least one of it and the Gompertz constant is irrational, which Lagarias calls an observation of A. I. Aptekarev in 2009 and Wikipedia credits to Shidlovski (1962 on its Gompertz constant page, 1959 as Shidlovsky on its Euler–Mascheroni page), and at least one of them is transcendental, which Tanguy Rivoal proved in 2012; and the Meissel–Mertens constant is its analogue over the primes. The Meissel–Mertens constant is not known to be irrational. Le Lionnais named the Gompertz constant for its role in survival analysis; Euler met it in studying divergent series, and it is the Borel-summed value of 1 − 1 + 2 − 6 + 24 − 120 + ⋯.", "sources": [ { "title": "Euler–Mascheroni constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant", "own_work": false, "note": "limit definition, value, and open irrationality; “In 1959 Andrei Shidlovsky proved that at least one of Euler’s constant γ and the Gompertz constant δ is irrational.”", "fetched": [ "2026-09-09" ] }, { "title": "Gompertz constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gompertz_constant", "own_work": false, "note": "the value 0.596347362323194074…; “In 1962, Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational,” improved “in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental”; “When Euler studied divergent infinite series, he encountered δ via, for example, the above integral representation. Le Lionnais called δ the Gompertz constant because of its role in survival analysis”; and δ is “the regularized value of the summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by Borel summation.”", "fetched": [ "2026-09-09" ] }, { "title": "Meissel–Mertens constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Meissel%E2%80%93Mertens_constant", "own_work": false, "note": "M = 0.261497212847642783…, the limiting difference between the harmonic series over the primes and log log; γ “has an analogous definition involving a sum over all integers (not just the primes).” No arithmetic status is stated.", "fetched": [ "2026-09-09" ] }, { "title": "Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 (Bull. AMS 50 (2013) 527–628)", "url": "https://arxiv.org/abs/1303.1856", "own_work": false, "note": "read raw via the ar5iv rendering of arXiv:1303.1856. Section 3.16: “In 2012 Rivoal proved a result which, as a special case, implies the transcendence of at least one of the Euler constant γ and the Euler-Gompertz constant δ. It uses the Shidlovskii method and improves on an observation of Aptekarev that at least one of γ and δ must be irrational.” Aptekarev’s paper is On linear forms containing the Euler constant, arXiv:0902.1768, 28 February 2009.", "fetched": [ "2026-09-17" ] }, { "title": "List of mathematical constants, Wikipedia", "url": "https://en.wikipedia.org/wiki/List_of_mathematical_constants", "own_work": false, "note": "the Meissel–Mertens constant row reads “?” under ℚ (rational), 𝔸 (algebraic) and 𝒫: status unknown; the row’s year column reads “1866 & 1873.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Corrected 17 September 2026. The entry credited the irrationality result to Shidlovski in 1962, following Wikipedia’s Gompertz constant article, while the #gamma and #gompertz cards credited Aptekarev in 2009. The cards were right. Lagarias’s survey, which is the source that Wikipedia article itself footnotes for the sentence, says Rivoal’s 2012 theorem “improves on an observation of Aptekarev that at least one of γ and δ must be irrational”. Wikipedia’s Euler’s constant article makes the same attribution to Shidlovski and dates it 1959 rather than 1962, so the two tertiary pages disagree with each other as well as with the survey they cite. This is not a tie to record: a refereed survey in the Bulletin of the AMS is the better source and is the one the others are quoting. Shidlovskii is not absent from the story — Rivoal’s proof uses his method, and Lagarias dates his breakthrough on E-functions to 1959 — but the at-least-one-is-irrational observation is Aptekarev’s." }, { "id": "r-5.7", "url": "https://www.teclamairme.com/#r-5.7", "chapter": 5, "status": "confirmed", "claim": "Feigenbaum’s constants describe period-doubling scaling; signed α is negative while the displayed magnitude is positive. The signed value is α ≈ −2.502907875.", "sources": [ { "title": "Feigenbaum constants, Wikipedia", "url": "https://en.wikipedia.org/wiki/Feigenbaum_constants", "own_work": false, "note": "δ = 4.669201609102990671…, α = 2.502907875095892822…, and “Both numbers are believed to be transcendental, although they have not been proven to be so. In fact, there is no known proof that either constant is even irrational.”", "fetched": [ "2026-09-09" ] }, { "title": "Feigenbaum Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/FeigenbaumConstant.html", "own_work": false, "note": "“For the logistic map,” followed by displayed equation (8): “alpha = -2.502907875....”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.8", "url": "https://www.teclamairme.com/#r-5.8", "chapter": 5, "status": "judgement", "claim": "Catalan’s constant has the stated series and value and unknown irrationality. Gauss’s constant, also written G, was proven transcendental by Theodor Schneider in 1941.", "sources": [ { "title": "Catalan’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Catalan%27s_constant", "own_work": false, "note": "(article last edited 7 September 2026): “It is not known whether G is irrational, let alone transcendental,” and G described as “arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven.”", "fetched": [ "2026-09-09" ] }, { "title": "Zhi-Wei Sun, Catalan’s constant is irrational (preprint)", "url": "https://arxiv.org/abs/2609.04176", "own_work": false, "note": "posted 3 September 2026 to the arXiv’s unvetted math.GM section, “In this paper we prove the irrationality of G via using suitable weights”; not refereed. Re-read raw on 17 September 2026: the record still shows “[Submitted on 3 Sep 2026]” and a single version, v1, in math.GM. There is no v2, no withdrawal and no journal reference, so nothing supersedes the reference literature and the book’s “irrationality unknown” stands.", "fetched": [ "2026-09-09" ] }, { "title": "Lemniscate constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lemniscate_constant", "own_work": false, "note": "“Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "A claimed proof that Catalan’s constant is irrational was posted as a preprint on 3 September 2026, in the arXiv section reserved for unvetted work. It has not been refereed, objections were raised within days, and the reference literature is unchanged, so Catalan’s constant stays among the numbers not known to be irrational. If the proof stands, G rises one brace." }, { "id": "r-5.9", "url": "https://www.teclamairme.com/#r-5.9", "chapter": 5, "status": "judgement", "claim": "None of the constants in the second brace — Khinchin’s constant, the Glaisher–Kinkelin, Golomb–Dickman, Landau–Ramanujan, Lévy, Fransén–Robinson, Bernstein, Viswanath, Niven, Artin and Sierpiński constants, the twin prime constant, the MRB constant, Somos’ nested root, the Kepler–Bouwkamp radius and Backhouse’s constant — has been proved irrational. Nor have Catalan’s, Brun’s, the Gompertz or the Meissel–Mertens constant, which the same field note names with them. Whether Khinchin’s constant itself has Khinchin’s geometric mean in its own continued fraction has not been established either.", "sources": [ { "title": "Khinchin’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Khinchin%27s_constant", "own_work": false, "note": "K₀ = 2.685452001065306445…, the geometric mean of the partial quotients of almost every real number, and “The algebraic properties of Khinchin’s constant itself, e. g. whether it is a rational, algebraic irrational, or transcendental number, are also not known.”; “Many well-known numbers, such as π, the Euler–Mascheroni constant γ, and Khinchin’s constant itself, based on numerical evidence, are thought to be among the numbers for which the limit … converges to Khinchin’s constant. However, none of these limits has been rigorously established.”", "fetched": [ "2026-09-09" ] }, { "title": "Lévy’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/L%C3%A9vy%27s_constant", "own_work": false, "note": "e^(π²/(12 ln 2)) = 3.275822918721811159…, the growth rate of the denominators of almost every continued fraction; no arithmetic status is stated.", "fetched": [ "2026-09-09" ] }, { "title": "Somos’ quadratic recurrence constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Somos%27_quadratic_recurrence_constant", "own_work": false, "note": "σ = 1.661687949633594121… as the nested root √(1√(2√(3…))); no arithmetic status is stated.", "fetched": [ "2026-09-09" ] }, { "title": "Twin prime, Wikipedia", "url": "https://en.wikipedia.org/wiki/Twin_prime", "own_work": false, "note": "“Define the twin prime constant C₂ as C₂ = ∏(1−1/(p−1)²) ≈ 0.660161815846869573…,” the factor in the first Hardy–Littlewood conjecture.", "fetched": [ "2026-09-09" ] }, { "title": "List of mathematical constants, Wikipedia", "url": "https://en.wikipedia.org/wiki/List_of_mathematical_constants", "own_work": false, "note": "the table marks the arithmetic status of each constant in this group with a question mark, and its rows for Catalan’s constant, Brun’s constant, the Gompertz constant and the Meissel–Mertens constant also read “?” in the rational column. Its year column gives 1922 for the twin prime constant, where MathWorld cites the 1923 paper.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Khinchin’s constant is the one of this group whose arithmetic status a reference work states in as many words. For the others no source asserts a status either way: “status open” on their cards records the absence of a proof, not a theorem, which is the honest reading of a chapter about numbers nobody has classified." }, { "id": "r-5.10", "url": "https://www.teclamairme.com/#r-5.10", "chapter": 5, "status": "confirmed", "claim": "Brun’s constant is the convergent sum over twin primes. Its decimal value is an extrapolated estimate; the proved bounds are 1.840503 < B₂ < 2.288513.", "sources": [ { "title": "Brun’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Brun%27s_constant", "own_work": false, "note": "twin-prime reciprocal sum, convergence, and rationality conditional on finiteness; “While the exact value of B 2 is unknown, it is thought to be around 1.9,” and of the 2002 value 1.902160583104: “The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 10 16.”", "fetched": [ "2026-09-09" ] }, { "title": "D. J. Platt and T. S. Trudgian, Improved bounds on Brun’s constant, arXiv:1803.01925", "url": "https://arxiv.org/abs/1803.01925", "own_work": false, "note": "“We improve the unconditional bounds on Brun’s constant to 1.840503 < B < 2.288513.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "" }, { "id": "r-5.11", "url": "https://www.teclamairme.com/#r-5.11", "chapter": 5, "status": "confirmed", "claim": "Mills’ constant is irrational, proved unconditionally by Kota Saito in 2024; its transcendence is open, and only its digits depend on the Riemann Hypothesis.", "sources": [ { "title": "Saito, Mills’ constant is irrational", "url": "https://arxiv.org/abs/2404.19461", "own_work": false, "note": "“We define Mills’ constant as the smallest real number ξ satisfying this property… we show that Mills’ constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.” Submitted 30 April 2024, revised 8 May 2025; no hypothesis is assumed. Journal reference on the page: “Mathematika 71 (2025), no. 3, Paper No. e70027.” Read in the paper’s PDF: “Theorem 1.2. Either ξ3 is transcendental, or there exists a positive integer m such that ξ3^(3^m) is a Pisot number of degree 3”; “assuming Cramér’s conjecture, ξ3 is transcendental.”", "fetched": [ "2026-09-09" ] }, { "title": "Mills’ constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mills%27_constant", "own_work": false, "note": "“it is known to be irrational,” citing Saito, and “Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926…”; “Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.”", "fetched": [ "2026-09-09" ] }, { "title": "Saito, Transcendency of variants of Mills’ constant", "url": "https://arxiv.org/abs/2508.16068", "own_work": false, "note": "a 2025 sequel proving Mills’ constant transcendental under the Density Hypothesis for the Riemann zeta function, and transcendental outright for several variants; the unconditional question is left open.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The irrationality proof assumes nothing; the Riemann Hypothesis enters only in computing the constant’s digits, where it supplies a prime between consecutive cubes. Transcendence is genuinely open: Saito proves that either the constant is transcendental or some power of it is a cubic Pisot number, and that transcendence would follow from Cramér’s conjecture or from the Density Hypothesis." }, { "id": "r-5.12", "url": "https://www.teclamairme.com/#r-5.12", "chapter": 5, "status": "confirmed", "claim": "At least one of π+e and πe is transcendental, but which one is unknown. π + e ≈ 5.8599, π·e ≈ 8.5397 and π^e ≈ 22.459. π^π ≈ 36.4622 and e^e ≈ 15.1543.", "sources": [ { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "“Since the field of algebraic numbers is algebraically closed and e and π are roots of the polynomial x²−(e+π)x+eπ, at least one of the numbers eπ and e+π is transcendental”; π+e, π·e, π^π, e^e and π^e are listed as conjectured transcendental, none of them proved even irrational.", "fetched": [ "2026-09-09" ] }, { "title": "A059742, OEIS", "url": "https://oeis.org/A059742", "own_work": false, "note": "“Decimal expansion of e + Pi.”: 5.8598744820488384738… (value checked by computation; OEIS blocks automated fetches)", "fetched": [ "2026-09-22" ] }, { "title": "A019609, OEIS", "url": "https://oeis.org/A019609", "own_work": false, "note": "“Decimal expansion of Pi*e.”: 8.5397342226735670654… (value checked by computation; OEIS blocks automated fetches)", "fetched": [ "2026-09-22" ] }, { "title": "Gelfond’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gelfond%27s_constant", "own_work": false, "note": "“The decimal expansion of π^e is given by π^e = 22.45915771836104547342...”", "fetched": [ "2026-09-22" ] }, { "title": "A073233, OEIS", "url": "https://oeis.org/A073233", "own_work": false, "note": "“Decimal expansion of Pi^Pi.”: 36.462159607207911770… (value checked by computation; OEIS blocks automated fetches)", "fetched": [ "2026-09-22" ] }, { "title": "A073226, OEIS", "url": "https://oeis.org/A073226", "own_work": false, "note": "“Decimal expansion of e^e.”: 15.154262241479264189… (value checked by computation; OEIS blocks automated fetches)", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.13", "url": "https://www.teclamairme.com/#r-5.13", "chapter": 5, "status": "confirmed", "claim": "At least one of ζ(5), ζ(7), ζ(9), and ζ(11) is irrational without identifying which. ζ(5) ≈ 1.0369278 and is not itself known to be irrational; extending Apéry’s proof has produced no result on any specific odd value; each even value ζ(2n) is a rational multiple of π^(2n).", "sources": [ { "title": "Zudilin odd zeta result", "url": "https://doi.org/10.1070/RM2001v056n04ABEH000427", "own_work": false, "note": "theorem statement says at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational.", "fetched": [ "2026-09-09" ] }, { "title": "Particular values of the Riemann zeta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function", "own_work": false, "note": "table row ζ(5): “1.0369277551433699263...”; “The relationship between zeta at the positive even integers and powers of pi may be written as a_n ζ(2n) = π^(2n) b_n where a_n and b_n are coprime positive integers for all n.”", "fetched": [ "2026-09-22" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "“in particular Apéry’s constant ζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.”", "fetched": [ "2026-09-22" ] }, { "title": "Apéry’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant", "own_work": false, "note": "“Many people have tried to extend Apéry’s proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.15", "url": "https://www.teclamairme.com/#r-5.15", "chapter": 5, "status": "confirmed", "claim": "Rayo’s number is a fixed finite integer defined by a googol-symbol set-theory description, beyond feasible calculation. Rayo’s opponent in the duel was Adam Elga, and Rayo’s entry won; like every fixed finite integer the number is computable in the technical sense. Its size vastly exceeds TREE(3).", "sources": [ { "title": "Rayo’s number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rayo%27s_number", "own_work": false, "note": "records the 2007 MIT duel, googol-symbol definition, and fixed-finite-integer distinction; “At the time, it was claimed to be the largest named number, as its size vastly exceeds even other notable large numbers such as Graham’s number and TREE(3).”", "fetched": [ "2026-09-09" ] }, { "title": "Big Number Duel, Agustín Rayo (MIT)", "url": "https://web.mit.edu/arayo/www/bignums.html", "own_work": false, "note": "read raw: “On January 26th 2007, Adam Elga and I got together to see who could come up with a bigger finite number.” Under the heading “The winning entry” it sets out the definition by at most a googol symbols of first-order set theory that is now called Rayo’s number.", "fetched": [ "2026-09-22" ] }, { "title": "Computable number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Computable_number", "own_work": false, "note": "read as raw wikitext: “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.” Every integer is algebraic, so every fixed integer is computable.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.16", "url": "https://www.teclamairme.com/#r-5.16", "chapter": 5, "status": "confirmed", "claim": "TREE(3) is the TREE-function value associated with Kruskal’s tree theorem.", "sources": [ { "title": "Kruskal’s tree theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem", "own_work": false, "note": "confirms the theorem context behind TREE(3).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-5.17", "url": "https://www.teclamairme.com/#r-5.17", "chapter": 5, "status": "confirmed", "claim": "For 0 1.”", "fetched": [ "2026-09-22" ] }, { "title": "Extended real number line, Wikipedia", "url": "https://en.wikipedia.org/wiki/Extended_real_number_line", "own_work": false, "note": "read as raw wikitext: “the extended real number system is obtained from the real number system ℝ by adding two elements denoted +∞ and −∞ that are respectively greater and lower than every real number.” and “Adjoining the elements +∞ and −∞ to ℝ enables a definition of “limits at infinity””; and, with its math as rendered: “The expressions ∞ − ∞, 0 × (±∞), and ±∞/±∞ (called indeterminate forms) are usually left undefined.” and “However, in the context of probability or measure theory, 0 × ±∞ is often defined as 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.18", "url": "https://www.teclamairme.com/#r-5.18", "chapter": 5, "status": "confirmed", "claim": "Chaitin’s Ω, introduced by Gregory Chaitin in 1975, is the halting probability of a fixed optimal universal prefix-free computer and is uncomputable, algorithmically random and transcendental; normality follows from the randomness rather than being separately stated. Ω is a real number between 0 and 1, and its first N bits would settle the halting problem for all programs up to N bits long. Even so, Calude and colleagues computed the first 64 bits of one such Ω, for a particular universal machine, in 2002.", "sources": [ { "title": "Chaitin’s Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/ChaitinsConstant.html", "own_work": false, "note": "defines the halting probability and states Chaitin constants are uncomputable and transcendental; and “Calude et al. (2002) computed the first 64 bits of Chaitin’s constant for a certain universal Turing machine.”", "fetched": [ "2026-09-09" ] }, { "title": "Gregory Chaitin, The Limits of Reason, free preprint", "url": "https://arxiv.org/pdf/math/0611740", "own_work": false, "note": "The technical discussion on PDF page 11 defines the fixed universal self-delimiting Turing machine: no extension of a valid program is valid. Prefix-free programs make the probability sum converge; the precise value depends on the chosen machine.", "fetched": [ "2026-09-30" ] }, { "title": "Chaitin’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chaitin%27s_constant", "own_work": false, "note": "read as raw wikitext: “this sum converges to a real number between 0 and 1”; “Knowing the first N bits of Ω, one could calculate the halting problem for all programs of a size up to N.”; and, after “It is algorithmically random,” “As a consequence, it is a normal number, which means that its digits are equidistributed as if they were generated by tossing a fair coin.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.19", "url": "https://www.teclamairme.com/#r-5.19", "chapter": 5, "status": "confirmed", "claim": "BB(n) is the maximum halting time over n-state machines; BB(6) is fixed but currently unknown. As one fixed natural number, BB(6) is computable in the standard sense, though its value is unknown; only the function n ↦ BB(n) is uncomputable.", "sources": [ { "title": "Antihydra, bbchallenge", "url": "https://bbchallenge.org/antihydra", "own_work": false, "note": "Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.", "fetched": [ "2026-09-09" ] }, { "title": "BB(6), BusyBeaverWiki", "url": "https://wiki.bbchallenge.org/wiki/BB(6)", "own_work": false, "note": "BB(6) is the unsolved sixth value of the Busy Beaver function; the best lower bound, as of 24 September 2026, is: the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4.", "fetched": [ "2026-09-09" ] }, { "title": "Computable number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Computable_number", "own_work": false, "note": "read as raw wikitext: “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.” Every integer is algebraic, so every fixed integer is computable.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-5.20", "url": "https://www.teclamairme.com/#r-5.20", "chapter": 5, "status": "judgement", "claim": "Antihydra is unresolved rather than proved unknowable.", "sources": [ { "title": "Antihydra, bbchallenge", "url": "https://bbchallenge.org/antihydra", "own_work": false, "note": "Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.", "fetched": [ "2026-09-09" ] }, { "title": "Antihydra, BusyBeaverWiki", "url": "https://wiki.bbchallenge.org/wiki/Antihydra", "own_work": false, "note": "Antihydra is classified as a BB(6) Cryptid and carries the unsolved problem “Does Antihydra run forever?”; the page was last edited 9 May 2026 and records no resolution.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Antihydra halts or does not; nobody has proved which. The machine runs a Collatz-like process, every attempt at a proof either way has failed so far, and if it halts it may be the machine that fixes BB(6). That is an open problem, not a proved impossibility, and the chapter says nobody yet knows rather than nobody ever will." }, { "id": "r-5.21", "url": "https://www.teclamairme.com/#r-5.21", "chapter": 5, "status": "confirmed", "claim": "BB(5)=47,176,870 has been proved while BB(6) remains unresolved.", "sources": [ { "title": "Antihydra, bbchallenge", "url": "https://bbchallenge.org/antihydra", "own_work": false, "note": "Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.", "fetched": [ "2026-09-09" ] }, { "title": "BB(6), BusyBeaverWiki", "url": "https://wiki.bbchallenge.org/wiki/BB(6)", "own_work": false, "note": "BB(6) is “the unsolved 6th value of the Busy Beaver function” (BB(5) = 47,176,870 is carried by the Antihydra page of this entry). Re-read on 24 September 2026 (these counts change often): mxdys’s holdout list “has 855 machines up to equivalence and 1802 machines not considering equivalence as of mid-September 2026,” with Antihydra one of the cryptids among them; the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4. BB(6) is not waiting on Antihydra alone.", "fetched": [ "2026-09-09", "2026-09-11" ] } ], "editorial_note": "" }, { "id": "r-5.22", "url": "https://www.teclamairme.com/#r-5.22", "chapter": 5, "status": "confirmed", "claim": "Ω lies between 0 and 1 and BB(6) is positive, so the denominator is positive and finite.", "sources": [ { "title": "Chaitin’s Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/ChaitinsConstant.html", "own_work": false, "note": "defines the halting probability and states Chaitin constants are uncomputable and transcendental.", "fetched": [ "2026-09-09" ] }, { "title": "Antihydra, bbchallenge", "url": "https://bbchallenge.org/antihydra", "own_work": false, "note": "Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "The displayed factor is 0 divided by positive sums; its value is 0. The limit uses 0 n, it is common to define” the coefficient to be 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.14", "url": "https://www.teclamairme.com/#r-6.14", "chapter": 6, "status": "confirmed", "claim": "R(3, 3)=6: among six people there are three mutual acquaintances or three mutual strangers, while five do not suffice. R(4, 4)=18, and R(5, 5) is known only to lie between 43 and 46. The witness for five is K5 drawn as a pentagon in one color and the pentagram inside it in the other; Ramsey proved the general theorem in a paper of 1930. The only two-coloring of seventeen points with no four-clique of one color is the Paley graph of order 17; Erdős’s aliens story, told by Joel Spencer, sets R(5, 5) against R(6, 6).", "sources": [ { "title": "Wolfram MathWorld, Ramsey Number", "url": "https://mathworld.wolfram.com/RamseyNumber.html", "own_work": false, "note": "gives the party-problem definition, R(3, 3)=6 and R(4, 4)=18, credited in its table to Greenwood and Gleason (1955), and the bounds for R(5, 5), 43 from Exoo (1989) and 46 from Angeltveit and McKay (2024).", "fetched": [ "2026-09-09" ] }, { "title": "Theorem on friends and strangers, Wikipedia", "url": "https://en.wikipedia.org/wiki/Theorem_on_friends_and_strangers", "own_work": false, "note": "“We draw K5 as a pentagon surrounding a star (a pentagram). We color the edges of the pentagon red and the edges of the star blue.”; “In 1930, in a paper entitled ‘On a Problem of Formal Logic,’ Frank Ramsey proved a very general theorem (now known as Ramsey’s theorem) of which this theorem is a simple case.”", "fetched": [ "2026-09-22" ] }, { "title": "Ramsey’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ramsey%27s_theorem", "own_work": false, "note": "on R(4, 4): “only one (4, 4, 17) graph (the Paley graph of order 17) among 2.46 × 10^26 colourings. It follows that R(4, 4) = 18.”; and, quoting Joel Spencer, “Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.15", "url": "https://www.teclamairme.com/#r-6.15", "chapter": 6, "status": "confirmed", "claim": "The third Catalan number is 5 and counts, among other things, bracketings of four factors and triangulations of a pentagon. Stanley’s monograph Catalan Numbers (2015) lists 214 combinatorial interpretations of C_n.", "sources": [ { "title": "Wolfram MathWorld, Catalan Number", "url": "https://mathworld.wolfram.com/CatalanNumber.html", "own_work": false, "note": "lists the sequence and its bracketing and polygon-triangulation interpretations.", "fetched": [ "2026-09-09" ] }, { "title": "Richard P. Stanley, Catalan Numbers (talk slides, 13 March 2024), author’s MIT page", "url": "https://math.mit.edu/~rstan/transparencies/wellesley.pdf", "own_work": false, "note": "“R. Stanley, Catalan Numbers, Cambridge University Press, 2015. Includes 214 combinatorial interpretations of Cn and 68 additional problems.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.16", "url": "https://www.teclamairme.com/#r-6.16", "chapter": 6, "status": "confirmed", "claim": "The third Bell number is 5 and counts partitions of a three-element set into nonempty blocks. S(4, 1), S(4, 2), S(4, 3), S(4, 4) = 1, 7, 6, 1, and their sum is the fourth Bell number, B₄ = 15.", "sources": [ { "title": "Wolfram MathWorld, Bell Number", "url": "https://mathworld.wolfram.com/BellNumber.html", "own_work": false, "note": "defines Bell numbers as set partitions and gives B3=5; from n = 1 “the first few Bell numbers” “are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (OEIS A000110),” so B₄ = 15.", "fetched": [ "2026-09-09" ] }, { "title": "Stirling numbers of the second kind, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind", "own_work": false, "note": "“The 15 partitions of a 4-element set ordered in a Hasse diagram”; “There are S(4, 1), ..., S(4, 4) = 1, 7, 6, 1 partitions containing 1, 2, 3, 4 sets.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-6.17", "url": "https://www.teclamairme.com/#r-6.17", "chapter": 6, "status": "confirmed", "claim": "The number of derangements of four objects is !4=9.", "sources": [ { "title": "Wolfram MathWorld, Derangement", "url": "https://mathworld.wolfram.com/Derangement.html", "own_work": false, "note": "defines derangements, gives the recurrence relations, and lists them from n = 1: “The first few are 0, 1, 2, 9, 44, 265, 1854, ... (OEIS A000166).”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-6.18", "url": "https://www.teclamairme.com/#r-6.18", "chapter": 6, "status": "confirmed", "claim": "The chromatic polynomial counts proper vertex colorings; for a triangle it is k(k−1)(k−2), giving 0 at k=2 and 6 at k=3. An equilateral triangle has six symmetries, the dihedral group D3, the same number as its proper 3-colorings.", "sources": [ { "title": "Wolfram MathWorld, Chromatic Polynomial", "url": "https://mathworld.wolfram.com/ChromaticPolynomial.html", "own_work": false, "note": "defines the chromatic polynomial and gives the triangle polynomial k(k−1)(k−2).", "fetched": [ "2026-09-09" ] }, { "title": "Dihedral group of order 6, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dihedral_group_of_order_6", "own_work": false, "note": "the triangle’s symmetries number six: “One way to interpret the dihedral group D3 geometrically is by observing the symmetry of an equilateral triangle.” and “Six transformations are the results.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.19", "url": "https://www.teclamairme.com/#r-6.19", "chapter": 6, "status": "confirmed", "claim": "The handshaking lemma says the sum of degrees is twice the edge count; the Petersen graph has ten degree-three vertices and fifteen edges. Euler first proved the handshaking lemma in his 1736 paper on the Königsberg bridges, where it governs when an Euler path or tour exists.", "sources": [ { "title": "Wolfram MathWorld, Handshaking Lemma", "url": "https://mathworld.wolfram.com/HandshakingLemma.html", "own_work": false, "note": "states that the sum of vertex degrees is twice the number of edges.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Petersen Graph", "url": "https://mathworld.wolfram.com/PetersenGraph.html", "own_work": false, "note": "records Petersen’s 1898 introduction and the graph’s standard structural properties.", "fetched": [ "2026-09-09" ] }, { "title": "Handshaking lemma, Wikipedia", "url": "https://en.wikipedia.org/wiki/Handshaking_lemma", "own_work": false, "note": "ties the lemma to Euler and the bridges: “Leonhard Euler first proved the handshaking lemma in his work on the Seven Bridges of Königsberg” and “Euler stated the fundamental results for this problem in terms of the number of odd vertices in the graph, which the handshaking lemma restricts to be an even number.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.20", "url": "https://www.teclamairme.com/#r-6.20", "chapter": 6, "status": "confirmed", "claim": "Gauss’s congruence notation records equality of remainders modulo n, and modular addition, subtraction and multiplication are well-defined. Gauss introduced the sign ≡ in print in the Disquisitiones Arithmeticae of 1801, article 2.", "sources": [ { "title": "Wolfram MathWorld, Modular Arithmetic", "url": "https://mathworld.wolfram.com/ModularArithmetic.html", "own_work": false, "note": "defines congruence and modular arithmetic.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Uses of Symbols of Number Theory, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/nth/", "own_work": false, "note": "“The congruent symbol used in number theory ≡ was introduced in print in 1801 by Carl Friedrich Gauss (1777-1855) in Disquisitiones arithmeticae,” citing “Disquisitiones arithmeticae (Leipzig, 1801), art. 2.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.21", "url": "https://www.teclamairme.com/#r-6.21", "chapter": 6, "status": "confirmed", "claim": "The Chinese remainder theorem combines congruences to pairwise coprime moduli into one solution modulo their product.", "sources": [ { "title": "Wolfram MathWorld, Chinese Remainder Theorem", "url": "https://mathworld.wolfram.com/ChineseRemainderTheorem.html", "own_work": false, "note": "states unique simultaneous solutions for pairwise coprime moduli.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-6.22", "url": "https://www.teclamairme.com/#r-6.22", "chapter": 6, "status": "confirmed", "claim": "Factorials count orderings, and binomial coefficients count unordered choices.", "sources": [ { "title": "Wolfram MathWorld, Factorial", "url": "https://mathworld.wolfram.com/Factorial.html", "own_work": false, "note": "defines n! and its counting interpretation.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Binomial Coefficient", "url": "https://mathworld.wolfram.com/BinomialCoefficient.html", "own_work": false, "note": "defines C(n, k) as the number of k-subsets and gives the zero case when k exceeds n.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-6.23", "url": "https://www.teclamairme.com/#r-6.23", "chapter": 6, "status": "judgement", "claim": "The historical attribution of Wilson’s theorem is stable, but the dates and priority claims around Ibn al-Haytham, Leibniz, Wilson, Waring and Lagrange are reported differently across reference works. Lagrange’s proof is dated 1771 by the Encyclopedia of Mathematics and MacTutor but 1773 by MathWorld, so the card gives no year for it. Corrections welcome.", "sources": [ { "title": "Wolfram MathWorld, Wilson’s Theorem", "url": "https://mathworld.wolfram.com/WilsonsTheorem.html", "own_work": false, "note": "gives the factorial congruence and describes it as a necessary and sufficient primality test, and gives 1773: “It was proved by Lagrange in 1773.”", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Wilson theorem", "url": "https://encyclopediaofmath.org/wiki/Wilson_theorem", "own_work": false, "note": "states the iff theorem, Waring 1770, and Lagrange’s 1771 proof.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Chronology for 1760–1780", "url": "https://mathshistory.st-andrews.ac.uk/Chronology/16/", "own_work": false, "note": "records Lagrange proving Wilson’s theorem in 1771.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "We follow the 1770 Waring / 1771 Lagrange chronology because the Encyclopedia of Mathematics and MacTutor agree on it, while retaining the older attributions only as qualified historical context." }, { "id": "r-6.24", "url": "https://www.teclamairme.com/#r-6.24", "chapter": 6, "status": "judgement", "claim": "The Petersen graph, ten vertices of degree three and fifteen edges, is a standard counterexample in graph theory; Julius Petersen constructed it in 1898 as the smallest bridgeless cubic graph with no three-edge-coloring, and the same graph had already appeared in a paper of A. B. Kempe in 1886.", "sources": [ { "title": "Wolfram MathWorld, Petersen Graph", "url": "https://mathworld.wolfram.com/PetersenGraph.html", "own_work": false, "note": "records Petersen’s 1898 introduction and the graph’s standard structural properties.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Petersen graph", "url": "https://en.wikipedia.org/wiki/Petersen_graph", "own_work": false, "note": "gives Petersen’s 1898 construction, quotes Knuth calling the graph “a counterexample to many optimistic predictions about what might be true for graphs in general,” records its first appearance twelve years earlier in Kempe (1886) as the ten lines of the Desargues configuration, and cites Holton and Sheehan, The Petersen Graph (Cambridge, 1993).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "“The counterexample to nearly every conjecture in graph theory” is a figure of speech: no source states it, and there is no list of conjectures to check it against. The page uses Knuth’s description instead — a counterexample to many optimistic predictions about graphs in general — which is a named claim by a named authority, and which Holton and Sheehan’s book-length study of this one graph bears out." }, { "id": "r-6.25", "url": "https://www.teclamairme.com/#r-6.25", "chapter": 6, "status": "confirmed", "claim": "Euclid’s algorithm computes a gcd by repeated division, and Knuth calls it the oldest nontrivial algorithm that has survived to the present day. It is Proposition 2 of Book VII of the Elements, which r-12.12 dates to Euclid’s lifetime, about 325 to 265 BC.", "sources": [ { "title": "Wolfram MathWorld, Euclidean Algorithm", "url": "https://mathworld.wolfram.com/EuclideanAlgorithm.html", "own_work": false, "note": "describes repeated division as the gcd algorithm.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Euclidean algorithm", "url": "https://en.wikipedia.org/wiki/Euclidean_algorithm", "own_work": false, "note": "quotes Donald Knuth, The Art of Computer Programming, Vol. 2, 2nd edition (1981), p. 318, calling the algorithm “the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "No source states the unqualified superlative “the oldest algorithm still in daily use,” and there is no register of ancient algorithms against which to check it. Knuth’s qualified form is what the page claims instead, in his own words." }, { "id": "r-6.26", "url": "https://www.teclamairme.com/#r-6.26", "chapter": 6, "status": "confirmed", "claim": "The Mobius function is 0 when its argument is divisible by the square of a prime and (-1)^k otherwise, where k is the number of prime factors; A. Mobius introduced it in 1832.", "sources": [ { "title": "Encyclopedia of Mathematics, Mobius function", "url": "https://encyclopediaofmath.org/wiki/M%C3%B6bius_function", "own_work": false, "note": "defines mu(1)=1, mu(n)=0 if n is divisible by the square of a prime and mu(n)=(-1)^k otherwise, and states that the function was introduced by A. Mobius in 1832.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added because r-6.8’s only source is a DLMF inversion-formula section that uses the Mobius function without defining it; this entry carries the zero-on-repeated-prime-factor definition the chapter and the mobius card both state, and the 1832 date the prose gives." }, { "id": "r-6.27", "url": "https://www.teclamairme.com/#r-6.27", "chapter": 6, "status": "confirmed", "claim": "Euler answered the Konigsberg bridge problem in the negative in 1736, and that paper is the beginning of graph theory.", "sources": [ { "title": "Wolfram MathWorld, Koenigsberg Bridge Problem", "url": "https://mathworld.wolfram.com/KoenigsbergBridgeProblem.html", "own_work": false, "note": "states that the problem “was answered in the negative by Euler (1736),” that the work “represented the beginning of graph theory,” and cites Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Acad. Sci. U. Petrop. 8, 128-140, 1736.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-6.29", "url": "https://www.teclamairme.com/#r-6.29", "chapter": 6, "status": "confirmed", "claim": "Gauss, born on 30 April 1777, published the Disquisitiones Arithmeticae in the summer of 1801, aged twenty-four; the theory of congruences, in which congruences to one and the same modulus can be added, subtracted and multiplied like equalities, was first systematically developed by him. Gauss defines primitive roots in article 57 of the Disquisitiones and credits the term itself to Euler; article 56 records that Lambert and Euler knew of them, and Gauss’s own contribution, in articles 54 and 55, is the first rigorous proof that a primitive root exists for every prime. Article 1 of the Disquisitiones defines congruence as divisibility of the difference, article 2 introduces the sign ≡, and Gauss’s footnote says he chose it for the great analogy between equality and congruence. Euler introduced the totient function in 1763; the notation φ comes from Gauss’s Disquisitiones Arithmeticae of 1801.", "sources": [ { "title": "MacTutor, Carl Friedrich Gauss", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/", "own_work": false, "note": "gives the birth date 30 April 1777 and states that “He published the book Disquisitiones Arithmeticae in the summer of 1801,” which puts Gauss at twenty-four.", "fetched": [ "2026-09-10" ] }, { "title": "Encyclopedia of Mathematics, Congruence", "url": "https://encyclopediaofmath.org/wiki/Congruence", "own_work": false, "note": "defines a congruence modulo m as the divisibility of a - b by m, states that congruences to the same modulus “can be added, subtracted and multiplied in the same way as normal equalities,” and records that the theory of congruences “was first systematically developed by C.F. Gauss.”", "fetched": [ "2026-09-10" ] }, { "title": "Disquisitiones arithmeticae, Sectio prima, Latin Wikisource", "url": "https://la.wikisource.org/wiki/Disquisitiones_arithmeticae/Sectio_prima", "own_work": false, "note": "the primary text: article 1, “Si numerus a numerorum b,c differentiam metitur, b et c secundum a congrui dicuntur”; article 2, “Numerorum congruentiam hoc signo, ≡, in posterum denotabimus”; and its footnote, “Hoc signum propter magnam analogiam quae inter aequalitatem atque congruentiam invenitur adoptavimus.”", "fetched": [ "2026-09-22" ] }, { "title": "Primitive root modulo n, Wikipedia", "url": "https://en.wikipedia.org/wiki/Primitive_root_modulo_n", "own_work": false, "note": "“Gauss defined primitive roots in Article 57 of his Disquisitiones Arithmeticae (1801), where he credited Leonhard Euler with coining the term. In Article 56 he stated that Johann Heinrich Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime number n.”", "fetched": [ "2026-09-22" ] }, { "title": "Euler’s totient function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_totient_function", "own_work": false, "note": "“Leonhard Euler introduced the function in 1763.”; “The now-standard notation φ(A) comes from Gauss’s 1801 treatise Disquisitiones Arithmeticae, although Gauss did not use parentheses around the argument and wrote φA.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Neither source ties the congruence notation itself to the year 1801 in so many words: MacTutor dates the Disquisitiones and the Encyclopedia of Mathematics names Gauss as the first systematic developer of the theory. The two together carry the chapter’s sentence; the notation’s first appearance in the book’s opening lines is not separately sourced here." }, { "id": "r-6.30", "url": "https://www.teclamairme.com/#r-6.30", "chapter": 6, "status": "confirmed", "claim": "George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem. The four color theorem was proved by Appel and Haken in 1976, by way of reducible configurations.", "sources": [ { "title": "Wikipedia, Chromatic polynomial", "url": "https://en.wikipedia.org/wiki/Chromatic_polynomial", "own_work": false, "note": "states that “George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem,” and gives the triangle’s polynomial as x(x-1)(x-2).", "fetched": [ "2026-09-10" ] }, { "title": "Four color theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Four_color_theorem", "own_work": false, "note": "on the eventual proof: “the four color theorem resisted until 1976, when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof” and “The Appel–Haken proof proceeds by analyzing a very large number of “reducible configurations,” examples of maps with a particular property.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "r-6.18’s MathWorld page carries the triangle polynomial but names Birkhoff only in its bibliography (A Determinant Formula for the Number of Ways of Coloring a Map, Ann. Math. 14, 42-46, 1912) and never states the four-color motivation." }, { "id": "r-6.31", "url": "https://www.teclamairme.com/#r-6.31", "chapter": 6, "status": "confirmed", "claim": "Ramanujan’s 1919 paper proves the partition congruences modulo 5 and modulo 7; the proof of the modulo 11 congruence was extracted by G. H. Hardy from an unpublished manuscript after Ramanujan’s death in 1920 and published in 1921.", "sources": [ { "title": "Wikipedia, Ramanujan’s congruences", "url": "https://en.wikipedia.org/wiki/Ramanujan%27s_congruences", "own_work": false, "note": "states that “In his 1919 paper, he proved the first two congruences” and that “After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921).”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "The chapter’s prose and the partition5 card both say Ramanujan proved the congruence “with companion congruences modulo 7 and 11.” All three proofs are his, but only the 5 and 7 cases are in the 1919 paper; the 11 case appeared posthumously in 1921." }, { "id": "r-6.32", "url": "https://www.teclamairme.com/#r-6.32", "chapter": 6, "status": "confirmed", "claim": "Fermat stated the little theorem in a letter of 18 October 1640 to Frenicle de Bessy; the first published proof is Euler’s, in the 1736 paper Theorematum quorundam ad numeros primos spectantium demonstratio.", "sources": [ { "title": "Wikipedia, Fermat’s little theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_little_theorem", "own_work": false, "note": "states that Fermat first stated the theorem in a letter dated 18 October 1640 to Frenicle de Bessy, and that Euler provided the first published proof in 1736 in Theorematum quorundam ad numeros primos spectantium demonstratio; also “Fermat’s little theorem is the basis for the Fermat primality test,” and, of its generalization, “Euler’s theorem is used with n not prime in public-key cryptography, specifically in the RSA cryptosystem.”", "fetched": [ "2026-09-11" ] }, { "title": "Encyclopedia of Mathematics, Fermat little theorem", "url": "https://encyclopediaofmath.org/wiki/Fermat_little_theorem", "own_work": false, "note": "gives the congruence a^(p-1) = 1 (mod p) for a not divisible by the prime p and states that the theorem “was established by P. Fermat (1640).”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added for Q6.2. The fermatlittle card said Euler proved the theorem in 1736 while its only source, MathWorld, dates the first published proof to 1749. Two different Euler papers are in circulation under that description, so the card now says first published proof and names the 1736 one, and this entry carries both halves of the sentence. The Encyclopedia of Mathematics source was moved here on 17 September 2026 from the former entry 6.28, which said only “Fermat stated the little theorem in 1640,” was cited by nobody, and was retired as redundant with this entry; the move leaves the 1640 date resting on two independent sources instead of one." }, { "id": "r-6.33", "url": "https://www.teclamairme.com/#r-6.33", "chapter": 6, "status": "confirmed", "claim": "In the Königsberg graph one land mass is touched by 5 bridges and each of the other three by 3, so all four have odd degree; a walk crossing every edge exactly once needs a connected graph with exactly zero or two odd-degree nodes, which is why no such walk exists.", "sources": [ { "title": "Wikipedia, Seven Bridges of Königsberg", "url": "https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg", "own_work": false, "note": "states that “one is touched by 5 bridges, and each of the other three is touched by 3,” that “a necessary condition for the walk of the desired form is that the graph be connected and have exactly zero or two nodes of odd degree,” names Euler’s 1736 paper Solutio problematis ad geometriam situs pertinentis, and records that the solution “is considered to be the first theorem of graph theory.”", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Added with the konigsberg card. r-6.27 carries the negative answer, the year and the founding status, but not the degree count or the odd-vertex condition, which is the argument the card explains." }, { "id": "r-6.34", "url": "https://www.teclamairme.com/#r-6.34", "chapter": 6, "status": "confirmed", "claim": "The dated attributions on this chapter’s cards are as stated. Fermat stated the two-squares theorem in a letter to Mersenne of 25 December 1640 and Euler announced the first proof, by infinite descent, in letters to Goldbach of 6 May 1747 and 12 April 1749, publishing it in detail between 1752 and 1755. Legendre introduced his symbol in 1797 or 1798; the source gives both years, and corrections are welcome. Gauss first proved quadratic reciprocity, which Euler and Legendre had conjectured, on 8 April 1796 by his own diary, called it privately the golden theorem, published the first two proofs in the Disquisitiones Arithmeticae of 1801, and gave eight proofs in all, six published and two found in his posthumous papers. E. H. Moore proved in 1893 that the order of a finite field is a prime power and that fields of each prime-power order exist and are all isomorphic. Gabriel Lame proved in 1844 that Euclid’s algorithm never needs more steps than five times the number of decimal digits of the smaller integer. Pascal’s Traite du triangle arithmetique was written in 1654 and published posthumously in 1665, and the triangle was known earlier in India, Persia and China, appearing in Zhu Shijie’s Siyuan Yujian of 1303. Frank Ramsey’s theorem was published in 1930. Greenwood and Gleason established R(3, 3)=6, R(3, 4)=9 and R(4, 4)=18 in 1955. Euler described the Catalan sequence in 1751 through polygon triangulations, the numbers are named for Eugene Charles Catalan, and Ming Antu had reached them in China earlier. Dobinski gave his formula for the Bell numbers in 1877, Eric Temple Bell wrote about them in a paper of 1934 on Bell polynomials and one of 1938 in which he said they had been rediscovered many times, and Becker and Riordan gave them his name in 1948; the earliest exhaustive enumeration of set partitions was the medieval Japanese incense game genjiko. Pierre Remond de Montmort posed the derangement problem in the Essay d’analyse sur les jeux de hazard in 1708 and solved it in 1713, as did Nicholas Bernoulli, and the name subfactorial is William Allen Whitworth’s. Euler proved his partition theorem in 1748, Hardy and Ramanujan found the asymptotic size of p(n) in 1918, Rademacher gave a convergent exact series in 1937, and p(100) = 190,569,292. Euler introduced the generating function of the partition function in a paper presented in 1741 and printed in 1751, and discussed it in the Introductio of 1748. Any of these three years could be called the date of the generating function, so the card says the 1740s; corrections welcome.", "sources": [ { "title": "Fermat’s theorem on sums of two squares, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares", "own_work": false, "note": "read as raw HTML: Fermat wrote the statement “in a letter to Marin Mersenne dated December 25, 1640,” and of Euler, “The first proof was found by Euler after much effort and is based on infinite descent. He announced it in two letters to Goldbach, on May 6, 1747 and on April 12, 1749; he published the detailed proof in two articles (between 1752 and 1755).” The card’s “proved by Euler in 1749” is the second announcement; the printing is later.", "fetched": [ "2026-09-17" ] }, { "title": "Legendre symbol, Wikipedia", "url": "https://en.wikipedia.org/wiki/Legendre_symbol", "own_work": false, "note": "read as raw HTML: “The Legendre symbol was introduced by Adrien-Marie Legendre in 1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity.” The source hedges between the two years, and so does the card.", "fetched": [ "2026-09-17" ] }, { "title": "Quadratic reciprocity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Quadratic_reciprocity", "own_work": false, "note": "read as raw HTML: “The quadratic reciprocity theorem was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss”; Gauss “referred to it as the ‘fundamental theorem’ in his Disquisitiones Arithmeticae” and “Privately, Gauss referred to it as the ‘golden theorem’. He published six proofs for it, and two more were found in his posthumous papers,” which is the eight the card counts; the first proof is dated by “his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity),” and “Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125-146 and 262 of Disquisitiones Arithmeticae in 1801”; and, of a later proof, “This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.”", "fetched": [ "2026-09-17" ] }, { "title": "Finite field, Wikipedia", "url": "https://en.wikipedia.org/wiki/Finite_field", "own_work": false, "note": "read as raw HTML: “the following classification theorem first proved in 1893 by E. H. Moore: The order of a finite field is a prime power. For every prime power q there are fields of order q, and they are all isomorphic.”", "fetched": [ "2026-09-17" ] }, { "title": "Euclidean algorithm, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euclidean_algorithm", "own_work": false, "note": "read as raw HTML: “the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844 (Lamé’s Theorem), and marks the beginning of computational complexity theory.”", "fetched": [ "2026-09-17" ] }, { "title": "Pascal’s triangle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pascal%27s_triangle", "own_work": false, "note": "read as raw HTML: “Pascal’s Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665,” its page images being dated 1654; the page also records Tartaglia publishing six rows in 1556, Cardano in 1570, and the Chinese diagram in Zhu Shijie’s Siyuan Yujian of 1303.", "fetched": [ "2026-09-17" ] }, { "title": "Ramsey’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ramsey%27s_theorem", "own_work": false, "note": "read as raw HTML: the paper is given as Ramsey, F. P. (1930), “On a problem of formal logic,” Proceedings of the London Mathematical Society 30, 264-286. The page gives no 1928 date.", "fetched": [ "2026-09-17" ] }, { "title": "Wolfram MathWorld, Ramsey Number", "url": "https://mathworld.wolfram.com/RamseyNumber.html", "own_work": false, "note": "read as raw HTML: its table of known values credits “Greenwood and Gleason (1955)” for R(3, 3)=6, R(3, 4)=9, R(3, 5)=14 and R(4, 4)=18, and records that “Exoo (1989) proved that R(5, 5) ≥ 43 and Angeltveit and McKay (2024) proved that R(5, 5) ≤ 46.”", "fetched": [ "2026-09-17" ] }, { "title": "Catalan number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Catalan_number", "own_work": false, "note": "read as raw HTML: “The Catalan sequence was described in 1751 by Leonhard Euler, who was interested in the number of different ways of dividing a polygon into triangles. The sequence is named after Eugène Charles Catalan, who discovered the connection to parenthesized expressions during his exploration of the Towers of Hanoi puzzle.” The page carries no year for Catalan and links Larcombe, “The 18th century Chinese discovery of the Catalan numbers,” for Ming Antu.", "fetched": [ "2026-09-17" ] }, { "title": "OEIS A000108, the Catalan numbers", "url": "https://oeis.org/A000108", "own_work": false, "note": "read as raw HTML: “A very large number of combinatorial interpretations are known - see references, esp. R. P. Stanley, ‘Catalan Numbers’, Cambridge University Press, 2015,” which is the catalogue the card means. Neither this page nor Wikipedia’s states a count, so “more than two hundred families” is the book’s own reading of that book.", "fetched": [ "2026-09-17" ] }, { "title": "Bell number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bell_number", "own_work": false, "note": "read as raw HTML: “The Bell numbers are named after Eric Temple Bell, who wrote about them in 1938, following up a 1934 paper in which he studied the Bell polynomials”; “in his 1938 paper, he wrote that the Bell numbers ‘have been frequently investigated’ and ‘have been rediscovered many times’”; “Bell cites several earlier publications on these numbers, beginning with Dobiński 1877 which gives Dobiński’s formula”; “the name ‘Bell numbers’ and the notation Bₙ … was given to them by Becker & Riordan 1948”; and “The first exhaustive enumeration of set partitions appears to have occurred in medieval Japan, where … a parlor game called genjikō sprang up.”", "fetched": [ "2026-09-17" ] }, { "title": "Derangement, Wikipedia", "url": "https://en.wikipedia.org/wiki/Derangement", "own_work": false, "note": "read as raw HTML: “The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d’analyse sur les jeux de hazard in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time,” and “The name ‘subfactorial’ originates with William Allen Whitworth”; and, of e^−1 ≈ 0.367879, “This is the limit of the probability that a randomly selected permutation of a large number of objects is a derangement.”", "fetched": [ "2026-09-17" ] }, { "title": "Partition function (number theory), Wikipedia", "url": "https://en.wikipedia.org/wiki/Partition_function_(number_theory)", "own_work": false, "note": "read as raw HTML: “This asymptotic formula was first obtained by G. H. Hardy and Ramanujan in 1918 and independently by J. V. Uspensky in 1920,” and “In 1937, Hans Rademacher was able to improve on Hardy and Ramanujan’s results by providing a convergent series expression for p(n).”", "fetched": [ "2026-09-17" ] }, { "title": "Integer partition, Wikipedia", "url": "https://en.wikipedia.org/wiki/Integer_partition", "own_work": false, "note": "read as raw HTML, for the 1748 date: of the identity between partitions into odd parts and into distinct parts, “This result was proved by Leonhard Euler in 1748.” The page does not date the generating function; Jordan Bell’s paper below has Euler introduce it in a paper presented in 1741 and printed in 1751, and discuss it in the Introductio of 1748.", "fetched": [ "2026-09-17" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "p(100) computed by the standard coin-change recurrence over parts 1 to 100: p(100) = 190,569,292, and the same recurrence gives p(4) = 5 and p(9) = 30, which are the values r-6.11 cites from DLMF.", "fetched": [ "2026-09-17" ] }, { "title": "Traité du triangle arithmétique, Cambridge Digital Library", "url": "https://cudl.lib.cam.ac.uk/view/PR-CCB-00013-00024/1", "own_work": false, "note": "“In 1654 Blaise Pascal entered into correspondence with Pierre de Fermat of Toulouse about some problems in calculating the odds in games of chance, as a result of which he wrote the Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière, probably in August of that year. Not published until 1665”", "fetched": [ "2026-09-22" ] }, { "title": "Jordan Bell, Euler and the pentagonal number theorem, arXiv math/0510054", "url": "https://arxiv.org/abs/math/0510054", "own_work": false, "note": "“presented to the St. Petersburg Academy on April 6, 1741 and published in 1751”; “In this paper Euler introduces the generating function”; “In §323 (in ch. XVI, “De partitione numerorum”) of the Introductio in analysin infinitorum [Euler, 1748], E101, published in 1748, Euler notes in his discussion on the generating function”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 17 September 2026 for the B6 pass. Eleven cards in this chapter carried dated attributions on entries whose claim lines stop at the mathematics - MathWorld and DLMF definitions with no history in them. Four entries for exactly this pattern already existed and had been attached last pass (6.23, 6.26, 6.29, 6.30, 6.31), so this entry finishes the block rather than starting it. Three dates could not be sourced and were handled rather than asserted. Ramsey: no source found dates the proof to 1928, and the paper is 1930, so the card now says 1930 and the 1928 reading, which is the year the paper was read to the London Mathematical Society, is left out. Catalan: no source found gives 1838 for Catalan’s own work, so the card no longer gives a year for him. Stanley’s catalogue: no source states a count, so “more than two hundred families” is flagged above as the book’s own reading of Stanley’s Catalan Numbers (2015). One more nuance worth keeping: the Legendre symbol source hedges between 1797 and 1798, and Euler’s two-squares proof was announced in 1747 and 1749 and printed between 1752 and 1755, so “proved by Euler in 1749” is the announcement." }, { "id": "r-6.35", "url": "https://www.teclamairme.com/#r-6.35", "chapter": 6, "status": "confirmed", "claim": "The chapter epigraph is Gauss’s, as reported by Wolfgang Sartorius von Waltershausen in Gauss zum Gedaechtniss of 1856.", "sources": [ { "title": "Carl Friedrich Gauss, Wikiquote", "url": "https://en.wikiquote.org/wiki/Carl_Friedrich_Gauss", "own_work": false, "note": "read as raw HTML: “Mathematics is the queen of the sciences. As quoted in Gauss zum Gedächtniss (1856) by Wolfgang Sartorius von Waltershausen; Variants: Mathematics is the queen of sciences and number theory is the queen of mathematics,” which is the epigraph’s form.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. Another of the four secondhand ones: what survives is Sartorius’s report of 1856, not anything Gauss printed, and the claim line says so." }, { "id": "r-6.36", "url": "https://www.teclamairme.com/#r-6.36", "chapter": 6, "status": "confirmed", "claim": "RSA is a public-key cryptosystem publicly described in 1977 by Ron Rivest, Adi Shamir and Leonard Adleman, and described in Scientific American’s Mathematical Games column in August 1977, before the patent filing of December 1977. An equivalent system was developed secretly in 1973 at GCHQ by Clifford Cocks and declassified in 1997. Its arithmetic is exponentiation modulo the product of two primes, with the exponents paired modulo the totient.", "sources": [ { "title": "RSA cryptosystem, Wikipedia", "url": "https://en.wikipedia.org/wiki/RSA_cryptosystem", "own_work": false, "note": "read as raw HTML: “Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977… An equivalent system was developed secretly in 1973 at Government Communications Headquarters (GCHQ)… by the English mathematician Clifford Cocks. That system was declassified in 1997,” and “A detailed description of the algorithm was published in August 1977, in Scientific American’s Mathematical Games column… This preceded the patent’s filing date of December 1977.”", "fetched": [ "2026-09-17" ] } ], "editorial_note": "The chapter’s own sentence is a warning rather than a recipe — Fermat’s little theorem alone is not an encryption system — and this entry carries only the history and the shape of the arithmetic, not any claim about the security of any key size." }, { "id": "r-6.37", "url": "https://www.teclamairme.com/#r-6.37", "chapter": 6, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Ming Antu had found the Catalan numbers in China around 1730; Gauss restated the Chinese remainder theorem in 1801; Euler’s paper proving his theorem was presented to the Berlin Academy in 1758 and printed in 1763, and does not yet name the totient; Euler’s paper proving his theorem was printed in 1763; Gauss wrote [x] for the floor in his third proof of quadratic reciprocity (1808); Iverson named floor and ceiling in A Programming Language (1962); Euclid’s algorithm, Elements VII.2, about 300 BC; The notation n! is Christian Kramp’s, 1808; the Petersen graph is named for Julius Petersen’s paper of 1898 and had already appeared in a paper of A. B. Kempe in 1886. The Iverson bracket began in the same 1962 book, in parentheses; the square-bracket form was Knuth’s. Stirling stated his approximation in a 1729 letter to de Moivre and showed in 1730 that its constant is √(2π); the card gives both years rather than choose one. Corrections welcome.", "sources": [ { "title": "Catalan number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Catalan_number", "own_work": false, "note": "read raw: “In 1988, it came to light that the Catalan number sequence had been used in China by the Mongolian mathematician Mingantu by 1730, when he started to write his book Ge Yuan Mi Lu Jie Fa.”", "fetched": [ "2026-09-18" ] }, { "title": "Chinese remainder theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chinese_remainder_theorem", "own_work": false, "note": "read raw: “The Chinese remainder theorem appears in Gauss’s 1801 book Disquisitiones Arithmeticae.”; “Both names of the theorem refer to its earliest known statement that appeared in Sunzi Suanjing, a Chinese manuscript written during the 3rd to 5th century CE”; “a complete solution called Da-yan-shu (大衍術) in Qin Jiushao’s 1247 Mathematical Treatise in Nine Sections”; “Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption.”", "fetched": [ "2026-09-18" ] }, { "title": "Euler’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_theorem", "own_work": false, "note": "read raw: “This paper was first presented to the Berlin Academy on June 8, 1758 and to the St. Petersburg Academy on October 15, 1759.”; “Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is not prime.”; the theorem “states that, if n and a are coprime positive integers, then a φ ( n ) is congruent to 1 modulo n” (exponent as typeset on the page); of that paper, in its reference list: “In this paper, Euler’s totient function, φ(n), is not named but referred to as “numerus partium ad N primarum.””", "fetched": [ "2026-09-18" ] }, { "title": "Floor and ceiling functions, Wikipedia", "url": "https://en.wikipedia.org/wiki/Floor_and_ceiling_functions", "own_work": false, "note": "read raw: “Carl Friedrich Gauss introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808).”; “Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names “floor” and “ceiling” and the corresponding notations ⌊x⌋ and ⌈x⌉..”", "fetched": [ "2026-09-18" ] }, { "title": "Euclidean algorithm, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euclidean_algorithm", "own_work": false, "note": "read raw: “It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC)..”", "fetched": [ "2026-09-18" ] }, { "title": "Factorial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Factorial", "own_work": false, "note": "read raw: “The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808..”", "fetched": [ "2026-09-18" ] }, { "title": "Petersen graph, Wikipedia", "url": "https://en.wikipedia.org/wiki/Petersen_graph", "own_work": false, "note": "read raw: “Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886)..”", "fetched": [ "2026-09-18" ] }, { "title": "Iverson bracket, Wikipedia", "url": "https://en.wikipedia.org/wiki/Iverson_bracket", "own_work": false, "note": "cites Iverson (1962), A Programming Language, p. 11: “The notation was originally introduced by Kenneth E. Iverson in his programming language APL,” “though restricted to single relational operators enclosed in parentheses,” the square brackets being “advocated by Donald Knuth.”", "fetched": [ "2026-09-22" ] }, { "title": "Stirling’s approximation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stirling%27s_approximation", "own_work": false, "note": "dates the constant: “Stirling’s contribution in 1730 consisted of showing that the constant is precisely” √(2π); it also says “a related but less precise result was first stated by Abraham de Moivre.” The Factorial page above has “a 1729 letter from James Stirling to de Moivre stating what became known as Stirling’s approximation.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-6.38", "url": "https://www.teclamairme.com/#r-6.38", "chapter": 6, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Julius Petersen constructed the graph in 1898; Dirichlet used the pigeonhole principle in 1842, naming it the Schubfachprinzip only in later work; James Stirling introduced the numbers in Methodus Differentialis (1730); Jovan Karamata used the brace notation for Stirling numbers of the second kind in 1935, before Marx and Salmeri; The brace notation was used by Marx and Salmeri in 1962; Knuth adopted the brace notation in 1968; Ibn al-Haytham knew Wilson’s theorem around 1000; The Petersen graph is a ten-vertex graph from 1898. The year 1834 is often repeated, as on Wikipedia, but Rittaud and Heeffer (2014) find no source for it; Dirichlet’s first uses are in two papers of 1842, on diophantine approximation.", "sources": [ { "title": "Petersen graph, Wikipedia", "url": "https://en.wikipedia.org/wiki/Petersen_graph", "own_work": false, "note": "read raw: “Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by A. B. Kempe (1886).”; “The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring..”", "fetched": [ "2026-09-18" ] }, { "title": "Pigeonhole principle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pigeonhole_principle", "own_work": false, "note": "read raw: “it is commonly called Dirichlet’s box principle or Dirichlet’s drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip.”", "fetched": [ "2026-09-18" ] }, { "title": "Stirling number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stirling_number", "own_work": false, "note": "read raw: “They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730)..”", "fetched": [ "2026-09-18" ] }, { "title": "Stirling numbers of the second kind, Wikipedia", "url": "https://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind", "own_work": false, "note": "read raw: “According to the third edition of The Art of Computer Programming, this notation was also used earlier by Jovan Karamata in 1935.”; “The brace notation was used by Imanuel Marx and Antonio Salmeri in 1962 for variants of these numbers.”; “This led Knuth to use it, as shown here, in the first volume of The Art of Computer Programming (1968)..”", "fetched": [ "2026-09-18" ] }, { "title": "Wilson’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Wilson%27s_theorem", "own_work": false, "note": "read raw: “The theorem was first stated by Ibn al-Haytham c. 1000 AD.”; “There is evidence that Leibniz was also aware of the result a century earlier, but never published it.”; its footnote on Waring: “In the third (1782) edition of Waring’s Meditationes Algebraicae, Wilson’s theorem appears as problem 5 on page 380. On that page, Waring states: “Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger.” (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)”; its elementary proof: “the only values of a for which a ≡ a⁻¹ (mod p) are a ≡ ±1 (mod p). Therefore, with the exception of ±1, the factors in the expanded form of (p − 1)! can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo p.” (formulas typeset on the page)", "fetched": [ "2026-09-18" ] }, { "title": "Rittaud and Heeffer, The pigeonhole principle, two centuries before Dirichlet, Mathematical Intelligencer 36 (2014), Ghent repository copy", "url": "https://biblio.ugent.be/publication/4115264/file/4115265.pdf", "own_work": false, "note": "the historical study: “In the writings of Dirichlet, the application of the principle is to be found in 1842”; “It seems that no one knows any precise prior reference, even if the year 1834 is frequently mentioned as the year of the discovery”; “In [7] and [5], the pigeonhole principle is used to prove complex and multidimensional versions of Equation 1” (Dirichlet’s diophantine approximation); and “In a later work he calls it the “Schubfachprinzip.””", "fetched": [ "2026-09-22" ] }, { "title": "Ramsey’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ramsey%27s_theorem", "own_work": false, "note": "the proof of R(3, 3) = 6: “There are 5 edges incident to v and so (by the pigeonhole principle) at least 3 of them must be the same colour.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-6.39", "url": "https://www.teclamairme.com/#r-6.39", "chapter": 6, "status": "confirmed", "claim": "The Petersen graph is a ten-vertex, three-regular graph formed by connecting a pentagon to a pentagram; Julius Petersen constructed it in 1898 as the smallest bridgeless cubic graph with no three-edge-coloring, and it has served since as the standard counterexample to optimistic conjectures in graph theory. A. B. Kempe had already drawn the same graph in 1886.", "sources": [ { "title": "Petersen graph, Wikipedia", "url": "https://en.wikipedia.org/wiki/Petersen_graph", "own_work": false, "note": "describes the construction, Petersen’s 1898 paper, and its role as a standard counterexample.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.42", "url": "https://www.teclamairme.com/#r-6.42", "chapter": 6, "status": "confirmed", "claim": "Ramanujan found in 1919 that the partition function satisfies p(5n+4) congruent to 0 mod 5, p(7n+5) congruent to 0 mod 7, and p(11n+6) congruent to 0 mod 11 for every n, congruences later extended by Ken Ono and others to further prime moduli.", "sources": [ { "title": "Ramanujan’s congruences, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ramanujan%27s_congruences", "own_work": false, "note": "states the three original congruences and later extensions.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-6.45", "url": "https://www.teclamairme.com/#r-6.45", "chapter": 6, "status": "confirmed", "claim": "States the binomial theorem, that (x+y)^n expands as the sum of C(n, k) x^k y^(n-k), and notes Newton’s 1665 generalization to non-integer exponents. Notes that the convention 0! = 1 makes C(n, 0) = C(n, n) = 1.", "sources": [ { "title": "Binomial theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Binomial_theorem", "own_work": false, "note": "states the expansion (x+y)^n = sum C(n, k) x^k y^(n-k) and Newton’s 1665 generalized binomial series.", "fetched": [ "2026-09-22" ] }, { "title": "Factorial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Factorial", "own_work": false, "note": "carries why 0! = 1 matters for binomial coefficients: “the number of ways to choose all n elements from a set of n is C(n, n) = n!/(n!0!) = 1, a binomial coefficient identity that would only be valid with 0!=1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-7.1", "url": "https://www.teclamairme.com/#r-7.1", "chapter": 7, "status": "confirmed", "claim": "Charles L. Bouton of Harvard University published “Nim, A Game with a Complete Mathematical Theory” in the Annals of Mathematics, second series, volume 3 (1901–1902), pages 35–39.", "sources": [ { "title": "Bouton, “Nim, A Game with a Complete Mathematical Theory,” The Annals of Mathematics 3 (1901) 35–39 — Internet Archive item record", "url": "https://archive.org/details/jstor-1967631", "own_work": false, "note": "item metadata gives Title “Nim, A Game with a Complete Mathematical Theory,” Author “Bouton, Charles L.,” Journal “The Annals of Mathematics,” Volume 3, Year 1901, Pages 35-39, identifier urn:jstor-articleid:10.2307/1967631, ISSN 0003486X.", "fetched": [ "2026-09-09" ] }, { "title": "Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)", "url": "https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory", "own_work": false, "note": "byline reads “Charles L. Bouton, Harvard University, Cambridge, Massachusetts.” This is the source for the chapter’s “of Harvard.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Nim", "url": "https://en.wikipedia.org/wiki/Nim", "own_work": false, "note": "“Charles L. Bouton of Harvard University” developed the complete theory in 1901; paper cited as Annals of Mathematics vol. 3, pp. 35–39, 1901–1902.", "fetched": [ "2026-09-09" ] }, { "title": "Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange)", "url": "https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf", "own_work": false, "note": "text-extracted locally: it introduces Nim “taking as a reference the one introduced by Charles Leonard Bouton in his article, 1901,” calls Joseph Leonard Walsh “a student then a colleague of Bouton at Harvard University,” and its footnote 1 cites the paper as “The Annals of Mathematics, 2nd [series].”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.2", "url": "https://www.teclamairme.com/#r-7.2", "chapter": 7, "status": "confirmed", "claim": "Bouton’s theorem: in normal-play Nim the nim-sum is the bitwise exclusive-or of the heap sizes (write the heaps in binary and add each column without carrying), and the position is a loss for the player about to move exactly when the nim-sum is 0; when it is nonzero there is always a move making it 0. Two players alternately remove any positive number of counters from a single heap and whoever takes the last counter wins.", "sources": [ { "title": "Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)", "url": "https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory", "own_work": false, "note": "Bouton defines a “safe combination” as heap sizes which, written in binary and aligned by columns, have every column sum ≡ 0 (mod 2); he states “If A leaves a safe combination on the table, B cannot leave a safe combination on the table at his next move” and that “the player who can first leave a safe combination on the table should win.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Nim", "url": "https://en.wikipedia.org/wiki/Nim", "own_work": false, "note": "winning strategy is to “finish every move with a nim-sum of 0”; the nim-sum is the XOR of the heap sizes; the player facing a nim-sum of zero loses against perfect play.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.3", "url": "https://www.teclamairme.com/#r-7.3", "chapter": 7, "status": "confirmed", "claim": "3 ⊕ 5 ⊕ 6 = 0. In binary 3 = 011, 5 = 101, 6 = 110, and each of the three columns contains exactly two ones, so every column cancels. Among Nim positions with three nonempty heaps, 1, 2, 3 is the smallest position with nim-sum 0; smaller zero positions exist with fewer heaps (for example two equal heaps 1, 1).", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Python: 3^5^6 = 0; bin(3)='0b11', bin(5)='0b101', bin(6)='0b110', so the columns are (0, 1, 1), (1, 0, 1), (1, 1, 0) reading 4s, 2s, 1s — two ones in each. Exhaustive search over 1 <= a < b < c <= 10 with a^b^c == 0 returns (1, 2, 3) first, then (1, 4, 5), (1, 6, 7), (1, 8, 9). For three nonempty heaps a <= b <= c with a^b^c = 0, a = b would force c = 0, so all three must be distinct and (1, 2, 3) is minimal by total. Note that the two-heap position (1, 1) also has nim-sum 0 and a smaller total, so the minimality of 1, 2, 3 holds only among positions with three heaps.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-7.4", "url": "https://www.teclamairme.com/#r-7.4", "chapter": 7, "status": "confirmed", "claim": "1 ⊕ 2 ⊕ 4 = 7, so the position is a win for the mover; reducing the heap of 4 to 3 gives 1 ⊕ 2 ⊕ 3 = 0.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Python: 1^2^4 = 7 (001 | 010 | 100 -> 111); 1^2^3 = 0 (001 | 010 | 011). Since 7 has its 4-bit set and only the heap of 4 has that bit, the unique winning move is in that heap, taking it to 4 XOR 7 = 3.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-7.5", "url": "https://www.teclamairme.com/#r-7.5", "chapter": 7, "status": "confirmed", "claim": "Nim’s origin is not known. Variants have been played since ancient times, a Chinese origin has been proposed and disputed, and Bouton himself said he had “not been able to discover much concerning its history”; the name is usually derived from the German imperative nimm, “take.”", "sources": [ { "title": "Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)", "url": "https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory", "own_work": false, "note": "“The writer has not been able to discover much concerning its history, although certain forms of it seem to be played at a number of American colleges, and at some of the American fairs. It has been called Fan-Tan, but as it is not the Chinese game of that name, the name in the title is proposed for it.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Nim", "url": "https://en.wikipedia.org/wiki/Nim", "own_work": false, "note": "“Variants of nim have been played since ancient times”; a Chinese origin (jiǎn-shízǐ) is suggested but uncertain; earliest European references from the beginning of the 16th century; the OED derives the name from German nimm, ‘take’.", "fetched": [ "2026-09-09" ] }, { "title": "Rougetet, “The Prehistory of Nim Game”", "url": "https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf", "own_work": false, "note": "text extracted locally: the Fan-Tan link “was wrong, as chance is part of the Fan-Tan”; Ross (Mathematical Gazette, May 1953) denied a Chinese origin and proposed German nimm; Walsh, Bouton’s Harvard colleague, confirmed it in December 1953; “Games as simple as Nim ... are handed over verbally; therefore this orality is difficult to track.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.6", "url": "https://www.teclamairme.com/#r-7.6", "chapter": 7, "status": "confirmed", "claim": "Complete winning rules for the one-heap ancestor of Nim were in print four centuries before Bouton. Luca Pacioli’s De Viribus Quantitatis, written between 1496 and 1508, sets two players to reach 30 by adding a number from 1 to 6 in turn, names the safety steps 2, 9, 16 and 23, and gives a rule for finding them in any such game. Problem XIX of Claude-Gaspard Bachet’s Problèmes plaisans et delectables (Lyon, first edition 1612) does the same for reaching 100 by adding a number from 1 to 10, with the steps 1, 12, 23 and so on to 89. What was new in 1901 is generality: Bouton’s article is where the solution of a combinatorial game was first formalized and studied as a general case with any number of piles and any number of chips in each pile.", "sources": [ { "title": "Rougetet, “The Prehistory of Nim Game” (Gathering 4 Gardner G4G11 gift exchange)", "url": "https://www.gathering4gardner.org/g4g11gift/Rougetet_Lisa-Prehistory_of_Nim.pdf", "own_work": false, "note": "and text-extracted locally. Pacioli’s Effect XXXIIII asks two players to reach 30 by adding, in turn, a number from 1 to 6: “it is a simplified version of Bouton’s Nim game ... the solution remains based on the same principle as Bouton’s Nim: there exist ‘safe combinations’ that secure the win, provided that one plays correctly.” For the target 30 Pacioli “quickly gives the winning strategy to be applied: the four safety steps, 2, 9, 16 and 23 have to be reached,” and states a general rule — “always divide the number that you wish to arrive at by one more than has been taken and the remainder of the said division will always be the first [step of the] progression” — with a worked example for the target 35: “he takes whatever he wants up to 6, and you will take, or actually you will make 7 the first degree, and then 14, 21, 28, and 35.” The paper then says exactly what is new in 1901: “it was not before 1901, with Bouton’s article, that the solution of a combinatorial game was to be formalized and studied as a general case with any number of piles and any number of chips in each pile.” Problem XIX of Bachet’s Problemes plaisans et delectables, qui se font par les nombres (Lyon, 1st edition, 1612) proposes “a version of Nim that is similar to the one formulated by Pacioli”: two players must reach 100 by adding numbers from 1 to 10, and Bachet’s rule is to “add 1 to the number that cannot be exceeded, here 10; you obtain 11 and then, always remove 11 of the number to be reached, 100; you will obtain these numbers 89, 78, 67, 56, 45, 34, 23, 12, 1.”", "fetched": [ "2026-09-09" ] }, { "title": "Nim, A Game with a Complete Mathematical Theory — Wikisource transcription of Bouton (1901)", "url": "https://en.wikisource.org/wiki/Nim,_A_Game_with_a_Complete_Mathematical_Theory", "own_work": false, "note": "Bouton makes no priority claim of any kind. He says only that the game “has interested the writer on account of its seeming complexity, and its extremely simple and complete mathematical theory.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Bouton makes no priority claim himself, and the unqualified form of one — that in solving Nim he did something never done for any game — fails on Pacioli and Bachet, whose one-heap game was solved completely and in print. So the chapter and the ⊕ card claim generality rather than priority: not the first game solved, but the first solved for any number of heaps of any size, which is the distinction Rougetet draws." }, { "id": "r-7.7", "url": "https://www.teclamairme.com/#r-7.7", "chapter": 7, "status": "confirmed", "claim": "In On Numbers and Games (1976) John Horton Conway defined a nimber multiplication ⊗ which, with the nim-sum ⊕, makes the non-negative integers a field of characteristic 2 in which every element is its own additive inverse; the multiplication rule is a ⊗ b = mex{(a′ ⊗ b) ⊕ (a ⊗ b′) ⊕ (a′ ⊗ b′) : a′ < a, b′ < b}, giving 2 ⊗ 2 = 3; and for every n the nimbers below 2^(2^n) form the finite field GF(2^(2^n)), so a product of nonzero nimbers is never zero.", "sources": [ { "title": "Wikipedia, Nimber", "url": "https://en.wikipedia.org/wiki/Nimber", "own_work": false, "note": "nim-addition α ⊕ β = mex({α′ ⊕ β} ∪ {α ⊕ β′}), equal to bitwise XOR for finite ordinals; nim-multiplication α ⊗ β = mex{(α′ ⊗ β) ⊕ (α ⊗ β′) ⊕ (α′ ⊗ β′) : α′ < α, β′ < β}; “2² = 3” in nim-multiplication; “an algebraically closed field of characteristic 2” in which each element is “its own additive inverse”; “For all natural numbers n, the set of nimbers less than 2^(2^n) form the Galois field GF(2^(2^n))”; the field comes “with the nimber multiplicative inverse of a nonzero ordinal α,” so every nimber but 0 can be divided by; Conway, On Numbers and Games, 1976 (2nd edition 2000).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.8", "url": "https://www.teclamairme.com/#r-7.8", "chapter": 7, "status": "confirmed", "claim": "2 ⊗ 2 = 3 follows directly from the mex rule.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Recursive implementation of a⊗b = mex{(a′⊗b) xor (a⊗b′) xor (a′⊗b′) : a′ R > P > S (temptation, reward, punishment, sucker); mutual defection is the only strong Nash equilibrium, and “the collectively ideal result of mutual cooperation is irrational from a self-interested standpoint.”", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "With T = 5, R = 3, P = 1, S = 0: against a cooperator, defect pays 5 > 3; against a defector, defect pays 1 > 0. Defection strictly dominates, so (D, D) is the unique Nash equilibrium and u(D, D) = 1, while u(C, C) = 3. Also R > (T + S)/2 = 2.5, so these values satisfy the usual repeated-game condition as well.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-7.15", "url": "https://www.teclamairme.com/#r-7.15", "chapter": 7, "status": "confirmed", "claim": "The chapter’s epigraph is Conway’s own wording, as recorded by the MacTutor History of Mathematics archive.", "sources": [ { "title": "MacTutor History of Mathematics, Quotations by John Horton Conway", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Conway/quotations/", "own_work": false, "note": "verbatim: “You get surreal numbers by playing games. I used to feel guilty in Cambridge that I spent all day playing games, while I was supposed to be doing mathematics. Then, when I discovered surreal numbers, I realized that playing games IS mathematics.” The chapter quotes the second and third sentences exactly.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.16", "url": "https://www.teclamairme.com/#r-7.16", "chapter": 7, "status": "confirmed", "claim": "Conway arrived at the surreal numbers by analyzing Go endgames — that is, by asking what a game position is worth — and every surreal number is a game but not every game is a surreal number; the nimbers, star among them, are games that are not numbers.", "sources": [ { "title": "Wikipedia, Surreal number", "url": "https://en.wikipedia.org/wiki/Surreal_number", "own_work": false, "note": "“Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers”; “He was analyzing Go endgames, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames”; “Every surreal number is a game, but not all games are surreal numbers, e.g. the game {0 | 0} is not a surreal number” — {0 | 0} being star. Conway used the surreals for game analysis in On Numbers and Games (1976).", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Nimber", "url": "https://en.wikipedia.org/wiki/Nimber", "own_work": false, "note": "the nimbers form a field of characteristic 2 in which each element is its own additive inverse — the chapter’s “every one being its own negative.” Read again on 23 September 2026, its lead also says the nimbers “form a subclass of surreal numbers”; the book does not follow that phrase (the Star line below: ∗ = {0 | 0} is a value but not a number) and keeps the link for the field facts.", "fetched": [ "2026-09-09" ] }, { "title": "Star (game theory), Wikipedia", "url": "https://en.wikipedia.org/wiki/Star_(game_theory)", "own_work": false, "note": "read as raw wikitext: star, “written as ∗ or ∗1,” may be denoted as “{0|0}” and “is a value, but not a number in the traditional sense”; it is “confused with” 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-7.17", "url": "https://www.teclamairme.com/#r-7.17", "chapter": 7, "status": "confirmed", "claim": "Exclusive-or is the arithmetic of a binary adder: the sum bit of a half adder is A ⊕ B and of a full adder A ⊕ B ⊕ C_in.", "sources": [ { "title": "Wikipedia, Adder (electronics)", "url": "https://en.wikipedia.org/wiki/Adder_(electronics)", "own_work": false, "note": "“The Boolean logic for the sum (in this case S) will be A ⊕ B whereas for the carry (C) will be A ⋅ B”; for the full adder “S = A ⊕ B ⊕ C_in.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-7.18", "url": "https://www.teclamairme.com/#r-7.18", "chapter": 7, "status": "judgement", "claim": "The combinatorial-game denominator 1 ⊕ 2 ⊕ 4 + 2 ⊗ 2 is nonzero on either reading, but the two readings give different numbers: under the book’s homecoming rule, where each term returns an integer and the + is ordinary addition, it is 7 + 3 = 10; if the + were read as a further nim-sum it would be 7 ⊕ 3 = 4.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "1^2^4 = 7; 2 ⊗ 2 = 3 by the mex rule (verified by recursion, see 7.8); 7 + 3 = 10 and 7 XOR 3 = 4. Both are nonzero, so the value of the chapter’s factor (0 over a nonzero denominator, times a second such fraction) is 0 either way.", "fetched": [] }, { "title": "Wikipedia, Nimber", "url": "https://en.wikipedia.org/wiki/Nimber", "own_work": false, "note": "nim-addition of finite nimbers is bitwise XOR, and the multiplication table gives 2 ⊗ 2 = 3 — the two facts that make both readings available.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Three symbols sit side by side in this denominator — ⊕, ⊗ and + — so the + could be read as a third nimber operation. The book’s homecoming rule settles it: each term returns an ordinary integer and the + is ordinary addition, so the denominator is 7 + 3 = 10. The tip on both formula copies, the chapter key and the 2 ⊗ 2 card all say so. Under the nim reading it would be 7 ⊕ 3 = 4; both values are nonzero, so the chapter’s factor is 0 either way and nothing but the reading is at stake." }, { "id": "r-7.19", "url": "https://www.teclamairme.com/#r-7.19", "chapter": 7, "status": "judgement", "claim": "Dr. Nim, the marble-powered plastic computer that plays Nim, was invented by John Thomas Godfrey and manufactured by E.S.R., Inc. in the mid-1960s.", "sources": [ { "title": "Wikipedia, Dr. Nim", "url": "https://en.wikipedia.org/wiki/Dr._Nim", "own_work": false, "note": "“Dr. Nim is a toy invented by John Thomas Godfrey and manufactured by E.S.R., Inc. in the mid-1960s. It consists of a marble-powered plastic computer capable of playing the game of Nim.” No specific year is given.", "fetched": [ "2026-09-09" ] }, { "title": "Dr. Nim (Instruction Manual), E.S.R. Inc. — Internet Archive item record", "url": "https://archive.org/details/dr-nim", "own_work": false, "note": "the scanned manual is cataloged with a publication date of 1 January 1966.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "No reference work pins a year. Wikipedia says only “mid-1960s”; the Internet Archive catalogues the scanned instruction manual as 1966, which is also the year collectors’ listings give, while some listings say 1965. The chapter says “mid-1960s,” the one description all of them support, and the toy is mentioned only to describe a video about it." }, { "id": "r-7.20", "url": "https://www.teclamairme.com/#r-7.20", "chapter": 7, "status": "judgement", "claim": "No zero-sum game smaller than two strategies apiece can lack a pure-strategy equilibrium, and matching pennies is a 2×2 zero-sum game that lacks one, so it is a smallest game of its kind. It is also the standard textbook example of one.", "sources": [ { "title": "Wikipedia, Matching pennies", "url": "https://en.wikipedia.org/wiki/Matching_pennies", "own_work": false, "note": "describes matching pennies as the basic two-player zero-sum game with no pure-strategy Nash equilibrium, but does not call it the smallest such game.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "A zero-sum game needs at least two strategies per player for a pure equilibrium to fail (any 1×n or m×1 game has one), so 2×2 is the smallest possible shape and matching pennies is a 2×2 game with none. In that sense the claim is exact; it is not exact in the sense of a unique smallest game, since the 2×2 game obtained by relabeling or rescaling the payoffs behaves the same.", "fetched": [] } ], "editorial_note": "The literature treats matching pennies as the standard example but does not use the superlative, so the page rests “smallest” on the argument rather than on an attribution. It means smallest in shape: any 1×n or m×1 zero-sum game has a pure equilibrium, so 2×2 is the least shape in which one can fail. It does not mean unique, since relabeling or rescaling the payoffs gives another 2×2 game that behaves the same." }, { "id": "r-8.1", "url": "https://www.teclamairme.com/#r-8.1", "chapter": 8, "status": "confirmed", "claim": "The Iverson bracket [P] is 1 when P is true and 0 when P is false; Iverson introduced it in 1962, restricted to a single relation in parentheses, and Knuth introduced the square brackets and the generality, campaigning for them in Two Notes on Notation, American Mathematical Monthly 99 (1992), 403–422.", "sources": [ { "title": "Wikipedia, Iverson bracket", "url": "https://en.wikipedia.org/wiki/Iverson_bracket", "own_work": false, "note": "defines the bracket as the indicator of a proposition and attributes its 1962 introduction to Kenneth Iverson; the history section records Knuth’s later advocacy.", "fetched": [ "2026-09-09" ] }, { "title": "Donald E. Knuth, Two notes on notation, arXiv:math/9205211", "url": "https://arxiv.org/abs/math/9205211", "own_work": false, "note": "Knuth’s article free on arXiv, advocating the notation [P] and extending Iverson’s idea; journal reference Amer. Math. Monthly 99 (1992), no. 5, 403–422.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.2", "url": "https://www.teclamairme.com/#r-8.2", "chapter": 8, "status": "confirmed", "claim": "George Boole published the algebra of logic in 1847 and 1854, with Boolean operations represented arithmetically by 0 and 1.", "sources": [ { "title": "Wikipedia, Boolean algebra", "url": "https://en.wikipedia.org/wiki/Boolean_algebra", "own_work": false, "note": "records Boole’s The Mathematical Analysis of Logic (1847) and The Laws of Thought (1854), and the Boolean operations and two-element algebra.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.3", "url": "https://www.teclamairme.com/#r-8.3", "chapter": 8, "status": "confirmed", "claim": "Claude Shannon showed in 1937 that Boolean algebra models switching circuits.", "sources": [ { "title": "Wikipedia, Boolean algebra", "url": "https://en.wikipedia.org/wiki/Boolean_algebra", "own_work": false, "note": "history section identifies Shannon’s 1937 master’s thesis as the application of Boolean algebra to electrical switching circuits.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.4", "url": "https://www.teclamairme.com/#r-8.4", "chapter": 8, "status": "confirmed", "claim": "Fermat’s Last Theorem says that for every integer n>2 there are no positive integers x,y,z with x^n+y^n=z^n; Fermat wrote the claim in a margin around 1637 and the 1670 edition was published by his son. Whether Fermat had a valid general proof is not known and appears unlikely; the proof by infinite descent he did leave behind covers the exponent 4.", "sources": [ { "title": "Wikipedia, Fermat’s Last Theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "infobox and overview give the integer statement, c. 1637 margin, and posthumous 1670 publication in Samuel Fermat’s edition of Arithmetica; on Fermat’s own proof, “It is not known whether Fermat had actually found a valid proof for all exponents n, but it appears unlikely,” and the descent proof he did leave is the one for fourth powers.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.5", "url": "https://www.teclamairme.com/#r-8.5", "chapter": 8, "status": "confirmed", "claim": "Euler proved the n=3 case, Sophie Germain proved the first case of the theorem for a broad class of prime exponents, and Kummer proved the theorem for regular primes in ~1847. The two standard histories date it differently and both are given below.", "sources": [ { "title": "Wikipedia, Fermat’s Last Theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "historical-development section lists the proof of exponent 3, Germain’s theorem for a class of primes, and Kummer’s proof for regular primes, without a year.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Fermat’s last theorem", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem/", "own_work": false, "note": "supports 1847 — “By September 1847 Kummer sent to Dirichlet and the Berlin Academy a paper proving that a prime p is regular … if”.", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Regular prime", "url": "https://en.wikipedia.org/wiki/Regular_prime", "own_work": false, "note": "supports 1850 — “In 1850, Kummer proved that Fermat’s Last Theorem is true for a prime exponent p if p is regular”, i.e. the published general proof, Crelle 40.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-8.6", "url": "https://www.teclamairme.com/#r-8.6", "chapter": 8, "status": "confirmed", "claim": "Andrew Wiles worked on the theorem alone for seven years, in the attic office of his house in Princeton, and announced a proof in three lectures on 21–23 June 1993. Peer review exposed a gap: the Euler system used to extend the Kolyvagin–Flach method was incomplete. He spent almost a year on the repair, by himself and then with his former student Richard Taylor, and found the way around the gap on 19 September 1994. The corrected proof fills the May 1995 issue of the Annals of Mathematics as two papers: Wiles, “Modular elliptic curves and Fermat’s Last Theorem,” 141(3), 443–551, and Taylor and Wiles, “Ring theoretic properties of certain Hecke algebras,” 141(3), 553–572.", "sources": [ { "title": "Wikipedia, Fermat’s Last Theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "the three lectures “on 21–23 June 1993 at the Isaac Newton Institute for Mathematical Sciences”; the error in peer review; Nick Katz “alerted Wiles on 23 August 1993”; the two manuscripts submitted on 24 October 1994; “the entirety of the May 1995 issue” of the Annals of Mathematics.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Wiles’s proof of Fermat’s Last Theorem", "url": "https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem", "own_work": false, "note": "Katz, as referee, “asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap”; “the Euler system used to extend Kolyvagin and Flach’s method was incomplete”; Wiles “spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor,” with the breakthrough on 19 September 1994; the papers are cited as Annals of Mathematics 141(3), 443–551 and 553–572.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia.com, Andrew J. Wiles", "url": "https://www.encyclopedia.com/history/encyclopedias-almanacs-transcripts-and-maps/andrew-j-wiles", "own_work": false, "note": "“He spent seven years working on it in the attic office of his home in Princeton.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.7", "url": "https://www.teclamairme.com/#r-8.7", "chapter": 8, "status": "confirmed", "claim": "In Boolean algebra x∧¬x=0 and x∨¬x=1; these express non-contradiction and excluded middle, both of them Aristotle’s, and Aristotle calls non-contradiction the firmest of all principles in Metaphysics Book IV. L. E. J. Brouwer challenged the unrestricted validity of the classical laws in 1908, in a paper whose title translates as The untrustworthiness of the principles of logic. Tradition names three laws of thought: identity, non-contradiction and excluded middle. ⊤ (tee, verum) denotes the constant true and ⊥ (up tack, falsum) the constant false.", "sources": [ { "title": "Wikipedia, Boolean algebra", "url": "https://en.wikipedia.org/wiki/Boolean_algebra", "own_work": false, "note": "gives complement identities and identifies Boolean algebra as an algebra of truth values.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Law of excluded middle", "url": "https://en.wikipedia.org/wiki/Law_of_excluded_middle", "own_work": false, "note": "identifies excluded middle as a classical logical law, discusses its Aristotelian formulation, and gives Brouwer’s intuitionist refusal of it, but no year for it.", "fetched": [ "2026-09-09" ] }, { "title": "L. E. J. Brouwer, Wikipedia", "url": "https://en.wikipedia.org/wiki/L._E._J._Brouwer", "own_work": false, "note": "read as raw HTML, quoting Kleene: “in 1908: ‘… Brouwer, in a paper titled “The untrustworthiness of the principles of logic,” challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied’.”", "fetched": [ "2026-09-17" ] }, { "title": "Stanford Encyclopedia of Philosophy, Contradiction", "url": "https://plato.stanford.edu/entries/contradiction/", "own_work": false, "note": "introduces Aristotle’s law of non-contradiction as “the firmest of all principles,” quoting Metaphysics 1005b19–23 in Kirwan’s translation: “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Law of noncontradiction", "url": "https://en.wikipedia.org/wiki/Law_of_noncontradiction", "own_work": false, "note": "gives Aristotle’s formulations in Metaphysics Book IV, including “The most certain of all basic principles is that contradictory propositions are not true simultaneously” (1011b13–14).", "fetched": [ "2026-09-09" ] }, { "title": "Law of thought, Wikipedia", "url": "https://en.wikipedia.org/wiki/Law_of_thought", "own_work": false, "note": "names the three: “the law of identity (LOI), the law of non-contradiction (LNC), and the law of excluded middle (LEM)”; and, quoting Russell, “for no very good reason, three of these principles have been singled out by tradition under the name of ‘Laws of Thought’.”", "fetched": [ "2026-09-22" ] }, { "title": "Tee (symbol), Wikipedia", "url": "https://en.wikipedia.org/wiki/Tee_(symbol)", "own_work": false, "note": "“The tee (⊤, \\top in LaTeX), also called down tack (as opposed to the up tack) or verum, is a symbol used to represent: The truth value of being true in logic, or a sentence (e.g., formula in propositional calculus) which is unconditionally true.”", "fetched": [ "2026-09-22" ] }, { "title": "Up tack, Wikipedia", "url": "https://en.wikipedia.org/wiki/Up_tack", "own_work": false, "note": "“The truth value ‘false’, or a logical constant denoting a proposition in logic that is always false. (The names “falsum,” “absurdum” and “absurdity” come from this context.)”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.8", "url": "https://www.teclamairme.com/#r-8.8", "chapter": 8, "status": "confirmed", "claim": "The four denominator brackets evaluate to 1: 1+1=2, √2 is irrational, there are infinitely many primes, and ℵ₀<𝔠; therefore their ordinary sum is 4 and adding excluded middle gives 5.", "sources": [ { "title": "Wikipedia, Irrational number", "url": "https://en.wikipedia.org/wiki/Irrational_number", "own_work": false, "note": "states that √2 is irrational.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Euclid’s theorem", "url": "https://en.wikipedia.org/wiki/Euclid%27s_theorem", "own_work": false, "note": "states that there are infinitely many primes.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Continuum hypothesis", "url": "https://en.wikipedia.org/wiki/Continuum_hypothesis", "own_work": false, "note": "states that the cardinality of the integers is strictly below that of the reals.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Each displayed proposition is true, so its Iverson bracket is 1; 1+1+1+1=4 and the Boolean excluded-middle term contributes 1, giving denominator 5.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-8.9", "url": "https://www.teclamairme.com/#r-8.9", "chapter": 8, "status": "judgement", "claim": "The irrationality of √2 is credited by tradition to the Pythagoreans, in the fifth century BC; no original proof survives, and the reference works that report the tradition also record that neither the discoverer nor the date can be fixed. √2 ≈ 1.4142 is the diagonal of a unit square and probably the first number known to be irrational; historians do not agree on how the original proof went.", "sources": [ { "title": "Wikipedia, Irrational number", "url": "https://en.wikipedia.org/wiki/Irrational_number", "own_work": false, "note": "gives the traditional story, hedged: “The first proof of the existence of irrational numbers is attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.” The historians’ disagreement over how the original proof went is on the arXiv line.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Pythagoras", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Pythagoras/", "own_work": false, "note": "presents the Pythagorean historical tradition but does not establish a precise pre-400 BC document date.", "fetched": [ "2026-09-09" ] }, { "title": "Square root of 2, Wikipedia", "url": "https://en.wikipedia.org/wiki/Square_root_of_2", "own_work": false, "note": "value, diagonal and history: “The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2.”; “Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational.”; “Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned.”", "fetched": [ "2026-09-22" ] }, { "title": "Negrepontis et al., The anthyphairetic reconstruction of the original Pythagorean proof of incommensurability, arXiv:2501.08708", "url": "https://arxiv.org/abs/2501.08708", "own_work": false, "note": "the original proof is reconstructed, not extant: “Unquestionably the greatest discovery of the Pythagoreans is the existence of incommensurable magnitudes, most probably the incommensurability of the diameter to the side of a square, but there is no agreement among historians of Greek mathematics on their method of proof.”; “In this chapter we present novel arguments not only for an anthyphairetic reconstruction of the original Pythagorean proof of incommensurability”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The tradition is universally reported and nowhere dated to a document. Wikipedia gives the Pythagorean discovery story together with the uncertainty about who proved it and how; MacTutor gives the tradition without a text behind it. So the page attributes the proof rather than dating it, and the entry is a judgement because “before 400 BC” rests on that tradition. The mathematics is not at issue: √2 is irrational, which is all the bracket in the denominator asserts." }, { "id": "r-8.10", "url": "https://www.teclamairme.com/#r-8.10", "chapter": 8, "status": "confirmed", "claim": "Euclid’s proof that there are infinitely many primes appears in Elements IX.20 and is conventionally dated to about 300 BC.", "sources": [ { "title": "Wikipedia, Euclid’s theorem", "url": "https://en.wikipedia.org/wiki/Euclid%27s_theorem", "own_work": false, "note": "“Euclid offered a proof in his work Elements (Book IX, Proposition 20)”; the c. 300 BC date is on the Euclid’s Elements line; the proof step, “Since q is either prime or not: If q is prime, then there is at least one more prime that is not in the list, namely, q itself. If q is not prime, then some prime factor p divides q”; and “Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method.”", "fetched": [ "2026-09-09" ] }, { "title": "Euclid’s Elements, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euclid%27s_Elements", "own_work": false, "note": "“The Elements … is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.11", "url": "https://www.teclamairme.com/#r-8.11", "chapter": 8, "status": "confirmed", "claim": "Cantor proved the real numbers uncountable in his article of 1874, and again by the diagonal argument of 1891, so ℵ₀<𝔠; he stated the continuum hypothesis in his paper of 1878.", "sources": [ { "title": "Wikipedia, Continuum hypothesis", "url": "https://en.wikipedia.org/wiki/Continuum_hypothesis", "own_work": false, "note": "the hypothesis “was first introduced by Georg Cantor in his 1878 paper,” and “Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Cantor’s first set theory article", "url": "https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article", "own_work": false, "note": "dates the article to 1874 and gives its proof that the set of real numbers is uncountable.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Cantor’s diagonal argument", "url": "https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument", "own_work": false, "note": "dates the diagonal proof to 1891.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.12", "url": "https://www.teclamairme.com/#r-8.12", "chapter": 8, "status": "confirmed", "claim": "Gödel’s incompleteness theorems, published in 1931, say that sufficiently strong consistent effectively axiomatized theories have undecidable arithmetic truths and cannot prove their own consistency.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems", "url": "https://plato.stanford.edu/entries/goedel-incompleteness/", "own_work": false, "note": "explains the 1931 first and second incompleteness theorems and their hypotheses: “The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F”; the standard statement is given “Accommodating an improvement due to J. Barkley Rosser in 1936.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.13", "url": "https://www.teclamairme.com/#r-8.13", "chapter": 8, "status": "confirmed", "claim": "The continuum hypothesis is the first of the twenty-three problems Hilbert set in 1900. Gödel announced in 1938 and showed in full in 1940 that it is consistent with ZFC (relative to consistency), and Cohen showed in 1963 that it is independent of ZFC. Gödel’s model of ZFC satisfies CH and Cohen’s forcing model does not.", "sources": [ { "title": "Wikipedia, Continuum hypothesis", "url": "https://en.wikipedia.org/wiki/Continuum_hypothesis", "own_work": false, "note": "attributes relative consistency to Gödel’s constructible universe and independence/non-provability to Cohen’s forcing work; its references give Gödel’s announcement as “Gödel, Kurt (1938). “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.” Proceedings of the National Academy of Sciences. 24 (12): 556–557.”", "fetched": [ "2026-09-09" ] }, { "title": "Hilbert’s problems, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_problems", "own_work": false, "note": "read as raw HTML: the 1902 Bulletin of the AMS translation heads the list with “Cantor’s problem of the cardinal number of the continuum,” and the table of problems gives “1st: The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers.”", "fetched": [ "2026-09-17" ] }, { "title": "List of statements independent of ZFC, Wikipedia", "url": "https://en.wikipedia.org/wiki/List_of_statements_independent_of_ZFC", "own_work": false, "note": "“Gödel produced a model of ZFC in which CH is true, showing that CH cannot be disproven in ZFC; Paul Cohen later invented the method of forcing to exhibit a model of ZFC in which CH fails, showing that CH cannot be proven in ZFC.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.14", "url": "https://www.teclamairme.com/#r-8.14", "chapter": 8, "status": "confirmed", "claim": "Peano published his arithmetic axioms in 1889; Zermelo axiomatized set theory in 1908, and Fraenkel and Skolem supplied the later 1922 refinements represented in ZFC. Whitehead and Russell reach 1 + 1 = 2 by way of proposition *54.43, on page 379 of the first volume of Principia Mathematica in 1910, which remarks only that it will follow once addition has been defined; the proof is completed in the second volume, of 1912. From Peano’s axioms, with addition defined by recursion, 1 + 1 = 2 follows in three steps.", "sources": [ { "title": "Wikipedia, Peano axioms", "url": "https://en.wikipedia.org/wiki/Peano_axioms", "own_work": false, "note": "gives Peano’s 1889 publication and the basic successor/induction framework; and “The Peano axioms can be derived from set theoretic constructions of the natural numbers and axioms of set theory such as ZF..”", "fetched": [ "2026-09-22" ] }, { "title": "Wikipedia, Zermelo–Fraenkel set theory", "url": "https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory", "own_work": false, "note": "gives the history of Zermelo’s 1908 axiomatization, Fraenkel’s and Skolem’s 1922 work, and ZFC.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Principia Mathematica", "url": "https://en.wikipedia.org/wiki/Principia_Mathematica", "own_work": false, "note": "proposition *54.43, “From this proposition it will follow, when arithmetical addition has been defined, that 1 + 1 = 2,” stands on page 379 of the first volume of the first edition and page 362 of the second; the proof is completed in the second volume; the three volumes appeared in 1910, 1912 and 1913; and “The Principia Mathematica (often abbreviated PM) is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell.”", "fetched": [ "2026-09-09" ] }, { "title": "11.2 Arithmetic, MATH1001 Introduction to Number Theory, University of Southampton", "url": "https://personal.soton.ac.uk/jhr/math1001/arithmetic.html", "own_work": false, "note": "Theorem 11.2 derives it from the successor function and the recursive definition of addition: “1+1 = 1+s(0) = s(1+0) = s(1) = 2.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.15", "url": "https://www.teclamairme.com/#r-8.15", "chapter": 8, "status": "confirmed", "claim": "The single and double turnstiles express syntactic and semantic consequence; Gödel’s completeness theorem of 1929 makes them coincide for first-order logic, while incompleteness still separates arithmetic truth from formal proof. The quantifier and connective signs are datable too: ∃ is Peano’s, of 1897, and ∀ is Gentzen’s, of 1935, made by analogy with it; ∨ is Russell’s of 1908, ¬ and ∧ are Heyting’s of 1930, and ⇒ and ⇔ are Bourbaki’s of 1954; and Frege’s Begriffsschrift of 1879, where quantified variables first appear, has left nothing behind but the turnstile.", "sources": [ { "title": "Wikipedia, Turnstile (symbol)", "url": "https://en.wikipedia.org/wiki/Turnstile_(symbol)", "own_work": false, "note": "describes ⊢ as syntactic derivability and ⊨ as semantic consequence.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Gödel’s completeness theorem", "url": "https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem", "own_work": false, "note": "dates and states the completeness theorem for first-order logic.", "fetched": [ "2026-09-09" ] }, { "title": "Stanford Encyclopedia of Philosophy, Gödel’s Incompleteness Theorems", "url": "https://plato.stanford.edu/entries/goedel-incompleteness/", "own_work": false, "note": "distinguishes completeness from incompleteness and explains the latter’s arithmetic consequences.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Quantifier (logic)", "url": "https://en.wikipedia.org/wiki/Quantifier_(logic)", "own_work": false, "note": "“In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano’s ∃ symbol,” which Peano introduced “in 1897”; ∀ “did not become canonical until the 1960s.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Logical connective", "url": "https://en.wikipedia.org/wiki/Logical_connective", "own_work": false, "note": "the history of notations gives ∨ in Russell 1908, ¬ and ∧ in Heyting 1930, → in Hilbert 1918, and ⇒ and ⇔ in Bourbaki 1954.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Begriffsschrift", "url": "https://en.wikipedia.org/wiki/Begriffsschrift", "own_work": false, "note": "the 1879 calculus “contains the first appearance of quantified variables,” and “Some vestige of Frege’s notation survives in the ‘turnstile’ symbol ⊢.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-8.16", "url": "https://www.teclamairme.com/#r-8.16", "chapter": 8, "status": "confirmed", "claim": "De Morgan’s laws were published in modern form by Augustus De Morgan in 1847 and interchange negated conjunction and disjunction. The observation is older: Aristotle made a similar one, William of Ockham wrote the laws out in words in the fourteenth century, and Jean Buridan gives rules of conversion along the same lines in his Summulae de Dialectica. In circuit design, De Morgan’s laws underlie building every Boolean function from NAND gates alone.", "sources": [ { "title": "Wikipedia, De Morgan’s laws", "url": "https://en.wikipedia.org/wiki/De_Morgan%27s_laws", "own_work": false, "note": "states both identities and gives De Morgan’s 1847 Formal Logic attribution. Re-read raw on 17 September 2026 for the precursors: “a similar observation was made by Aristotle, and was known to Greek and Medieval logicians. For example, in the 14th century, William of Ockham wrote down the words that would result by reading the laws out. Jean Buridan, in his Summulae de Dialectica, also describes rules of conversion that follow the lines of De Morgan’s laws. Still, De Morgan is given credit for stating the laws in the terms of modern formal logic.”", "fetched": [ "2026-09-09" ] }, { "title": "NAND logic, Wikipedia", "url": "https://en.wikipedia.org/wiki/NAND_logic", "own_work": false, "note": "“The NAND Boolean function has the property of functional completeness. This means that any Boolean expression can be re-expressed by an equivalent expression utilizing only NAND operations.”; and “noting from de Morgan’s law that a NAND gate is an inverted-input OR gate.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-8.17", "url": "https://www.teclamairme.com/#r-8.17", "chapter": 8, "status": "confirmed", "claim": "The therefore sign ∴ carries its modern meaning in Johann Rahn’s Teutsche Algebra of 1659; the exclusive pairing of ∴ for therefore with the inverted ∵ for because dates from the nineteenth century; Q.E.D. translates the ὅπερ ἔδει δεῖξαι of Euclid and Archimedes; and the solid tombstone that now closes a proof is Paul Halmos’s, taken from the end marks of magazine typography.", "sources": [ { "title": "Wikipedia, Therefore sign", "url": "https://en.wikipedia.org/wiki/Therefore_sign", "own_work": false, "note": "gives the 1659 Rahn attribution for ∴ and the inverse because sign’s history.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Q.E.D.", "url": "https://en.wikipedia.org/wiki/Q.E.D.", "own_work": false, "note": "explains Q.E.D. and the modern tombstone/halmos symbol, including Halmos’s adoption.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Two of these are attributions rather than datings. Rahn used the triangle in both orientations, and so did writers after him; what the nineteenth century added was the exclusive convention. The tombstone is credited to Halmos on his own account — he “claims to have pioneered” it — which is how the reference works put it, and how the page takes it." }, { "id": "r-8.18", "url": "https://www.teclamairme.com/#r-8.18", "chapter": 8, "status": "confirmed", "claim": "The chapter’s displayed factor evaluates to (0+0)/5=0, with Boolean truth values returning as ordinary integers.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Fermat’s bracket is 0 by 8.4–8.6; x∧¬x=0 by 8.7; the denominator is 4+1=5 by 8.8; hence (0+0)/5=0. An Iverson bracket has values 0 and 1, which are ordinary integers by 8.1.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-8.19", "url": "https://www.teclamairme.com/#r-8.19", "chapter": 8, "status": "confirmed", "claim": "The chapter epigraph is Fermat’s marginal note, written about 1637 (the note is undated, and every source gives the year as approximate) and printed by his son Clement-Samuel in the 1670 edition of Diophantus’s Arithmetica; the margin itself does not survive.", "sources": [ { "title": "Fermat’s Last Theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "read as raw HTML: the Latin is given as “Cubum autem in duos cubos … cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet” and translated “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”; and “After Fermat’s death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father’s comments.”", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker, and to make the provenance explicit: the margin itself does not survive, and what exists is the 1670 edition his son Clément-Samuel printed. The chapter’s footer was reworded to say so, and to give the son’s full name, in the pass of 18 September 2026." }, { "id": "r-8.20", "url": "https://www.teclamairme.com/#r-8.20", "chapter": 8, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Peano built on Dedekind’s 1888 analysis; Boole’s algebra (1847, 1854); Goedel’s incompleteness theorems of 1931; Fermat’s marginal note in his copy of Diophantus’s Arithmetica, 1637.", "sources": [ { "title": "Peano axioms, Wikipedia", "url": "https://en.wikipedia.org/wiki/Peano_axioms", "own_work": false, "note": "read raw: “In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new method.”", "fetched": [ "2026-09-18" ] }, { "title": "Boolean algebra, Wikipedia", "url": "https://en.wikipedia.org/wiki/Boolean_algebra", "own_work": false, "note": "read raw: “Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854)..”", "fetched": [ "2026-09-18" ] }, { "title": "Gödel’s incompleteness theorems, Wikipedia", "url": "https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems", "own_work": false, "note": "read raw: “These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in philosophy of mathematics..”", "fetched": [ "2026-09-18" ] }, { "title": "Fermat’s Last Theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "read raw: “The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of Arithmetica..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-9.1", "url": "https://www.teclamairme.com/#r-9.1", "chapter": 9, "status": "confirmed", "claim": "The untyped lambda calculus is a formal model of computation; Church first published it in 1932, in A set of postulates for the foundation of logic, and beta-reduction applies a function to an argument. Church numerals encode n as a function that applies its argument n times, so zero is λf.λx.x; the λ-definition of the predecessor function is Kleene’s.", "sources": [ { "title": "Wikipedia, Lambda calculus", "url": "https://en.wikipedia.org/wiki/Lambda_calculus", "own_work": false, "note": "defines lambda calculus as a formal system based on abstraction and application, attributes its introduction to Alonzo Church in the 1930s, and gives beta-reduction for (λx.t)s.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Church encoding", "url": "https://en.wikipedia.org/wiki/Church_encoding", "own_work": false, "note": "describes Church numerals as repeated function application and gives zero as λf.λx.x.", "fetched": [ "2026-09-09" ] }, { "title": "Internet Encyclopedia of Philosophy, Lambda Calculi", "url": "https://iep.utm.edu/lambda-calculi/", "own_work": false, "note": "‘Alonzo Church first introduced the λ-calculus as A set of postulates for the foundation of logic in two papers of that title published in 1932 and 1933’; says Church gave the numeral encoding at the end of the 1933 paper, and that Kleene found how to define the predecessor function.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.2", "url": "https://www.teclamairme.com/#r-9.2", "chapter": 9, "status": "confirmed", "claim": "Turing’s 1936 work establishes that no general mechanical procedure can decide whether an arbitrary machine and input will halt; the diagonal/self-reference argument is the standard proof idea.", "sources": [ { "title": "A. M. Turing, On Computable Numbers (interactive edition)", "url": "https://biomorphbuilder.com/turing-paper/", "own_work": false, "note": "gives the paper’s 1936 receipt/read dates, describes machines with finite control and unbounded tape, states that no general process can determine which machines are circle-free, and connects this to the modern halting problem.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.3", "url": "https://www.teclamairme.com/#r-9.3", "chapter": 9, "status": "judgement", "claim": "Church’s lambda-definability, Herbrand and Gödel’s general recursion, and Turing computability are extensionally equivalent; the Church–Turing thesis is a thesis about what counts as an effective procedure, not a theorem proved from a formal definition of ‘effective’. Gödel set out the general recursive functions in his 1934 Princeton lectures, on a suggestion of Herbrand. Church’s answer to the Entscheidungsproblem was published earlier in 1936 than Turing’s, which was received on 28 May 1936 and printed in two parts on 30 November and 23 December 1936, so Church’s was in print first; the appendix Turing added shows that the lambda-definable and the machine-computable functions are the same class. No model of computation proposed since has escaped that class: a quantum computer can always be simulated by a Turing machine, and changes what is efficient rather than what is computable. Quantum Turing machines are equivalent to deterministic Turing machines; the models that compute non-computable functions are the theoretical hypercomputers.", "sources": [ { "title": "A. M. Turing, On Computable Numbers (interactive edition)", "url": "https://biomorphbuilder.com/turing-paper/", "own_work": false, "note": "Turing says Church’s effective calculability is equivalent to his computability and discusses the equivalence in an appendix.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Lambda calculus", "url": "https://en.wikipedia.org/wiki/Lambda_calculus", "own_work": false, "note": "states that untyped lambda calculus can simulate any Turing machine and vice versa.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Church–Turing thesis", "url": "https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis", "own_work": false, "note": "Gödel ‘with Jacques Herbrand, formalized the definition of the class of general recursive functions’ and modified his approach on Herbrand’s suggestion in his 1934 Princeton lectures; Turing’s appendix ‘showed that the classes of functions defined by λ-calculus and Turing machines coincided’; the thesis ‘cannot be formally proven, as the concept of effective calculability is only informally defined’; and (read as raw HTML, 17 September 2026) a strict containment of BPP in BQP “would not however invalidate the original Church–Turing thesis, since a quantum computer can always be simulated by a Turing machine, but it would invalidate the classical complexity-theoretic Church–Turing thesis for efficiency reasons.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Entscheidungsproblem", "url": "https://en.wikipedia.org/wiki/Entscheidungsproblem", "own_work": false, "note": "‘Church’s paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936’; Turing’s paper ‘was received by the Proceedings of the London Mathematical Society on 28 May 1936, read on 12 November 1936, and published in series 2, volume 42 (1936–7)’, in parts issued 30 November and 23 December 1936, with corrections in volume 43.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "We call the equivalence a confirmed mathematical result for the formal models, but retain ‘thesis’ for the broader claim about every possible machine or effective procedure: that extension is methodological and not a theorem from an agreed formal definition of human computation." }, { "id": "r-9.4", "url": "https://www.teclamairme.com/#r-9.4", "chapter": 9, "status": "confirmed", "claim": "A Turing machine has finite control, a tape of cells, and a scanned head whose transition rules specify what to write, which way to move, and the next state; a universal machine can simulate machines described on its tape.", "sources": [ { "title": "A. M. Turing, On Computable Numbers (interactive edition)", "url": "https://biomorphbuilder.com/turing-paper/", "own_work": false, "note": "section 1 describes finite m-configurations, an unbounded tape divided into squares, and a scanned square.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Turing machine", "url": "https://en.wikipedia.org/wiki/Turing_machine", "own_work": false, "note": "describes the transition function and universal-machine simulation.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.5", "url": "https://www.teclamairme.com/#r-9.5", "chapter": 9, "status": "confirmed", "claim": "Radó introduced the Busy Beaver problem in 1962; in the step-count convention BB(n) is the maximum number of steps made by a halting n-state, two-symbol machine started on a blank tape.", "sources": [ { "title": "Busy Beaver Challenge, July 2024 announcement", "url": "https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237", "own_work": false, "note": "history line attributes the busy-beaver game and functions S and Σ to Radó in 1962.", "fetched": [ "2026-09-09" ] }, { "title": "Busy Beaver Challenge paper", "url": "https://github.com/bbchallenge/bbchallenge-paper", "own_work": false, "note": "defines S(n) as the maximum steps of an n-state, two-symbol machine from the all-zero tape and identifies Radó in 1962.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.6", "url": "https://www.teclamairme.com/#r-9.6", "chapter": 9, "status": "confirmed", "claim": "For the step-count convention used here, BB(1)=1, BB(2)=6, BB(3)=21, and BB(4)=107; S(3)=21 was settled by Lin and Radó in 1965 and S(4)=107 by Brady in 1983. Radó’s own paper left the two-state step count open; Michel’s survey credits S(2) = 6, probably, to Lin in 1963, and Lin’s 1963 proof of S(3) = 21 was published by Lin and Radó in 1965.", "sources": [ { "title": "Wikipedia, Busy beaver", "url": "https://en.wikipedia.org/wiki/Busy_beaver", "own_work": false, "note": "gives the standard small step-count values 1, 6, 21, and 107, distinguishes step-count from productivity conventions, and credits Lin and Radó (Computer Studies of Turing Machine Problems, 1965) with S(3) = 21 and Brady with S(4) = 107 in 1983.", "fetched": [ "2026-09-09" ] }, { "title": "Pascal Michel, Historical survey of Busy Beavers", "url": "https://bbchallenge.org/~pascal.michel/ha.html", "own_work": false, "note": "“Rado (1963) claimed that Sigma(2, 2) = 4, but that S(2, 2) was yet unknown. The value S(2, 2) = 6 was probably set by Lin in 1963.”; “Lin (1963) proved this conjecture and this proof was eventually published by Lin and Rado (1965).”; “Brady (1974, 1975) proved this conjecture, and the proof was eventually published in Brady (1983).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.7", "url": "https://www.teclamairme.com/#r-9.7", "chapter": 9, "status": "confirmed", "claim": "BB(5)=47,176,870 was proved in July 2024 by the collaborative Busy Beaver Challenge, using a formally checked Coq proof; the five-state machine that runs the 47,176,870 steps was itself found by Heiner Marxen and Jürgen Buntrock in 1989 and published in 1990, and BB(6) remains a frontier problem.", "sources": [ { "title": "Busy Beaver Challenge, We have proved BB(5) = 47,176,870", "url": "https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237", "own_work": false, "note": "dated July 2, 2024; states the value, describes the Coq proof, reports the symmetry-reduced enumeration, and says BB(6) is a next target.", "fetched": [ "2026-09-09" ] }, { "title": "The bbchallenge Collaboration, Determination of the fifth Busy Beaver value", "url": "https://arxiv.org/abs/2509.12337", "own_work": false, "note": "abstract states S(5)=47,176,870 and that the proof enumerates 181,385,789 five-state machines, deciding halting for each; posted 15 September 2025, revised 23 March 2026.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Busy beaver", "url": "https://en.wikipedia.org/wiki/Busy_beaver", "own_work": false, "note": "‘The 5-state busy beaver was discovered by Heiner Marxen and Jürgen Buntrock in 1989, but only proved to be the winning fifth busy beaver in 2024 by an online amateur mathematical collective, using a proof formalized in Rocq.’", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.8", "url": "https://www.teclamairme.com/#r-9.8", "chapter": 9, "status": "confirmed", "claim": "Busy Beaver grows faster than every computable function and is uncomputable, because a sufficiently effective exact computation of its values would decide halting for bounded-size machines.", "sources": [ { "title": "Wikipedia, Busy beaver", "url": "https://en.wikipedia.org/wiki/Busy_beaver", "own_work": false, "note": "states that the Busy Beaver function is non-computable and eventually dominates every computable function, with the halting-problem reduction.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.9", "url": "https://www.teclamairme.com/#r-9.9", "chapter": 9, "status": "confirmed", "claim": "Ackermann’s function is a total computable function that is not primitive recursive; the common two-variable recurrence has A(0, n)=n+1, A(m, 0)=A(m−1, 1), and A(m, n)=A(m−1, A(m, n−1)).", "sources": [ { "title": "NIST Dictionary of Algorithms and Data Structures, Ackermann’s function", "url": "https://xlinux.nist.gov/dads/HTML/ackermann.html", "own_work": false, "note": "gives the recurrence, attributes the 1928 observation to Wilhelm Ackermann, and states that the function is recursive but not primitive recursive.", "fetched": [ "2026-09-09" ] }, { "title": "Stanford Encyclopedia of Philosophy, History of the Ackermann and Péter functions", "url": "https://seop.illc.uva.nl/entries/recursive-functions/ackermann-peter.html", "own_work": false, "note": "explains that Ackermann’s result was growth beyond functions defined by ordinary recursion and discusses the historical qualification.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.10", "url": "https://www.teclamairme.com/#r-9.10", "chapter": 9, "status": "confirmed", "claim": "Under the displayed Péter-style recurrence, A(3, 3)=61 and A(4, 2)=2^65536−3, a 19,729-digit integer; A(4, 3)=2^(2^65536)−3, which has far too many digits to write out.", "sources": [ { "title": "Wikipedia, Ackermann function", "url": "https://en.wikipedia.org/wiki/Ackermann_function", "own_work": false, "note": "gives the common recurrence, A(3, 3)=61, and A(4, 2)=2^65536−3 with 19,729 decimal digits.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Dictionary of Algorithms and Data Structures, Ackermann’s function", "url": "https://xlinux.nist.gov/dads/HTML/ackermann.html", "own_work": false, "note": "provides the recurrence used to recompute A(3, 3).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Own arithmetic check: the recurrence gives A(3, 3)=61; 2^65536−3 has floor(65536 log10 2)+1 = 19,729 digits; A(4, n) = 2↑↑(n+3) − 3, so A(4, 3) = 2^(2^65536) − 3, whose decimal expansion has about 10^19,728 digits." }, { "id": "r-9.11", "url": "https://www.teclamairme.com/#r-9.11", "chapter": 9, "status": "confirmed", "claim": "P is contained in NP: a deterministic polynomial-time solver is also a polynomial-time verifier that can ignore its certificate; the status of P versus NP remains open. In the notation those classes are written in, f = O(g) means |f| is eventually bounded by a constant multiple of g; Bachmann proposed the notation in 1894, Landau extended it in 1909, and Knuth fixed Ω and Θ for the family in 1976. Comparison sorting needs Ω(n log n) comparisons, and Harvey and van der Hoeven multiply two n-digit integers in O(n log n) bit operations. Harvey and van der Hoeven announced their O(n log n) multiplication algorithm in March 2019; it was published in 2021.", "sources": [ { "title": "Wikipedia, P versus NP problem", "url": "https://en.wikipedia.org/wiki/P_versus_NP_problem", "own_work": false, "note": "defines P and NP through polynomial-time solving and verification and states the open P-versus-NP question.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Big O notation", "url": "https://en.wikipedia.org/wiki/Big_O_notation", "own_work": false, "note": "‘Bachmann proposed the notation in 1894 and Landau extended it in 1909’; ‘In 1976, Donald Knuth defined f(x) = Ω(g(x)) ⟺ g(x) = O(f(x))’; f(x) = O(g(x)) if there is a positive real M with |f(x)| ≤ M·g(x) throughout the domain. Its table of orders gives n log n as the “fastest possible comparison sort; heapsort and merge sort.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Comparison sort", "url": "https://en.wikipedia.org/wiki/Comparison_sort", "own_work": false, "note": "‘A comparison sort must have an average-case lower bound of Ω(n log n) comparison operations’.", "fetched": [ "2026-09-09" ] }, { "title": "David Harvey and Joris van der Hoeven, Integer multiplication in time O(n log n)", "url": "https://doi.org/10.4007/annals.2021.193.2.4", "own_work": false, "note": "Annals of Mathematics 193 (2021), 563–617 (the March 2019 preprint announcement is on the Multiplication algorithm line). Multiplies two n-bit integers in O(n log n) bit operations, confirming the 1971 conjecture of Schönhage and Strassen.", "fetched": [] }, { "title": "Multiplication algorithm, Wikipedia", "url": "https://en.wikipedia.org/wiki/Multiplication_algorithm", "own_work": false, "note": "“In March 2019, David Harvey and Joris van der Hoeven announced their discovery of an O(n log n) multiplication algorithm.”; “It was published in the Annals of Mathematics in 2021.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.12", "url": "https://www.teclamairme.com/#r-9.12", "chapter": 9, "status": "confirmed", "claim": "The historical attributions in the card are Cook (1971) for NP-completeness of satisfiability, Levin (1973) for the independent theorem, and Karp (1972) for a list of 21 NP-complete problems; Cobham and Edmonds supplied early polynomial-time formulations in 1965. Karp’s list includes the undirected Hamilton circuit, the round trip that visits every point of a network exactly once.", "sources": [ { "title": "Wikipedia, P versus NP problem", "url": "https://en.wikipedia.org/wiki/P_versus_NP_problem", "own_work": false, "note": "records Cook 1971, Levin 1973, and Karp’s 21 NP-complete problems (without a year; 1972 is on the Karp line) in the history of NP-completeness and the P-versus-NP problem.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Cobham’s thesis", "url": "https://en.wikipedia.org/wiki/Cobham%27s_thesis", "own_work": false, "note": "‘Alan Cobham’s 1965 paper entitled The intrinsic computational difficulty of functions is one of the earliest mentions of the concept of the complexity class P’, and ‘Jack Edmonds’s 1965 paper Paths, trees, and flowers is also credited with identifying P with tractable problems’.", "fetched": [ "2026-09-09" ] }, { "title": "Karp’s 21 NP-complete problems, Wikipedia", "url": "https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems", "own_work": false, "note": "“In his 1972 paper, ‘Reducibility Among Combinatorial Problems’, Richard Karp used Stephen Cook’s 1971 theorem that the Boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the Boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete.”; its list includes “Undirected Hamilton circuit (Karp’s name, now usually called Undirected Hamiltonian cycle),” and it calls the paper “one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.13", "url": "https://www.teclamairme.com/#r-9.13", "chapter": 9, "status": "confirmed", "claim": "IEEE binary64 uses 53 bits of significand precision; the next representable number above 1 is 1+2^−52, so the spacing immediately above 1 is 2^−52 ≈ 2.220446×10^−16. Decimal fractions such as 0.1 and 0.2 have no finite binary expansion, so they are stored rounded, which is a separate mechanism from the spacing at 1 and is why 0.1 + 0.2 does not equal 0.3 in binary64. The name machine epsilon has two prevailing definitions: in the mainstream one it is this gap, 2^−52; in the formal one (unit roundoff) it is half of it, 2^−53.", "sources": [ { "title": "Wikipedia, IEEE 754", "url": "https://en.wikipedia.org/wiki/IEEE_754", "own_work": false, "note": "describes binary64 as having 53 bits of precision and gives the binary64 spacing/precision facts: “The binary32 (single) and binary64 (double) formats are two of the most common formats used today.”", "fetched": [ "2026-09-09" ] }, { "title": "Double-precision floating-point format, Wikipedia", "url": "https://en.wikipedia.org/wiki/Double-precision_floating-point_format", "own_work": false, "note": "(§ IEEE 754 double-precision binary floating-point format: binary64, § Double-precision examples): “Significand precision: 53 bits (52 explicitly stored),” and the smallest number greater than 1 is 1.0000000000000002220…, so the spacing immediately above 1 is 2⁻⁵² ≈ 2.220446 × 10⁻¹⁶; “The maximum relative rounding error when rounding a number to the nearest representable one (the machine epsilon) is therefore 2⁻⁵³.” (exponent as typeset on the page)", "fetched": [ "2026-09-10" ] }, { "title": "Floating-point arithmetic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Floating-point_arithmetic", "own_work": false, "note": "read as raw wikitext: “The classical toy problem that illustrates the issue with decimals is 0.1 + 0.2 - 0.3, which evaluates to about 5.551e-17 in double precision instead of the expected 0, because as stated above in Representable numbers, conversion and rounding all three numbers cannot be represented by a finite binary expansion.”", "fetched": [ "2026-09-22" ] }, { "title": "Machine epsilon, Wikipedia", "url": "https://en.wikipedia.org/wiki/Machine_epsilon", "own_work": false, "note": "“There are two prevailing definitions, denoted here as rounding machine epsilon or the formal definition and interval machine epsilon or mainstream definition.”; “In the mainstream definition, machine epsilon is independent of rounding method, and is defined simply as the difference between 1 and the next larger floating point number.” Its table gives binary64 a rounding machine epsilon of 2⁻⁵³ ≈ 1.11e-16 and an interval machine epsilon of 2⁻⁵² ≈ 2.22e-16 (exponents as typeset on the page).", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-9.14", "url": "https://www.teclamairme.com/#r-9.14", "chapter": 9, "status": "confirmed", "claim": "With round-to-nearest, ties-to-even, binary64 evaluates fl(1+2^−53) to 1 because the exact value is halfway between 1 and 1+2^−52 and 1 has an even low significand bit; subtracting 1 therefore gives 0.", "sources": [ { "title": "Wikipedia, IEEE 754", "url": "https://en.wikipedia.org/wiki/IEEE_754", "own_work": false, "note": "describes round-to-nearest, ties-to-even and binary floating-point formats.", "fetched": [ "2026-09-09" ] }, { "title": "David Goldberg, What Every Computer Scientist Should Know About Floating-Point Arithmetic, ACM Computing Surveys 23(1), March 1991", "url": "https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html", "own_work": false, "note": "sets out the round-to-even rule and argues for it (Theorem 5), then adopts it — “Throughout the rest of this paper, round to even will be used” — and defines machine epsilon. It is the classic exposition, not the specification; the ties-to-even default itself is carried by the IEEE 754 entry above.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Own computation: 1+2^−53 is exactly the midpoint of 1 and 1+2^−52; the significand of 1 has an even trailing bit, so the rounded result is 1 and the subtraction is exactly zero." }, { "id": "r-9.15", "url": "https://www.teclamairme.com/#r-9.15", "chapter": 9, "status": "confirmed", "claim": "Kolmogorov complexity is the length of the shortest program in a fixed language that outputs a string; it is uncomputable, by an argument modeled on Berry’s paradox. Solomonoff published the idea in 1960 and described it more fully in 1964; Kolmogorov published it independently in 1965 and acknowledged Solomonoff’s priority once he became aware of that work; Chaitin’s Journal of the ACM paper appeared in 1966. Chaitin’s constant Ω is the halting probability of a universal machine, one for each encoding of programs.", "sources": [ { "title": "Wikipedia, Kolmogorov complexity", "url": "https://en.wikipedia.org/wiki/Kolmogorov_complexity", "own_work": false, "note": "defines the shortest-program measure and explains uncomputability through a construction ‘modeled on a self-referential construction used in Berry’s paradox’; says Solomonoff ‘published it in 1960’ in A Preliminary Report on a General Theory of Inductive Inference and ‘gave a more complete description in his 1964 publications’, that Kolmogorov ‘later independently published this theorem in Problems Inform. Transmission in 1965’, that Chaitin presents it in the Journal of the ACM, and that ‘When Kolmogorov became aware of Solomonoff’s work, he acknowledged Solomonoff’s priority’.", "fetched": [ "2026-09-09" ] }, { "title": "Algorithmic information theory, Wikipedia", "url": "https://en.wikipedia.org/wiki/Algorithmic_information_theory", "own_work": false, "note": "“Algorithmic information theory was later developed independently by Andrey Kolmogorov, in 1965 and Gregory Chaitin, around 1966.”; its references list Chaitin, “On the Length of Programs for Computing Finite Binary Sequences,” Journal of the ACM 13 (4): 547–569 (1966).", "fetched": [ "2026-09-22" ] }, { "title": "Chaitin’s constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Chaitin%27s_constant", "own_work": false, "note": "a real number that, “informally speaking, represents the probability that a randomly constructed program will halt”; “Although there are infinitely many halting probabilities, one for each (universal, see below) method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.16", "url": "https://www.teclamairme.com/#r-9.16", "chapter": 9, "status": "confirmed", "claim": "Rice’s theorem says every nontrivial semantic property of the partial function computed by a program is undecidable; Rice proved the theorem in his 1951 dissertation and published it in 1953.", "sources": [ { "title": "Wikipedia, Rice’s theorem", "url": "https://en.wikipedia.org/wiki/Rice%27s_theorem", "own_work": false, "note": "states the theorem for nontrivial semantic properties of programs and gives the 1951 dissertation/1953 publication history.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-9.17", "url": "https://www.teclamairme.com/#r-9.17", "chapter": 9, "status": "confirmed", "claim": "In combinatory logic, S=λxyz.xz(yz), K=λxy.x, and I=SKK; every lambda term can be represented using S and K, while Curry’s Y combinator satisfies Yf=f(Yf). Schönfinkel introduced the combinators in 1920 and published them in 1924; Curry rediscovered them at Princeton in 1927 and published Grundlagen der kombinatorischen Logik in 1930. Schönfinkel’s 1920 introduction was a talk to the Göttingen Mathematics Society.", "sources": [ { "title": "Wikipedia, Combinatory logic", "url": "https://en.wikipedia.org/wiki/Combinatory_logic", "own_work": false, "note": "gives the S, K, and I combinators, the S/K basis, and the fixed-point combinator construction; says the combinators ‘were introduced by Schönfinkel in 1920’ and published in 1924, and that Curry ‘rediscovered the combinators while working as an instructor at Princeton University in late 1927’, publishing Grundlagen der Kombinatorischen Logik in 1930.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Fixed-point combinator", "url": "https://en.wikipedia.org/wiki/Fixed-point_combinator", "own_work": false, "note": "gives Y = λf.(λx.f (x x)) (λx.f (x x)) as ‘Haskell Curry’s paradoxical combinator Y’, satisfying Y g = g (Y g).", "fetched": [ "2026-09-09" ] }, { "title": "Stephen Wolfram, Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel", "url": "https://writings.stephenwolfram.com/2020/12/where-did-combinators-come-from-hunting-the-story-of-moses-schonfinkel/", "own_work": false, "note": "“On Tuesday, December 7, 1920, the Göttingen Mathematics Society held its regular weekly meeting—at which a 32-year-old local mathematician named Moses Schönfinkel with no known previous mathematical publications gave a talk entitled ‘Elemente der Logik’ (‘Elements of Logic’).”; “Moses Schönfinkel described the result of his 1920 talk in an 11-page paper published in 1924”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.18", "url": "https://www.teclamairme.com/#r-9.18", "chapter": 9, "status": "confirmed", "claim": "Gödel numbering encodes symbols, formulas, and proofs as natural numbers so arithmetic can represent syntactic statements; Turing’s universal machine similarly encodes program descriptions on its tape. Under the numbering, Gödel’s sentence says that a certain calculation yields the Gödel number of an unprovable formula; ⌜φ⌝ is the standard notation for the numeral of φ’s code.", "sources": [ { "title": "Wikipedia, Gödel numbering", "url": "https://en.wikipedia.org/wiki/G%C3%B6del_numbering", "own_work": false, "note": "describes the encoding of symbols, formulas, and proofs by natural numbers and its role in incompleteness; “Gödel used a system based on prime factorization..”", "fetched": [ "2026-09-09" ] }, { "title": "A. M. Turing, On Computable Numbers (interactive edition)", "url": "https://biomorphbuilder.com/turing-paper/", "own_work": false, "note": "describes machine descriptions and universal-machine computation.", "fetched": [ "2026-09-09" ] }, { "title": "Gödel’s incompleteness theorems, Wikipedia", "url": "https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems", "own_work": false, "note": "“The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement.”", "fetched": [ "2026-09-22" ] }, { "title": "Diagonal lemma, Wikipedia", "url": "https://en.wikipedia.org/wiki/Diagonal_lemma", "own_work": false, "note": "“For n̄, the standard numeral of n (i.e. 0̄ =df 0 and n+1̄ =df S(n̄)), let ⌜φ⌝ be the standard numeral of the code of φ.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-9.19", "url": "https://www.teclamairme.com/#r-9.19", "chapter": 9, "status": "confirmed", "claim": "The displayed arithmetic checks are correct: (λf.λx.x) S 0 reduces in two beta steps to 0; 1+6+21+107=135, so 47,176,870 + 61 + 135 = 47,177,066; and the three denominator terms are positive step counts while [P⊆NP]=1 and the binary64 rounding numerator is 0.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Direct symbolic/arithmetic check: (λf.λx.x) S 0 → (λx.x) 0 → 0; 1+6+21+107=135 and 47,176,870 + 61 + 135 = 47,177,066; the cited Busy Beaver values are positive; P⊆NP is a true inclusion; and the midpoint tie calculation in binary64 gives fl(1+2^−53)−1=0.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-9.20", "url": "https://www.teclamairme.com/#r-9.20", "chapter": 9, "status": "confirmed", "claim": "The BB(5) proof enumerates 181,385,789 five-state, two-symbol Turing machines and decides halting for each; the July 2024 announcement describes that as the search space reduced by symmetry to about 180 million machines. The unreduced space is 21^10 = 16,679,880,978,201 machines, about 17 trillion, since each of the ten state-and-symbol pairs takes one of twenty transitions or halts.", "sources": [ { "title": "The bbchallenge Collaboration, Determination of the fifth Busy Beaver value", "url": "https://arxiv.org/abs/2509.12337", "own_work": false, "note": "abstract explicitly says the proof enumerates 181,385,789 five-state machines.", "fetched": [ "2026-09-09" ] }, { "title": "Busy Beaver Challenge, July 2024 announcement", "url": "https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237", "own_work": false, "note": "says the space was reduced to approximately 180 million machines using symmetries.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Own arithmetic: five states and two symbols give ten table entries, each of which writes one of two symbols, moves one of two ways and enters one of five states, or halts, so 21^10 = 16,679,880,978,201. Both counts are correct and count different things; the number the chapter gives is the one the proof actually enumerates." }, { "id": "r-9.21", "url": "https://www.teclamairme.com/#r-9.21", "chapter": 9, "status": "confirmed", "claim": "David Hilbert and Wilhelm Ackermann posed the Entscheidungsproblem in 1928; it asks for an algorithm that answers yes or no according to whether an input statement is universally valid, that is, valid in every structure. In 1936 Alonzo Church and Alan Turing published independent papers showing that a general solution is impossible.", "sources": [ { "title": "Entscheidungsproblem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Entscheidungsproblem", "own_work": false, "note": "attributes the problem to ‘David Hilbert and Wilhelm Ackermann in 1928’; states that it asks for ‘an algorithm that considers an input statement and answers yes or no according to whether it is universally valid, i.e., valid in every structure’; and that ‘In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible’; and ‘Turing reduced the question of the existence of an ‘algorithm’ or ‘general method’ able to solve the Entscheidungsproblem to the question of the existence of a ‘general method’ that decides whether any given Turing machine halts or not (the halting problem).’", "fetched": [ "2026-09-10" ] } ], "editorial_note": "The dates of the two 1936 papers, and Turing’s appendix showing the two answers describe the same class, are carried separately by 9.3." }, { "id": "r-9.22", "url": "https://www.teclamairme.com/#r-9.22", "chapter": 9, "status": "confirmed", "claim": "The first IEEE standard for floating-point arithmetic, IEEE 754-1985, was published in 1985, and round to nearest, ties to even is its default rounding for binary floating point. Almost all modern machines follow it, and on most systems the double type is its binary64 format.", "sources": [ { "title": "IEEE 754, Wikipedia", "url": "https://en.wikipedia.org/wiki/IEEE_754", "own_work": false, "note": "‘The first IEEE standard for floating-point arithmetic, IEEE 754-1985, was published in 1985’, and ‘Round to nearest, ties to even’ is given as the default for binary floating point.", "fetched": [ "2026-09-10" ] }, { "title": "Floating-point arithmetic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Floating-point_arithmetic", "own_work": false, "note": "“This first standard is followed by almost all modern machines.”", "fetched": [ "2026-09-22" ] }, { "title": "Double-precision floating-point format, Wikipedia", "url": "https://en.wikipedia.org/wiki/Double-precision_floating-point_format", "own_work": false, "note": "“Double precision is not required by the standards (except by the optional annex F of C99, covering IEEE 754 arithmetic), but on most systems, the double type corresponds to double precision.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added in the V14 prose pass: the chapter’s 1985 date and the card line ‘the double of every mainstream language since the 1985 standard’ previously stood on 9.13, whose claim covers only binary64 precision and spacing and carries no date." }, { "id": "r-9.23", "url": "https://www.teclamairme.com/#r-9.23", "chapter": 9, "status": "confirmed", "claim": "The chapter epigraph is the opening sentence of Turing’s On Computable Numbers, with an Application to the Entscheidungsproblem, received 28 May 1936 and read 12 November 1936.", "sources": [ { "title": "A. M. Turing, On Computable Numbers (full text)", "url": "https://biomorphbuilder.com/turing-paper/", "own_work": false, "note": "read as raw HTML: “Received 28 May, 1936 — Read 12 November, 1936. Proceedings of the London Mathematical Society, s2-42 (1937), pp. 230–265,” and the paper opens “The ‘computable’ numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means,” which is the epigraph word for word.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. The footer dates the paper 1936, which is the year it was received and read; the volume it appeared in is dated 1937, as r-9.3 already records." }, { "id": "r-9.24", "url": "https://www.teclamairme.com/#r-9.24", "chapter": 9, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Goedel’s 1931 proof introduced the numbering; Turing’s universal machine of 1936.", "sources": [ { "title": "Gödel numbering, Wikipedia", "url": "https://en.wikipedia.org/wiki/G%C3%B6del_numbering", "own_work": false, "note": "read raw: “Since the publishing of Gödel’s paper in 1931, the term “Gödel numbering” or “Gödel code” has been used to refer to more general assignments of natural numbers to mathematical objects..”", "fetched": [ "2026-09-18" ] }, { "title": "Universal Turing machine, Wikipedia", "url": "https://en.wikipedia.org/wiki/Universal_Turing_machine", "own_work": false, "note": "read raw: “Turing introduced the idea of such a machine in 1936–1937..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-9.25", "url": "https://www.teclamairme.com/#r-9.25", "chapter": 9, "status": "confirmed", "claim": "States the recursive definition of Ackermann’s function and that it is total and computable yet not primitive recursive. Notes Ackermann’s 1928 three-argument original, Péter’s two-argument form, A(3, 3) = 61, and that A(4, 2) has 19,729 decimal digits.", "sources": [ { "title": "Ackermann function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ackermann_function", "own_work": false, "note": "gives the Ackermann-Peter recurrence and states the function is computable but grows faster than any primitive recursive function.", "fetched": [ "2026-09-22" ] }, { "title": "Wilhelm Ackermann, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Ackermann/", "own_work": false, "note": "carries the two-argument simplification: “A(x, y, z) was simplified to a function P(x, y) of two variables by Rozsa Peter whose initial condition was simplified by Raphael Robinson.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.1", "url": "https://www.teclamairme.com/#r-10.1", "chapter": 10, "status": "confirmed", "claim": "The Nine Chapters used an array-based elimination method for systems of simultaneous linear equations, and Gauss later used the method in orbit determination.", "sources": [ { "title": "MacTutor, Nine Chapters on the Mathematical Art", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/", "own_work": false, "note": "Chapter 8 describes systems of up to six simultaneous linear equations, an augmented array, and reduction to triangular form by elementary column operations.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "says Gaussian elimination appeared in the Nine Chapters and was used by Gauss for the orbit of Pallas; “The first to use the term ‘matrix’ was Sylvester in 1850..”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.2", "url": "https://www.teclamairme.com/#r-10.2", "chapter": 10, "status": "confirmed", "claim": "Sylvester introduced the word matrix in 1850; Cayley published the first matrix inverse in 1853 and the first abstract definition of a matrix, with addition, multiplication, scalar multiplication and inverses, in his memoir of 1858.", "sources": [ { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "“The first to use the term ‘matrix’ was Sylvester in 1850”; “by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix”; and the 1858 memoir “is remarkable for containing the first abstract definition of a matrix,” giving addition, multiplication, scalar multiplication and inverses.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, James Joseph Sylvester", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Sylvester/", "own_work": false, "note": "links Sylvester’s first use of the word matrix to his 1850 paper.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.3", "url": "https://www.teclamairme.com/#r-10.3", "chapter": 10, "status": "confirmed", "claim": "For a linear map, the determinant is the area/volume scale factor; a zero determinant means the map is singular and has no inverse.", "sources": [ { "title": "Wikipedia, Determinant", "url": "https://en.wikipedia.org/wiki/Determinant", "own_work": false, "note": "describes the determinant as the signed scale factor for volume and states that a matrix is invertible exactly when its determinant is nonzero.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For A = [[1, 2], [2, 4]], det A = 1·4 − 2·2 = 0; the second row is twice the first, so the image is the line spanned by (1, 2), and no inverse can exist.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-10.4", "url": "https://www.teclamairme.com/#r-10.4", "chapter": 10, "status": "confirmed", "claim": "For A = [[1, 2], [2, 4]], the displayed determinant, trace, Frobenius norm, eigenvalues, rank, nullity and Cayley-Hamilton substitution have the stated values.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Direct calculation: det A = 0; tr A = 5; AᵀA has eigenvalues 25 and 0, so the Frobenius norm is 5; the characteristic polynomial is λ²−5λ, giving eigenvalues 5 and 0; im A = span(1, 2) has dimension 1 and ker A = span(2, −1) has dimension 1; A² = [[5, 10], [10, 20]] = 5A.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-10.5", "url": "https://www.teclamairme.com/#r-10.5", "chapter": 10, "status": "confirmed", "claim": "The Kronecker delta is 1 when its indices agree and 0 otherwise; the identity matrix has these delta entries; the notation is traced to Kronecker’s paper on bilinear forms of 1868. Sources differ on the year: the Encyclopedia of Mathematics dates Kronecker’s first use of the symbol to 1866, while the paper the notation is usually traced to was printed in 1868. Corrections welcome.", "sources": [ { "title": "Wikipedia, Kronecker delta", "url": "https://en.wikipedia.org/wiki/Kronecker_delta", "own_work": false, "note": "defines δᵢⱼ as 1 for i=j and 0 otherwise and gives the identity matrix as the matrix of Kronecker deltas, and carries no history section, so the date below rests on the paper itself.", "fetched": [ "2026-09-09" ] }, { "title": "L. Kronecker, Ueber bilineare Formen, Journal für die reine und angewandte Mathematik 68 (1868), 273–285", "url": "https://eudml.org/doc/148042", "own_work": false, "note": "the European Digital Mathematics Library record for the paper the delta notation is standardly traced to, which dates it 1868 (volume 68, pages 273–285).", "fetched": [ "2026-09-09" ] }, { "title": "Kronecker symbol, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Kronecker_symbol", "own_work": false, "note": "“The Kronecker symbol was first used by L. Kronecker (1866).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.6", "url": "https://www.teclamairme.com/#r-10.6", "chapter": 10, "status": "confirmed", "claim": "The trace is the sum of diagonal entries and, for a square matrix, the sum of the eigenvalues counted with multiplicity; the English word translates the German Spur, which survives in the older abbreviation sp A.", "sources": [ { "title": "Wikipedia, Trace (linear algebra)", "url": "https://en.wikipedia.org/wiki/Trace_(linear_algebra)", "own_work": false, "note": "defines trace as the sum of diagonal entries and states that it equals the sum of eigenvalues with algebraic multiplicity.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics, T", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/t/", "own_work": false, "note": "gives trace as a translation of the German die Spur, used in H. L. Brose’s 1922 translation of Weyl’s Raum, Zeit, Materie, and records Aitken’s “the spur or trace of A. We shall denote it by sp A.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.7", "url": "https://www.teclamairme.com/#r-10.7", "chapter": 10, "status": "confirmed", "claim": "The Frobenius norm is the square root of the sum of squared entries and also the square root of the sum of squared singular values.", "sources": [ { "title": "Wikipedia, Matrix norm", "url": "https://en.wikipedia.org/wiki/Matrix_norm", "own_work": false, "note": "gives the Frobenius norm as the square root of the sum of squared entries and as the Euclidean norm of the singular-value vector.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.8", "url": "https://www.teclamairme.com/#r-10.8", "chapter": 10, "status": "confirmed", "claim": "The eigenvalues of A are the roots of det(λI−A), and for this symmetric matrix the eigenvectors for 5 and 0 are perpendicular; the word eigen is Hilbert’s, from 1904, and the idea is older: Euler studied the rotation of a rigid body and found the principal axes, and Lagrange recognised them as the eigenvectors of the inertia matrix.", "sources": [ { "title": "Wikipedia, Eigenvalues and eigenvectors", "url": "https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors", "own_work": false, "note": "defines eigenvalues through Av=λv and the characteristic polynomial det(A−λI). Re-read raw on 17 September 2026 for the history: “In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.” It does not state the symmetric-matrix orthogonality property — it says “The eigenvectors need not be orthogonal to one another” — so that half is carried by the entry below; an eigenvalue is “the multiplying factor λ (possibly a negative or complex number),” and there is “a direct correspondence between linear transformations from the vector space into itself and n × n square matrices.”", "fetched": [ "2026-09-09" ] }, { "title": "Symmetric matrix, Wikipedia", "url": "https://en.wikipedia.org/wiki/Symmetric_matrix", "own_work": false, "note": "(§ Real symmetric matrices): “For every real symmetric matrix A, there exists a real orthogonal matrix Q such that D = QᵀAQ is a diagonal matrix,” and derives that eigenvectors for distinct eigenvalues satisfy ⟨x,y⟩ = 0.", "fetched": [ "2026-09-10" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "A(1, 2)ᵀ = 5(1, 2)ᵀ and A(2, −1)ᵀ = 0; their dot product is 2−2=0.", "fetched": [] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics, E", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/", "own_work": false, "note": "David Hilbert introduced Eigenfunktion and Eigenwert in 1904, in “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen”; Eigenvektor follows in Courant and Hilbert, 1924.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.9", "url": "https://www.teclamairme.com/#r-10.9", "chapter": 10, "status": "confirmed", "claim": "A nilpotent matrix has a power equal to zero; N=[[0, 1], [0, 0]] is nonzero with N²=0, and Jordan’s 1870 canonical-form work decomposes matrices using a nilpotent part. Jordan first stated his decomposition theorem in 1870; every complex matrix is D + N with D diagonalizable, N nilpotent and DN = ND.", "sources": [ { "title": "Wikipedia, Nilpotent matrix", "url": "https://en.wikipedia.org/wiki/Nilpotent_matrix", "own_work": false, "note": "defines nilpotence by Aᵏ=0 for some positive k and gives the 2×2 Jordan nilpotent block as the standard example.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "dates Jordan’s canonical form to 1870: “In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan.” The nilpotent part is on the Jordan normal form line.", "fetched": [ "2026-09-09" ] }, { "title": "Jordan normal form, Wikipedia", "url": "https://en.wikipedia.org/wiki/Jordan_normal_form", "own_work": false, "note": "“The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.”; “Any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.10", "url": "https://www.teclamairme.com/#r-10.10", "chapter": 10, "status": "confirmed", "claim": "Rank is the dimension of the image, nullity the dimension of the kernel, and rank plus nullity equals the dimension of the domain. Sources differ on the year of the word rank: MacTutor places Frobenius’s rank in his matrix paper of 1878, while Miller dates the word Rang to his paper in J. reine angew. Math. 86 (1879). Corrections welcome.", "sources": [ { "title": "MathWorld, Rank-Nullity Theorem", "url": "https://mathworld.wolfram.com/Rank-NullityTheorem.html", "own_work": false, "note": "states dim(V)=dim(Ker(T))+dim(Im(T)) and identifies these dimensions as nullity and rank.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "says Sylvester defined nullity in 1884 and Frobenius defined rank in his 1878 matrix work.", "fetched": [ "2026-09-09" ] }, { "title": "Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (R), MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/r/", "own_work": false, "note": "“RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper ‘Uber homogene totale Differentialgleichungen,’ J. reine angew. Math. Vol. 86 (1879) p.1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.11", "url": "https://www.teclamairme.com/#r-10.11", "chapter": 10, "status": "confirmed", "claim": "The Cayley-Hamilton theorem says every square matrix satisfies its characteristic polynomial; Hamilton proved a case for quaternions in 1853, Cayley handled the 2×2 case and stated the general one in 1858, and Frobenius proved the general result in 1878.", "sources": [ { "title": "MathWorld, Cayley-Hamilton Theorem", "url": "https://mathworld.wolfram.com/Cayley-HamiltonTheorem.html", "own_work": false, "note": "states A is annihilated by det(xI−A) and works out the 2×2 identity.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "dates Cayley’s 2×2/3×3 result and Frobenius’s general proof to 1878, and notes Hamilton’s special quaternion case.", "fetched": [ "2026-09-09" ] }, { "title": "Cayley–Hamilton theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem", "own_work": false, "note": "“A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions.”; “Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case.”; “The general case was first proved by Ferdinand Frobenius in 1878.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.12", "url": "https://www.teclamairme.com/#r-10.12", "chapter": 10, "status": "confirmed", "claim": "The identity matrix acts as a multiplicative identity; an inverse exists exactly for a nonsingular matrix; transposition swaps rows and columns, and this A is symmetric. The identity matrix has ones on the diagonal and zeros elsewhere, written δᵢⱼ; a real symmetric matrix’s eigenvectors for distinct eigenvalues are perpendicular.", "sources": [ { "title": "Wikipedia, Invertible matrix", "url": "https://en.wikipedia.org/wiki/Invertible_matrix", "own_work": false, "note": "states a square matrix is invertible exactly when its determinant is nonzero and defines the inverse equation AA⁻¹=I.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "I·A=A, Aᵀ=A because A is symmetric, and A has det A=0, so it has no inverse.", "fetched": [] }, { "title": "Identity matrix, Wikipedia", "url": "https://en.wikipedia.org/wiki/Identity_matrix", "own_work": false, "note": "“the identity matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere”; “When A is an m × n matrix, it is a property of matrix multiplication that I_m A = A I_n = A.”; “The identity matrix can also be written using the Kronecker delta notation: (I_n)_ij = δ_ij.”", "fetched": [ "2026-09-22" ] }, { "title": "Transpose, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transpose", "own_work": false, "note": "“transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, called the transpose of A.”", "fetched": [ "2026-09-22" ] }, { "title": "Symmetric matrix, Wikipedia", "url": "https://en.wikipedia.org/wiki/Symmetric_matrix", "own_work": false, "note": "“To see orthogonality, suppose x and y are eigenvectors corresponding to distinct eigenvalues λ₁, λ₂.” … “Since λ₁ and λ₂ are distinct, we have ⟨x, y⟩ = 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.13", "url": "https://www.teclamairme.com/#r-10.13", "chapter": 10, "status": "confirmed", "claim": "Singular values are square roots of eigenvalues of AᵀA; the SVD factors a matrix as UΣVᵀ, and for this rank-one A the sole nonzero singular value is 5 and equals the Frobenius norm; Beltrami and Jordan reached the factorization independently in 1873 and 1874, and Golub and Kahan gave the first stable algorithm for it in 1965.", "sources": [ { "title": "Wikipedia, Singular value decomposition", "url": "https://en.wikipedia.org/wiki/Singular_value_decomposition", "own_work": false, "note": "defines singular values through AᵀA and gives A=UΣVᵀ, with applications including least squares, principal component analysis and image compression.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "AᵀA = [[5, 10], [10, 20]] has eigenvalues 25 and 0, so the singular values are 5 and 0; the Frobenius norm is therefore 5.", "fetched": [] }, { "title": "Nicholas J. Higham, What Is the Singular Value Decomposition?", "url": "https://nhigham.com/2020/10/13/what-is-the-singular-value-decomposition/", "own_work": false, "note": "“The SVD was introduced independently by Beltrami in 1873 and Jordan in 1874,” and credits Golub with the first reliable algorithms for computing it.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Several names are in circulation for one idea. G. W. Stewart’s history of the decomposition (SIAM Review 35, 1993) treats Beltrami in 1873 and Jordan in 1874 as the independent first derivations, with Sylvester, Schmidt and Weyl arriving at it again before the numerical work of the 1960s. The card names the first two and the 1965 algorithm, and leaves the rediscoveries out." }, { "id": "r-10.14", "url": "https://www.teclamairme.com/#r-10.14", "chapter": 10, "status": "confirmed", "claim": "Cramer’s rule expresses each unknown using a determinant ratio, and it cannot give a unique solution when det A=0; Maclaurin was teaching it by ~1729 and Cramer published it in 1750. See 10.23. Corrected 11 September 2026: this entry previously gave Maclaurin 1748, which is the posthumous printing of his Treatise of Algebra, two years after his death.", "sources": [ { "title": "MacTutor, Gabriel Cramer", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Cramer.pdf", "own_work": false, "note": "discusses Cramer’s 1750 rule and Maclaurin’s equivalent idea in the 1748 Treatise of Algebra.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Cramer’s rule", "url": "https://en.wikipedia.org/wiki/Cramer%27s_rule", "own_work": false, "note": "gives xᵢ=det(Aᵢ)/det(A) and states the rule applies when the determinant is nonzero.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.15", "url": "https://www.teclamairme.com/#r-10.15", "chapter": 10, "status": "confirmed", "claim": "A vector space supports addition and scalar multiplication, a basis is a linearly independent spanning set, and dimension is the common cardinality of its bases; ℝ² has dimension 2 and polynomial space has countably infinite dimension; Peano gave the first axiomatic definition of a vector space in 1888. A spanning set is a basis exactly when it is minimal.", "sources": [ { "title": "Wikipedia, Vector space", "url": "https://en.wikipedia.org/wiki/Vector_space", "own_work": false, "note": "gives the vector-space axioms, basis definition and dimension, including the standard basis of R².", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "The monomials 1,x,x²,… form a countable basis of the polynomial vector space over a field, while (1, 0),(0, 1) is a basis of R².", "fetched": [] }, { "title": "MacTutor, Abstract linear spaces", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Abstract_linear_spaces/", "own_work": false, "note": "“The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888,” the Calcolo geometrico, which Peano wrote after Grassmann’s 1844 Ausdehnungslehre.", "fetched": [ "2026-09-09" ] }, { "title": "Basis (linear algebra), Wikipedia", "url": "https://en.wikipedia.org/wiki/Basis_(linear_algebra)", "own_work": false, "note": "“A generating set S is a basis of V if and only if it is minimal, that is, no proper subset of S is also a generating set of V.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.16", "url": "https://www.teclamairme.com/#r-10.16", "chapter": 10, "status": "confirmed", "claim": "The outer product u⊗v has entries uᵢvⱼ; A=(1, 2)⊗(1, 2), and the three-dimensional cross product vanishes for parallel vectors and has parallelogram-area magnitude; Gibbs introduced the dot and the cross in 1881, and Heaviside split the quaternion product the same way independently. The calculus of higher-rank tensors is Ricci-Curbastro’s, developed from 1887 to 1896 and set out with his pupil Levi-Civita in 1900. Gibbs’s 1881 notes were privately printed; the name dot product appears in Gibbs and Wilson’s Vector Analysis of 1901, and for two nonzero vectors the dot product is zero exactly when they are perpendicular. A matrix has rank one exactly when it is a nonzero outer product of two vectors.", "sources": [ { "title": "Wikipedia, Outer product", "url": "https://en.wikipedia.org/wiki/Outer_product", "own_work": false, "note": "defines the outer product as the matrix with entries uᵢvⱼ and notes that rank-one matrices arise this way.", "fetched": [ "2026-09-09" ] }, { "title": "Ricci calculus, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ricci_calculus", "own_work": false, "note": "read as raw HTML: the absolute differential calculus was “developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900,” the paper being Ricci and Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Mathematische Annalen 54 (March 1900), 125–201, doi 10.1007/BF01454201.", "fetched": [ "2026-09-17" ] }, { "title": "Wikipedia, Cross product", "url": "https://en.wikipedia.org/wiki/Cross_product", "own_work": false, "note": "defines the cross product in three dimensions, gives its magnitude as parallelogram area, states it is zero for parallel vectors, and dates the notation a · b and a × b to Gibbs in 1881, with Heaviside independently.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "(1, 2)⊗(1, 2)=[[1, 2], [2, 4]], and (1, 2, 0)×(2, 4, 0)=(0, 0, 0).", "fetched": [] }, { "title": "Josiah Willard Gibbs, Wikipedia", "url": "https://en.wikipedia.org/wiki/Josiah_Willard_Gibbs", "own_work": false, "note": "“Gibbs’s lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.”", "fetched": [ "2026-09-22" ] }, { "title": "Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (D), MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/d/", "own_work": false, "note": "DOT PRODUCT, found in 1901 in Gibbs and Wilson’s Vector Analysis: “A·B This is read A dot B and therefore may often be called the dot product instead of the direct product.”", "fetched": [ "2026-09-22" ] }, { "title": "Dot product, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dot_product", "own_work": false, "note": "“Two non-zero vectors a and b are orthogonal if and only if a ⋅ b = 0”", "fetched": [ "2026-09-22" ] }, { "title": "Rank (linear algebra), Wikipedia", "url": "https://en.wikipedia.org/wiki/Rank_(linear_algebra)", "own_work": false, "note": "“a matrix is defined to have rank 1 if and only if it can be written as a nonzero product c ⋅ r of a column vector c and a row vector r”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.17", "url": "https://www.teclamairme.com/#r-10.17", "chapter": 10, "status": "confirmed", "claim": "The characteristic polynomial p_A(λ)=det(λI−A) has eigenvalues as roots; for a 2×2 matrix its coefficients recover trace and determinant, and Cauchy used the polynomial in 1829 and named the characteristic equation in 1840.", "sources": [ { "title": "Wikipedia, Characteristic polynomial", "url": "https://en.wikipedia.org/wiki/Characteristic_polynomial", "own_work": false, "note": "defines det(λI−A), identifies its roots as eigenvalues, and gives the 2×2 coefficient relation λ²−tr(A)λ+det(A).", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "records the historical development of determinant and matrix theory, including Cauchy’s work on determinants.", "fetched": [ "2026-09-09" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics, E", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/e/", "own_work": false, "note": "Cauchy’s 1829 paper on symmetric determinants is “Sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des mouvements des planètes,” and he introduced the term l’équation caractéristique in the Exercices d’analyse et de physique mathématique of 1840.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.18", "url": "https://www.teclamairme.com/#r-10.18", "chapter": 10, "status": "confirmed", "claim": "A linear map is written f:V→W, composition means doing f then g, and matrix multiplication represents composition; A∘A=A²=5A here.", "sources": [ { "title": "Wikipedia, Linear map", "url": "https://en.wikipedia.org/wiki/Linear_map", "own_work": false, "note": "defines linear maps between vector spaces and their composition.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "“Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses” (1858); earlier Gauss “describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra),” for the arrays of quadratic forms, not linear maps.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "A²=[[5, 10], [10, 20]]=5A, so the composition A∘A is represented by 5A.", "fetched": [] } ], "editorial_note": "Cayley’s 1858 memoir is where matrix multiplication is defined so that it represents composition. The barred arrow ↦ is Bourbaki’s, but no reference work consulted dates it: the accounts that give a date put it in the Théorie des ensembles of the early 1960s rather than in Bourbaki’s first decade, so the card gives the arrow neither an author nor a year." }, { "id": "r-10.19", "url": "https://www.teclamairme.com/#r-10.19", "chapter": 10, "status": "judgement", "claim": "Two matrix-valued zeros in the numerator are evaluated in their own matrix algebra and then contribute the integer 0 to the scalar factor. Chapter ᚁ(B) applies the same editorial rule to its six laws: each zero-valued law contributes the plain number 0 to the displayed sum.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "N² and A²−5A are the zero matrices. The chapter’s displayed scalar factor treats each matrix equality as contributing its zero value to the surrounding numerical sum; this is an editorial evaluation rule, not a theorem about adding unlike mathematical types.", "fetched": [] } ], "editorial_note": "This is a declared convention of the book rather than a theorem, and the disclaimer page states it: every factor is computed inside its own framework and hands back an ordinary finite real before it joins the product. Adding a zero matrix to a zero integer is not an operation supplied by the frameworks used here, so the chapter says what it is doing where its first matrix-valued zeros appear; the later chapters whose zeros are vectors, forms, operators or numbers of some other algebra rest on the same rule." }, { "id": "r-10.20", "url": "https://www.teclamairme.com/#r-10.20", "chapter": 10, "status": "confirmed", "claim": "The determinant appears in Seki’s work of 1683 and in Leibniz’s letter to de l’Hôpital of 1693; Gauss introduced the word determinant in 1801 for a different object, and Cauchy used it in its modern sense in 1812.", "sources": [ { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "“In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables”; “In 1693 Leibniz wrote to de l’Hôpital” giving the condition for a system to have a solution; and “It was Cauchy in 1812 who used ‘determinant’ in its modern sense,” Gauss having introduced the term in 1801 while discussing quadratic forms.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-10.21", "url": "https://www.teclamairme.com/#r-10.21", "chapter": 10, "status": "confirmed", "claim": "The word matrix is Latin for womb, and Sylvester introduced it into mathematics in 1850 for the oblong array out of which systems of determinants are formed.", "sources": [ { "title": "Wiktionary, matrix", "url": "https://en.wiktionary.org/wiki/matrix", "own_work": false, "note": "the Latin entry glosses matrix as “uterus, womb,” then “dam (non-human female animal kept for breeding)” and “source, origin,” formed from mater; the English noun comes “from Old French matrice (“pregnant animal”), or from Latin mātrīx (“dam, womb”),” and its English sense “The womb” is marked “now rare.”", "fetched": [ "2026-09-10" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics, M", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/m/", "own_work": false, "note": "“The term MATRIX was introduced into mathematics by James Joseph Sylvester (1814-1897) in 1850. Matrix was a long-established word with the meaning of ‘the place from which something else originates.’ For Sylvester the ‘something else’ was a determinant of some description,” quoting his 1850 Philosophical Magazine paper: an oblong arrangement of m lines and n columns “will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-10.22", "url": "https://www.teclamairme.com/#r-10.22", "chapter": 10, "status": "confirmed", "claim": "Chapter 8 of the Nine Chapters, Rectangular Arrays, gives eighteen problems solved by column elimination on an array of coefficients; most historians date the text to about 200 BC, with 206 BC to 50 AD covering its different parts; and Gauss devised his notation for symmetric elimination in 1810, having obtained six linear equations in six unknowns from observations of the asteroid Pallas taken between 1803 and 1809.", "sources": [ { "title": "MacTutor, Nine Chapters on the Mathematical Art", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/", "own_work": false, "note": "“Most historians think that the text originated around 200 BC after the burning of the books by Shih Huang Ti,” while “others give dates between 100 BC and 50 AD,” the page settling on “206 BC to 50 AD” for its different parts; chapter 8 gives “18 problems which reduce to solving systems of simultaneous linear equations,” solved with “the augmented matrix of coefficients,” which “is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.”", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Gaussian elimination", "url": "https://en.wikipedia.org/wiki/Gaussian_elimination", "own_work": false, "note": "names the source as “Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art,” where “its use is illustrated in eighteen problems, with two to five equations,” and states that “Carl Friedrich Gauss in 1810 devised a notation for symmetric elimination that was adopted in the 19th century by professional hand computers to solve the normal equations of least-squares problems.”", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor, Matrices and determinants", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/", "own_work": false, "note": "“Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix”; the same page gives no year for that work, which is why the 1810 date is carried by the entry above.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-10.23", "url": "https://www.teclamairme.com/#r-10.23", "chapter": 10, "status": "confirmed", "claim": "Colin Maclaurin was teaching the rule now called Cramer’s rule to his students by 1729, more than twenty years before Cramer published it in 1750; the Treatise of Algebra that carries it was published in 1748, two years after Maclaurin’s death.", "sources": [ { "title": "Bruce A. Hedman, An Earlier Date for “Cramer’s Rule,” Historia Mathematica 26 (1999) 365-368", "url": "http://professorhedman.com/Cramers.Rule.pdf", "own_work": false, "note": "read in full here. Abstract: “Carl B. Boyer has shown that Colin Maclaurin published ‘Cramer’s Rule’ two years before Gabriel Cramer and conjectured that Maclaurin knew the technique as early as 1729. This note establishes Boyer’s conjecture through a heretofore unpublished manuscript in the Edinburgh University Library.” The manuscript is a handwritten copy of the Treatise of Algebra made by John Russell and dated 1729 (Edinburgh University Library, shelfmark Dc. 3.66); its twelfth chapter carries the same form of the rule as the 1748 edition, and Hedman concludes that it “provides conclusive evidence that Maclaurin was teaching his students ‘Cramer’s Rule’ over 20 years before Cramer published it.” Hedman also records that Patrick Murdoch prepared the Treatise for posthumous publication in 1748.", "fetched": [ "2026-09-11" ] }, { "title": "MacTutor, Colin Maclaurin", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Maclaurin/", "own_work": false, "note": "gives Maclaurin’s dates as February 1698 to 14 June 1746 and states that “Maclaurin’s Treatise on algebra was published in 1748, two years after his death.”", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Added 11 September 2026. The page previously said “Cramer published it in 1750 and Maclaurin had an equivalent rule in 1748,” which dates Maclaurin two years after his own death and names the later man first. Under D1 the prose now takes the earliest defensible year and the first name: Maclaurin, ~1729." }, { "id": "r-10.24", "url": "https://www.teclamairme.com/#r-10.24", "chapter": 10, "status": "confirmed", "claim": "An orthogonal projection is a linear map P with P² = P whose kernel is orthogonal to its range, equivalently a self-adjoint idempotent; it sends each vector to the nearest point of the subspace it projects onto, so the error is perpendicular to that subspace. For this chapter’s A = [[1, 2], [2, 4]], A/5 is the orthogonal projection onto the line y = 2x.", "sources": [ { "title": "Projection (linear algebra), Wikipedia", "url": "https://en.wikipedia.org/wiki/Projection_(linear_algebra)", "own_work": false, "note": "read as raw HTML: defines a projection as a linear map with P² = P, and an orthogonal projection as one whose range and kernel are orthogonal complements, equivalently a projection that is self-adjoint; gives the nearest-point and least-squares readings.", "fetched": [ "2026-09-17" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For A = [[1, 2], [2, 4]] and u = (1, 2): uuᵀ = A and |u|² = 5, so A/5 = uuᵀ/|u|² is symmetric and (A/5)² = A²/25 = 5A/25 = A/5; its range is the line y = 2x, spanned by (1, 2), and its kernel, spanned by (2, −1), is perpendicular to it. So A/5 is the orthogonal projection onto y = 2x, and A = 5·(A/5) has determinant 0, rank 1 and eigenvalues 5 and 0.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-10.25", "url": "https://www.teclamairme.com/#r-10.25", "chapter": 10, "status": "confirmed", "claim": "The spectral theorem for a real symmetric matrix says it is diagonalizable by an orthogonal matrix: there is an orthonormal basis of eigenvectors and the eigenvalues are real. Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, and John von Neumann generalized it to operators on a Hilbert space. Spectral theory underlies quantum measurement: it attaches a probability measure to an observable’s values in any state.", "sources": [ { "title": "Spectral theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Spectral_theorem", "own_work": false, "note": "read as raw HTML: “Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, i.e., that every real, symmetric matrix is diagonalizable,” and “The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory.”", "fetched": [ "2026-09-17" ] }, { "title": "Spectral Theorem, MathWorld", "url": "https://mathworld.wolfram.com/SpectralTheorem.html", "own_work": false, "note": "the statement for a normal bounded operator on a Hilbert space, via a unique resolution of the identity on the Borel subsets of its spectrum.", "fetched": [ "2026-09-17" ] }, { "title": "Mathematical formulation of quantum mechanics, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics", "own_work": false, "note": "“By spectral theory, we can associate a probability measure to the values of A in any state ψ.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.26", "url": "https://www.teclamairme.com/#r-10.26", "chapter": 10, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Sylvester’s ‘womb of a common parent’ definition of a matrix, in ‘On the relation between the minor determinants of linearly equivalent quadratic functions’, 1851.", "sources": [ { "title": "The Collected Mathematical Papers of James Joseph Sylvester, vol. I (Cambridge, 1904), Internet Archive full text", "url": "https://archive.org/download/collectedmathem01sylvrich/collectedmathem01sylvrich_djvu.txt", "own_work": false, "note": "read raw: “37. ON THE RELATION BETWEEN THE MINOR DETERMINANTS OF LINEARLY EQUIVALENT QUADRATIC FUNCTIONS. [Philosophical Magazine, I. (1851), pp. 295 — 305.].”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-10.27", "url": "https://www.teclamairme.com/#r-10.27", "chapter": 10, "status": "confirmed", "claim": "States that the Nine Chapters on the Mathematical Art used column-elimination methods on an array of coefficients, and that Gauss used the same method in 1810 for orbit computations. Notes that the Nine Chapters reached its final form in the 1st century AD, and that Gauss’s 1810 work solved six equations for the orbit of the asteroid Pallas.", "sources": [ { "title": "Gaussian elimination, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gaussian_elimination", "own_work": false, "note": "traces the method to the Chinese text Nine Chapters on the Mathematical Art and to Gauss’s 1810 notation for symmetric elimination; “The first reference to the book by this title is dated to 179 AD, but parts of it were written as early as approximately 150 BC.”", "fetched": [ "2026-09-22" ] }, { "title": "The Nine Chapters on the Mathematical Art, Wikipedia", "url": "https://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art", "own_work": false, "note": "carries the dating: “composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 1st century CE.”", "fetched": [ "2026-09-22" ] }, { "title": "Nine chapters on the Mathematical Art, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/", "own_work": false, "note": "carries the column operations: “The matrix is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination.”", "fetched": [ "2026-09-22" ] }, { "title": "The Quest for Linear Equation Solvers, John L. Gustafson", "url": "https://ed-thelen.org/ABC/QuestLinearSolvers-.html", "own_work": false, "note": "carries Gauss and Pallas: “To solve the problem of predicting the orbit of Pallas, he had to reduce astronomical data to a least-squares problem… involving six equations in six unknowns.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.30", "url": "https://www.teclamairme.com/#r-10.30", "chapter": 10, "status": "confirmed", "claim": "Defines least squares as minimizing the sum of squared residuals and states its characterization by the normal equations, crediting Legendre 1805 and Gauss’s earlier claimed use. Notes that the residual is orthogonal to the column space of A, that Gauss claimed use since 1795 while Legendre keeps priority of publication, and that orthogonal methods such as the SVD compute it more stably.", "sources": [ { "title": "Least squares, Wikipedia", "url": "https://en.wikipedia.org/wiki/Least_squares", "own_work": false, "note": "defines the least-squares solution, gives the normal equations, and credits Legendre’s 1805 publication and Gauss’s priority claim.", "fetched": [ "2026-09-22" ] }, { "title": "Linear least squares, Wikipedia", "url": "https://en.wikipedia.org/wiki/Linear_least_squares", "own_work": false, "note": "carries the orthogonality: “the matrix X(XᵀX)⁻¹Xᵀ represents orthogonal projection onto the column space of X.”", "fetched": [ "2026-09-22" ] }, { "title": "Legendre, Adrien-Marie, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Legendre,_Adrien-Marie", "own_work": false, "note": "carries the priority verdict: “The claim for priority made by Gauss at the time would seem to have been substantiated in the opinion of Laplace, Plackett (1972), Stigler (1977, 1981, 1986), and other authorities, but Legendre retains the priority of publication”", "fetched": [ "2026-09-22" ] }, { "title": "Numerical methods for linear least squares, Wikipedia", "url": "https://en.wikipedia.org/wiki/Numerical_methods_for_linear_least_squares", "own_work": false, "note": "carries the stability comparison: “Orthogonal decomposition methods of solving the least squares problem are slower than the normal equations method but are more numerically stable because they avoid forming the product XᵀX.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-10.33", "url": "https://www.teclamairme.com/#r-10.33", "chapter": 10, "status": "confirmed", "claim": "Defines a zero divisor as a ring element that multiplies with another nonzero element to give zero, and states that this fails in the integers, reals and complex numbers but occurs in rings like the integers mod 6. Notes that the rationals, quaternions and octonions also lack nonzero zero divisors, that 2 × 3 = 0 modulo 6, that they first appear at the sedenions in the Cayley–Dickson ladder and persist above, and that a nonzero matrix squaring to zero is one.", "sources": [ { "title": "Zero divisor, Wikipedia", "url": "https://en.wikipedia.org/wiki/Zero_divisor", "own_work": false, "note": "defines zero divisors, with the example Z/4Z, where the class of 2 is a zero divisor since 2 × 2 = 4 = 0, and “A nonzero commutative ring whose only zero divisor is 0 is called an integral domain.” The mod 6 example is on the Domain line.", "fetched": [ "2026-09-22" ] }, { "title": "Octonion, Wikipedia", "url": "https://en.wikipedia.org/wiki/Octonion", "own_work": false, "note": "carries the division algebras and the sedenions: “ℝ, ℂ, or ℍ, and 𝕆 are the only normed division algebras over the real numbers. … The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the sedenions) all fail to satisfy this property. They all have zero divisors.”", "fetched": [ "2026-09-22" ] }, { "title": "Domain (ring theory), Wikipedia", "url": "https://en.wikipedia.org/wiki/Domain_(ring_theory)", "own_work": false, "note": "carries the mod 6 example: “The ring Z/6Z is not a domain, because the images of 2 and 3 in this ring are nonzero elements with product 0.”", "fetched": [ "2026-09-22" ] }, { "title": "Sedenion, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "carries the persistence above the sedenions: “All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.1", "url": "https://www.teclamairme.com/#r-11.1", "chapter": 11, "status": "confirmed", "claim": "The fundamental theorem gives ∫_a^b f′(x) dx = f(b) − f(a) under the usual continuity hypotheses; for x² on [0, 1], the integral of 2x and the endpoint difference are both 1.", "sources": [ { "title": "Wolfram MathWorld, Fundamental Theorems of Calculus", "url": "https://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html", "own_work": false, "note": "states the endpoint formula ∫_a^b f(x)dx = F(b) − F(a) for a continuous integrand and antiderivative.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.2", "url": "https://www.teclamairme.com/#r-11.2", "chapter": 11, "status": "confirmed", "claim": "The prime derivative notation is associated with Lagrange and appears in his 1797 Théorie des fonctions analytiques; Newton used fluxions and Leibniz differential notation.", "sources": [ { "title": "MacTutor, Earliest Uses of Symbols of Calculus", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/", "own_work": false, "note": "records Lagrange’s f′ notation in 1797 and distinguishes earlier Newton/Leibniz notations.", "fetched": [ "2026-09-09" ] }, { "title": "Fluxion, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fluxion", "own_work": false, "note": "“Fluxions were introduced by Isaac Newton to describe his form of a time derivative (a derivative with respect to time).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.3", "url": "https://www.teclamairme.com/#r-11.3", "chapter": 11, "status": "confirmed", "claim": "For u=e^(−π²t)sin(πx), the heat operator (∂t−∂xx)u is zero; Fourier’s heat theory dates to his 1807 work and the separated modes on a unit rod decay at rates n²π². Two events: the 1807 memoir already solved the heat equation with trigonometric series (MacTutor), and the Heat equation page dates the separation-of-variables solution to the Théorie analytique de la chaleur of 1822.", "sources": [ { "title": "MacTutor, Joseph Fourier", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/", "own_work": false, "note": "Fourier completed his memoir On the Propagation of Heat in Solid Bodies by 1807 and read it to the Paris Institute on 21 December 1807; “He established the partial differential equation governing heat diffusion and solved it by using infinite series of trigonometric functions,” and Lagrange and Laplace’s first objection, in 1808, “was to Fourier’s expansions of functions as trigonometrical series.” (The review first cited here: T. N. Narasimhan, Fourier’s heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1999) 151–172, doi:10.1029/1998RG900006, paywalled).", "fetched": [ "2026-09-22" ] }, { "title": "Wikipedia, Heat equation", "url": "https://en.wikipedia.org/wiki/Heat_equation", "own_work": false, "note": "gives Fourier’s heat equation and his separation-of-variables solution, proposed in his Théorie analytique de la chaleur (1822).", "fetched": [ "2026-09-22" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Differentiate directly: ∂t u = −π²u and ∂xx u = −π²u, so their difference is 0; for sin(nπx), the same calculation gives decay rate n²π².", "fetched": [] } ], "editorial_note": "" }, { "id": "r-11.4", "url": "https://www.teclamairme.com/#r-11.4", "chapter": 11, "status": "confirmed", "claim": "π² is approximately 9.8696, is transcendental, and equals 6ζ(2).", "sources": [ { "title": "Wolfram MathWorld, Pi", "url": "https://mathworld.wolfram.com/Pi.html", "own_work": false, "note": "states that π is transcendental by Lindemann’s 1882 result.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions §25.6", "url": "https://dlmf.nist.gov/25.6", "own_work": false, "note": "equation 25.6.1 gives ζ(2)=π²/6, hence π²=6ζ(2).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.5", "url": "https://www.teclamairme.com/#r-11.5", "chapter": 11, "status": "confirmed", "claim": "The real minimum of x² is 0, attained uniquely at x=0, and the minimum of x²+1 is 1 at x=0. The framework of mathematical optimization names these parts: the objective function, the search space or choice set, convexity and duality.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "For every real x, x²≥0 with equality exactly at x=0; adding 1 shifts every value by 1, so the unique minimizer remains x=0 and the minimum becomes 1.", "fetched": [] }, { "title": "Mathematical optimization, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mathematical_optimization", "own_work": false, "note": "its notation example: “The minimum value in this case is 1, occurring at x = 0.”; and the vocabulary: “The function f is variously called an objective function,” “The domain A of f is called the search space or the choice set,” “When the objective function is a convex function, then any local minimum will also be a global minimum,” and “the theory of duality.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.6", "url": "https://www.teclamairme.com/#r-11.6", "chapter": 11, "status": "confirmed", "claim": "For fields whose second partial derivatives are continuous, div(curl v)=0 and curl(grad f)=0, because mixed partial derivatives commute. That mixed partials commute is Clairaut’s theorem, proposed by Alexis Clairaut in 1740 and first proved rigorously by H. A. Schwarz in 1873. Both identities are the vector-calculus form of d²=0. Vector calculus identities proves curl grad = 0 by the equality of mixed partials and calls div curl = 0 a special case of d² = 0. A conservative field, the gradient of a function, is irrotational and does zero work around any simple closed loop. The gradient points in the direction of most rapid increase, and a function whose Laplacian is 0 is called harmonic.", "sources": [ { "title": "Wolfram MathWorld, Vector Derivative", "url": "https://mathworld.wolfram.com/VectorDerivative.html", "own_work": false, "note": "lists the standard gradient, divergence, curl, and Laplacian operators and their Cartesian derivative identities.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Stokes’ Theorem", "url": "https://mathworld.wolfram.com/StokesTheorem.html", "own_work": false, "note": "identifies the gradient, curl, and divergence theorems as differential-form instances of Stokes’ theorem.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Symmetry of second derivatives", "url": "https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives", "own_work": false, "note": "records Clairaut’s proposed proof of 1740 and Schwarz’s first rigorous proof of 1873, and states the hypothesis as continuous second partial derivatives on a neighborhood of the point.", "fetched": [ "2026-09-09" ] }, { "title": "Vector calculus identities, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vector_calculus_identities", "own_work": false, "note": "curl of a gradient: “It can be easily proved by expressing ∇ × (∇ φ) in a Cartesian coordinate system with Schwarz’s theorem (also called Clairaut’s theorem on equality of mixed partials).”; divergence of a curl: “The divergence of the curl of any continuously twice-differentiable vector field A is always zero,” and “This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.”; gradient: “the gradient is proportional to, and points in the direction of, the function’s most rapid (positive) change”; Laplacian: “When the Laplacian is equal to 0, the function is called a harmonic function.”", "fetched": [ "2026-09-22" ] }, { "title": "Conservative vector field, Wikipedia", "url": "https://en.wikipedia.org/wiki/Conservative_vector_field", "own_work": false, "note": "“a conservative vector field is a vector field that is the gradient of some function”; irrotational: “if its curl is 0 everywhere in U”; and “the work W done in going around a simple closed loop C is 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.7", "url": "https://www.teclamairme.com/#r-11.7", "chapter": 11, "status": "confirmed", "claim": "The inverse-distance potential satisfies ∇²(1/r)=0 for r≠0 and equals −4πδ at the origin in the distributional formulation.", "sources": [ { "title": "Wolfram MathWorld, Laplacian", "url": "https://mathworld.wolfram.com/Laplacian.html", "own_work": false, "note": "derives that the Laplacian of 1/r vanishes away from the source and gives the −4π delta-function identity.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.8", "url": "https://www.teclamairme.com/#r-11.8", "chapter": 11, "status": "confirmed", "claim": "The divergence of the position field in three dimensions is 3 (n in n dimensions), so that the flux of r out of a closed surface is three times the volume inside, and the Laplacian of r²=x²+y²+z² is 6. As the radius R shrinks, the average of f over a sphere exceeds f at the center by ∇²f·R²/(2n) to leading order, R²∇²f/6 in three dimensions.", "sources": [ { "title": "Wolfram MathWorld, Vector Derivative", "url": "https://mathworld.wolfram.com/VectorDerivative.html", "own_work": false, "note": "equation 22 gives ∇·r=3.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "∇²(x²+y²+z²)=2+2+2=6; in n dimensions the corresponding value is 2n. Likewise ∇·r=1+1+…+1=n in n dimensions, and by the divergence theorem ∮ r·dA = ∫ ∇·r dV = 3V in three dimensions.", "fetched": [] }, { "title": "Laplace operator, Wikipedia", "url": "https://en.wikipedia.org/wiki/Laplace_operator", "own_work": false, "note": "“∇²f(x) = lim_{R→0} (2n/R²)(f_shell_R − f(x)),” where n is the dimension of the space and f_shell_R is the average of f over the surface of an n-sphere of radius R.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.9", "url": "https://www.teclamairme.com/#r-11.9", "chapter": 11, "status": "confirmed", "claim": "The flux of r̂/r² through a unit sphere is 4π, the solid angle of all directions, and Green’s line integral 1/2∮(x dy−y dx) around the unit circle is π. The solid angle of the whole sphere is 4π, and the flux of an inverse-square field through any closed surface is 4π times the enclosed source (for gravity, −4πGM).", "sources": [ { "title": "Wolfram MathWorld, Stokes’ Theorem", "url": "https://mathworld.wolfram.com/StokesTheorem.html", "own_work": false, "note": "states the surface-flux/line-circulation relation underlying these integrals.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "On S², r̂/r²·dA=sinθ dθ dφ, whose integral is 4π. Parametrizing the unit circle by x=cos t,y=sin t gives 1/2∫(x y’−y x’)dt=1/2∫_0^{2π}dt=π.", "fetched": [] }, { "title": "Solid angle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Solid_angle", "own_work": false, "note": "“The solid angle of a sphere measured from any point in its interior is 4π sr.”", "fetched": [ "2026-09-22" ] }, { "title": "Gauss’s law for gravity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity", "own_work": false, "note": "“The gravitational flux through any closed surface is proportional to the enclosed mass”: ∯∂V g · dA = −4πGM.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.10", "url": "https://www.teclamairme.com/#r-11.10", "chapter": 11, "status": "confirmed", "claim": "The divergence theorem relates total outward flux through a closed surface to the volume integral of divergence; the historical attributions in the card are three special cases in Gauss’s paper of 1813 and Ostrogradsky’s general proof, presented in 1826 and published in 1831. Corrected 11 September 2026: this entry previously carried “Lagrange 1762,” which Katz’s history does not support — see the note at 11.25. Katz 1979 (see 11.25): “In three special cases it occurs in an 1813 paper of Gauss.”", "sources": [ { "title": "Wikipedia, Divergence theorem", "url": "https://en.wikipedia.org/wiki/Divergence_theorem", "own_work": false, "note": "states the flux/divergence relation and documents Lagrange’s 1762 special case, Gauss’s 1813 cases, and Ostrogradsky’s 1826/1831 work.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.11", "url": "https://www.teclamairme.com/#r-11.11", "chapter": 11, "status": "confirmed", "claim": "Green’s essay was published in Nottingham in 1828, and the special line integral gives the area enclosed by a planar curve. Green’s theorem is the two-dimensional case of Stokes’ theorem, and ½∮(x dy − y dx) is the enclosed area; π is transcendental (Lindemann 1882).", "sources": [ { "title": "Cambridge Core, Green’s and allied theorems: a historical sketch", "url": "https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/greens-and-allied-theorems-a-historical-sketch/158EF85DB56287D429B35222BC9730D4", "own_work": false, "note": "states that Green’s essay was published at Nottingham in 1828 and discusses the transformation on which the theorem depends.", "fetched": [ "2026-09-09" ] }, { "title": "Green’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Green%27s_theorem", "own_work": false, "note": "“It is the two-dimensional special case of Stokes’ theorem (surface in R³).”; area: “A = ∮_{C} x dy = −∮_{C} y dx = ½∮_{C}(−y dx + x dy).”; history: Green “stated a similar result in an 1828 paper”; “In 1846, Augustin-Louis Cauchy published a paper stating Green’s theorem as the penultimate sentence. A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation.”", "fetched": [ "2026-09-22" ] }, { "title": "Pi, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/Pi.html", "own_work": false, "note": "“π is also transcendental (Lindemann 1882).”", "fetched": [ "2026-09-22" ] }, { "title": "J. Cufí and A. Reventós, A historical review of the Cauchy-Riemann equations and the Cauchy Theorem, arXiv:2311.16649", "url": "https://arxiv.org/pdf/2311.16649", "own_work": false, "note": "“Green’s formula (16) does not appear in Green’s paper”; “The first published proof of Green’s formula is due to Riemann in his 1851 dissertation.”", "fetched": [ "2026-10-01" ] }, { "title": "n-sphere, Wikipedia", "url": "https://en.wikipedia.org/wiki/N-sphere", "own_work": false, "note": "“A unit 1-sphere is the unit circle in the Euclidean plane.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "" }, { "id": "r-11.12", "url": "https://www.teclamairme.com/#r-11.12", "chapter": 11, "status": "confirmed", "claim": "Kelvin communicated the classical curl theorem to Stokes in 1850, Stokes used it in the 1854 Smith’s Prize examination, and the modern differential-form generalization is associated with Cartan after earlier work by Volterra, Goursat and Poincaré.", "sources": [ { "title": "Wikipedia, Generalized Stokes theorem", "url": "https://en.wikipedia.org/wiki/Generalized_Stokes_theorem", "own_work": false, "note": "gives Kelvin’s 1850 letter, the 1854 exam question, and Cartan’s modern formulation following earlier generalizations; its note on the exam: “Maxwell took this examination and tied for first place with Edward John Routh.”; and “Stokes set the theorem as a question on the 1854 Smith’s Prize exam, which led to the result bearing his name.”; and “Stokes’ theorem was formulated in its modern form by Élie Cartan in 1945, following earlier work on the generalization of the theorems of vector calculus by Vito Volterra, Édouard Goursat, and Henri Poincaré.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.13", "url": "https://www.teclamairme.com/#r-11.13", "chapter": 11, "status": "confirmed", "claim": "Maxwell published twenty scalar equations in 1865, while Heaviside and Gibbs-era vector notation condensed the system to the four equations now commonly taught; the equations imply electromagnetic waves with speed 1/√(μ₀ε₀). The identification of light as an electromagnetic wave is Maxwell’s own, made in the 1865 paper from his own twenty equations, and not a consequence of the later four-equation form; Heaviside and Gibbs supplied the notation, two decades after the prediction. Gradient, divergence and curl were first studied in the classical theory of electromagnetism.", "sources": [ { "title": "Wikipedia, History of Maxwell’s equations", "url": "https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations", "own_work": false, "note": "documents Maxwell’s 1865 formulation, the twenty-equation presentation, and the 1884 vector condensation associated with Heaviside and Gibbs; it also records the electromagnetic-wave result.", "fetched": [ "2026-09-09" ] }, { "title": "Nabla symbol, Wikipedia", "url": "https://en.wikipedia.org/wiki/Nabla_symbol", "own_work": false, "note": "“They were all originally studied in the context of the classical theory of electromagnetism, and contemporary university physics curricula typically treat the material using approximately the concepts and notation found in Gibbs and Wilson’s Vector Analysis.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.14", "url": "https://www.teclamairme.com/#r-11.14", "chapter": 11, "status": "confirmed", "claim": "A divergence-free field that is twice continuously differentiable on all of ℝ³ and decays at least as fast as 1/‖x‖ is the curl of a vector potential. On a smaller region, whether a divergence-free field is a curl depends on the shape of the region, and the inverse-square field on punctured space is the chapter’s own obstruction. In the Aharonov–Bohm effect a charged particle is affected by the potential A where the magnetic field B is zero.", "sources": [ { "title": "Wikipedia, Vector potential", "url": "https://en.wikipedia.org/wiki/Vector_potential", "own_work": false, "note": "gives the existence theorem for a solenoidal field on ℝ³ that is twice continuously differentiable and decreases at least as fast as 1/‖x‖ at infinity.", "fetched": [ "2026-09-09" ] }, { "title": "MIT 18.022, When is a Vector Field the Curl of Another?", "url": "https://math.mit.edu/~djk/18_022/chapter08/section03.html", "own_work": false, "note": "states the result for a region R that is simply connected. See 11.18: simple connectivity is not by itself enough.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For r̂/r² on ℝ³∖{0}, the divergence is 0 everywhere the field is defined, but the flux through every enclosing sphere is 4π; the flux of a curl through a closed surface is 0, so the field is not a curl there.", "fetched": [] }, { "title": "Aharonov–Bohm effect, Wikipedia", "url": "https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect", "own_work": false, "note": "“an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.15", "url": "https://www.teclamairme.com/#r-11.15", "chapter": 11, "status": "judgement", "claim": "The nabla operator is the vector of partial derivatives, but the date of Hamilton’s first use of it is not the same in every source. The name comes from the Hellenistic Greek νάβλα for a Phoenician harp; the Nabla symbol page calls the often-claimed Hebrew derivation mistaken. MacTutor credits Tait’s Elementary Treatise on Quaternions of 1867 with establishing ∇ as the conventional symbol, and William Robertson Smith proposed the name to Tait in a letter of 10 November 1870.", "sources": [ { "title": "Wikipedia, Nabla symbol", "url": "https://en.wikipedia.org/wiki/Nabla_symbol", "own_work": false, "note": "says Hamilton introduced the operator in 1831, writing it as a rotated ◁, and that the name nabla was proposed by William Robertson Smith in a letter to Peter Guthrie Tait of 10 November 1870; its footnote: “Letter from Smith to Tait, 10 November 1870: My dear Sir, The name I propose for ∇ is, as you will remember, Nabla.”", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Earliest Uses of Symbols of Calculus", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/", "own_work": false, "note": "dates Hamilton’s use of the vector differential operator to the Royal Irish Academy meeting of 20 July 1846, published in 1847, and credits Tait’s Elementary Treatise on Quaternions of 1867 with establishing ∇ and the term nabla.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Wikipedia dates Hamilton’s first use to 1831 and MacTutor to 1846, so we give no first-use date at all. The card names instead the two dates the sources agree on: Tait’s treatise of 1867 for the symbol, and Smith’s letter of 1870 for the name." }, { "id": "r-11.16", "url": "https://www.teclamairme.com/#r-11.16", "chapter": 11, "status": "confirmed", "claim": "The exterior derivative satisfies d∘d=0, and in three dimensions this packages curl grad=0 and div curl=0; Stokes’ theorem relates ∫_M dω to ∫_{∂M}ω. Élie Cartan first described the exterior derivative in its present form in 1899. d² = 0 comes from the symmetry of second derivatives; its dual, ∂∂ = 0, is required of every chain complex in homology.", "sources": [ { "title": "Wolfram MathWorld, Stokes’ Theorem", "url": "https://mathworld.wolfram.com/StokesTheorem.html", "own_work": false, "note": "identifies the differential-form form of Stokes’ theorem and its gradient/curl/divergence instances.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Exterior derivative", "url": "https://en.wikipedia.org/wiki/Exterior_derivative", "own_work": false, "note": "“The exterior derivative was first described in its current form by Élie Cartan in 1899”, and gives d(dα)=0 for a k-form α.", "fetched": [ "2026-09-09" ] }, { "title": "Closed and exact differential forms, Wikipedia", "url": "https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms", "own_work": false, "note": "“Because d² = 0, every exact form is necessarily closed.”; “The implication from ‘exact’ to ‘closed’ is then a consequence of the symmetry of second derivatives, with respect to x and y.”", "fetched": [ "2026-09-22" ] }, { "title": "Homology (mathematics), Wikipedia", "url": "https://en.wikipedia.org/wiki/Homology_(mathematics)", "own_work": false, "note": "“It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n, ∂ₙ ∘ ∂ₙ₊₁ = 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.17", "url": "https://www.teclamairme.com/#r-11.17", "chapter": 11, "status": "confirmed", "claim": "Fermat’s method of maxima and minima, which locates an extremum by setting the rate of change to zero, circulated in a Latin treatise of about 1636, before there was a differential calculus to put it in. Fermat’s theorem puts unconstrained optima at stationary points, and Fermat and Lagrange found the calculus-based formulae for identifying optima.", "sources": [ { "title": "Wikipedia, Fermat’s method of adequality", "url": "https://en.wikipedia.org/wiki/Adequality", "own_work": false, "note": "dates the treatise Methodus ad disquirendam maximam et minimam to about 1636 and describes Fermat’s method for maxima and minima before modern differential calculus.", "fetched": [ "2026-09-09" ] }, { "title": "Mathematical optimization, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mathematical_optimization", "own_work": false, "note": "“One of Fermat’s theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero”; “Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-11.18", "url": "https://www.teclamairme.com/#r-11.18", "chapter": 11, "status": "confirmed", "claim": "Simple connectivity is not sufficient for a divergence-free field to be a curl. ℝ³ with the origin removed is simply connected, and the inverse-square field r̂/r² has zero divergence on it and no vector potential. What suffices is that the region be star-shaped or contractible, by the Poincaré lemma, and in general that its second de Rham cohomology vanish.", "sources": [ { "title": "Wikipedia, Poincaré lemma", "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_lemma", "own_work": false, "note": "every closed p-form on a star-shaped open subset of ℝⁿ is exact, so the de Rham cohomology of such a region vanishes in every positive degree.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Vector potential", "url": "https://en.wikipedia.org/wiki/Vector_potential", "own_work": false, "note": "gives the existence theorem on ℝ³ and the restriction of the integral domain to a region.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Every loop in ℝ³∖{0} contracts, so the region is simply connected. The flux of r̂/r² through a sphere about the origin is 4π, while the flux of any curl through a closed surface is 0 by the divergence theorem and div curl = 0. So r̂/r² is not a curl there, and simple connectivity cannot be the sufficient condition.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-11.19", "url": "https://www.teclamairme.com/#r-11.19", "chapter": 11, "status": "confirmed", "claim": "The epigraph is Feynman’s: “From a long view of the history of mankind — seen from, say, ten thousand years from now — there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics.” It is in the lecture on electromagnetism that opens Volume II, published in 1964.", "sources": [ { "title": "Wikiquote, Richard Feynman", "url": "https://en.wikiquote.org/wiki/Richard_Feynman", "own_work": false, "note": "gives the passage — “the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics” — located in The Feynman Lectures on Physics, vol. II, lecture 1, section 1-6, p. 1-11.", "fetched": [ "2026-09-22" ] }, { "title": "John D. Cook, What was the most important event of the 19th century?", "url": "https://www.johndcook.com/blog/2011/05/30/feynman-maxwell-equations/", "own_work": false, "note": "quotes the sentence in full, with the dashes, and attributes it to The Feynman Lectures on Physics, Volume 2.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-11.20", "url": "https://www.teclamairme.com/#r-11.20", "chapter": 11, "status": "confirmed", "claim": "Green, almost entirely self-taught, worked as a miller at the mill his father built in Nottingham. He printed his 1828 essay at his own expense and sold it by subscription to fifty-one subscribers, most of them members of the Nottingham Subscription Library. It went almost unnoticed until William Thomson, later Lord Kelvin, was given three copies in January 1845, showed the essay to Liouville and Sturm in Paris that summer, and had it reprinted between 1850 and 1854.", "sources": [ { "title": "MacTutor, George Green", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Green/", "own_work": false, "note": "the essay was advertised in the Nottingham Review on 14 December 1827 and published in March 1828 by subscription at seven shillings and sixpence, with 51 subscribers, most of them members of the Nottingham Subscription Library; after Thomson took his degree in January 1845 his coach William Hopkins gave him three copies, and Thomson showed the work to Liouville and Sturm in Paris in the summer of 1845 and republished it in 1850–54.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, George Green (mathematician)", "url": "https://en.wikipedia.org/wiki/George_Green_(mathematician)", "own_work": false, "note": "the essay was sold on a subscription basis to 51 people; Thomson had seen a reference to Green’s work in an 1833 paper by Robert Murphy but could find no bookseller who remembered it; the essay was not reprinted in Britain until 1871; “As a mathematician, he was almost entirely self-taught, receiving only about one year of formal schooling as a child, between the ages of eight and nine. Green then worked his father’s bakery from a young age, and later at the mill his father constructed in 1807, now known as Green’s Windmill. Whilst working as a miller, he became a member of Nottingham Subscription Library.”", "fetched": [ "2026-09-09" ] }, { "title": "Cambridge Core, Green’s and allied theorems: a historical sketch", "url": "https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/greens-and-allied-theorems-a-historical-sketch/158EF85DB56287D429B35222BC9730D4", "own_work": false, "note": "states that Green’s essay was published at Nottingham in 1828 and was unknown to continental mathematicians until the reprints of 1850, 1852 and 1854.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Two claims have to be separated here. That the essay went entirely unread is too strong: it had fifty-one subscribers, and Robert Murphy cited it in 1833. That Thomson was handed three copies in 1845 and carried the essay to Paris is documented by both biographies, so the card states it plainly rather than gesturing at reprints." }, { "id": "r-11.21", "url": "https://www.teclamairme.com/#r-11.21", "chapter": 11, "status": "judgement", "claim": "Laplace made the equation for the gravitational potential the working tool of potential theory in his memoir on the attraction of spheroids, writing it in polar rather than Cartesian coordinates; the equation was known before him. Bibm@th dates the memoir to 1782; Rouse Ball says it was read in 1784. Siméon Denis Poisson showed in 1813 that inside the attracting matter the potential satisfies ∇²V = −4πρ rather than ∇²V = 0, so that the whole of the missing divergence sits where the mass or charge is. Sources differ on Poisson’s year: MacTutor and Rouse Ball give 1813, while Wikipedia’s Poisson’s equation says he published it in 1823. Corrections welcome. The Poisson equation for gravity follows from Gauss’s law in differential form.", "sources": [ { "title": "W. W. Rouse Ball, A Short Account of the History of Mathematics, Laplace", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Laplace/RouseBall/RB_Laplace.html", "own_work": false, "note": "the memoir on the attraction of spheroids was read in 1784 and reprinted in the third volume of the Mécanique céleste; it develops the use of the potential, which satisfies ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 0.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Pierre-Simon Laplace", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Laplace/", "own_work": false, "note": "“although we now name this equation after Laplace, it was in fact known before the time of Laplace”.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Siméon-Denis Poisson", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Poisson/", "own_work": false, "note": "“In 1813 Poisson studied the potential in the interior of attracting masses, producing results which would find application in electrostatics.”", "fetched": [ "2026-09-09" ] }, { "title": "W. W. Rouse Ball, A Short Account of the History of Mathematics, Poisson", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Poisson/RouseBall/RB_Poisson.html", "own_work": false, "note": "Poisson substituted ∇²V = −4πρ for Laplace’s ∇²V = 0, and this was first published in 1813.", "fetched": [ "2026-09-09" ] }, { "title": "Laplacien, Bibm@th dictionnaire", "url": "https://www.bibmath.net/dico/index.php?action=affiche&quoi=.%2Fl%2Flaplacien.html", "own_work": false, "note": "gives 1782 and the polar coordinates: “La première apparition du laplacien apparait dans un mémoire de Laplace sur l’attraction des sphéroïdes en 1782, sous une forme compliquée due au système de coordonnées polaires qu’il utilisait.”", "fetched": [ "2026-09-22" ] }, { "title": "Poisson’s equation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Poisson%27s_equation", "own_work": false, "note": "gives 1823: “The equation is named after French mathematician and physicist Siméon Denis Poisson who published it in 1823.”; and “Gauss’s law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The date of the spheroids memoir is not agreed. Rouse Ball says it was read in 1784; Wikipedia’s article on the equation dates Laplace’s study of its properties to 1786; the Mémoires volume the memoir belongs to is the one for 1782, printed in 1785. Ruled, 10 September 2026. The page names the volume rather than asserting a year: the memoir is the one in the Mémoires volume for 1782, which is checkable, and the bare date has been dropped from the chapter prose, where it carried no weight. The three accounts are not in conflict about a fact — an Académie volume is named for its session year while the memoir inside it is read and printed later — so recency cannot and should not separate them. The page claims nothing about who had the equation first, and nothing in the mathematics turns on the date. A later pass should not restore a bare 1782 to the prose: the volume year is not the year the memoir was read." }, { "id": "r-11.22", "url": "https://www.teclamairme.com/#r-11.22", "chapter": 11, "status": "confirmed", "claim": "Gauss’s law for an inverse-square field is the first of the lemmas Carl Friedrich Gauss proved in his Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata, published in 1813 in Commentationes societatis regiae scientiarium Gottingensis recentiores vol. II, pp. 355–378; the text in the fifth volume of his collected works is a reprint, not a posthumous first appearance. What Gauss proves there is a special case; he came back to inverse-square forces in general later, in his Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte, Leipzig, 1840. That work’s title and year are carried by the Carl Friedrich Gauss line; no source read here says which of its theorems is the general flux law, so the book does not say that it is.", "sources": [ { "title": "Wikipedia, Gauss’s law", "url": "https://en.wikipedia.org/wiki/Gauss%27s_law", "own_work": false, "note": "“The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1835, both in the context of the attraction of ellipsoids”, citing Gauss’s Werke, Band 5. The book no longer follows this page’s 1773 and 1835 (the Werke text is a reprint of the 1813 paper; see 11.25), but keeps the link as the record of the older account. The same page says Gauss’s law “is one of Maxwell’s equations” and that “Gauss’s law itself is essentially equivalent to Coulomb’s law, and Gauss’s law for gravity is essentially equivalent to Newton’s law of gravity, both of which are inverse-square laws.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Divergence theorem", "url": "https://en.wikipedia.org/wiki/Divergence_theorem", "own_work": false, "note": "read as raw HTML: its history footnote gives the printing exactly — “C. F. Gauss (1813) ‘Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,’ Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378; Gauss considered a special case of the theorem” — and its running text says Gauss was “working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem”, that Lagrange “discovered the divergence theorem in 1762”, and that the first proof of the general theorem is Ostrogradsky’s, of 1826.", "fetched": [ "2026-09-17" ] }, { "title": "Carl Friedrich Gauss, Wikipedia", "url": "https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss", "own_work": false, "note": "read as raw HTML: its list of works gives “1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte … Leipzig: Weidmannsche Buchhandlung, 1840,” translated there as General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "The fifth volume of Gauss’s Werke was first published in 1867 and reissued in 1877, and sources cite both years, so the card names the volume and not the year. Extended 17 September 2026. The entry previously asserted the 1813 printing while its only source said “followed by Carl Friedrich Gauss in 1835”; the Divergence theorem article gives the 1813 volume and pages outright, so the printing is settled and the entry no longer disputes its own evidence. Corrected later the same day: the Lagrange question is settled, and the evidence was in the book. r-11.25’s note records a deep-research pass of 11 September 2026 in which Katz 1979 was read in full and Lagrange’s memoirs were read in the original, and which concluded that “Lagrange 1773 is not a statement of Gauss’s law: the memoir was read end to end and contains no closed surface, no flux and no 4π; what it supplies is the change of variables Gauss’s proof runs on, which is a real priority for a method and not for the law.” The #flux card’s clause was briefly cut on 17 September 2026 for want of a source and has been restored, now citing 11.25, which the card had never cited. So the general-reference sentence that Lagrange “first formulated” the law is not followed here, and the reason is a reading of the primary document, not a preference between encyclopedias." }, { "id": "r-11.23", "url": "https://www.teclamairme.com/#r-11.23", "chapter": 11, "status": "judgement", "claim": "The fundamental theorem of calculus was reached by stages: James Gregory published the first statement and proof of a rudimentary form, Isaac Barrow proved a more generalized version, Isaac Newton stated the theorem explicitly in his tract on fluxions of October 1666 and completed the surrounding theory, and Gottfried Leibniz, working in Paris in the 1670s from quite different foundations, systematized the subject and introduced the notation still used, publishing his integral calculus in 1684 and 1686.", "sources": [ { "title": "MacTutor, The rise of calculus", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/", "own_work": false, "note": "records that ‘Newton wrote a tract on fluxions in October 1666’ and that ‘this work contains the first clear statement of the Fundamental Theorem of the Calculus’; that ‘the idea that integral and derivative were inverses to each other were familiar to Barrow’ although ‘Barrow never explicitly stated the fundamental theorem of the calculus’; and that ‘On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton’, with ‘His results on the integral calculus were published in 1684 and 1686’.", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Fundamental theorem of calculus", "url": "https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus", "own_work": false, "note": "the History section states that ‘The first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory (1638-1675)’, that ‘Isaac Barrow (1630-1677) proved a more generalized version of the theorem, while his student Isaac Newton (1642-1727) completed the development of the surrounding mathematical theory’, and that ‘Gottfried Leibniz (1646-1716) systematized the knowledge into a calculus for infinitesimal quantities and introduced the notation used today’.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass. The chapter sentence ‘which Newton and Leibniz each reached in the late seventeenth century’ previously stood on 11.2, whose claim covers only derivative notation and carries no history of the theorem. Neither source here divides the credit the way that sentence does: Gregory and Barrow come before Newton, and the sources describe Newton as stating the theorem and Leibniz as systematizing the calculus rather than as two men each arriving at the same theorem. The wording is left for the author." }, { "id": "r-11.24", "url": "https://www.teclamairme.com/#r-11.24", "chapter": 11, "status": "confirmed", "claim": "Newton and Leibniz were the parties to a priority dispute over who had first invented the calculus; the modern consensus is that they invented and described it independently in the seventeenth century.", "sources": [ { "title": "Wikipedia, Leibniz-Newton calculus controversy", "url": "https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy", "own_work": false, "note": "opens ‘the calculus controversy was a priority dispute between mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus’, and states ‘Today, the consensus is that Leibniz and Newton independently invented and described calculus in Europe in the 17th century’.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass to carry the chapter’s clause ‘the argument about whose calculus it was outlived them both’, which previously stood on 11.2. The source carries the dispute and the independence, but its narrative ends around 1716 and it does not state that the argument continued past the deaths of the two men. That half of the clause is still uncarried." }, { "id": "r-11.25", "url": "https://www.teclamairme.com/#r-11.25", "chapter": 11, "status": "judgement", "claim": "The divergence theorem carries at least six names and four dates in the reference literature — Lagrange 1762, Gauss 1813 (with further special cases in 1833 and 1839), Ostrogradsky 1826 for the first general proof, published 1831, and Green 1828 for special cases — and Gauss’s law has its own pair, Lagrange 1773 and Gauss 1835. The book gives Gauss 1813 for the special cases and Ostrogradsky 1826 for the general theorem, and for Gauss’s law it follows Gauss’s paper of 1813: Lagrange’s 1773 memoir contains no flux theorem, and the 1835 date rests on the Werke reprint of that paper (see 11.22).", "sources": [ { "title": "Wikipedia, Divergence theorem (History)", "url": "https://en.wikipedia.org/wiki/Divergence_theorem", "own_work": false, "note": "Lagrange ‘introduced the notion of surface integrals in 1760 and again in more general terms in 1811’ and ‘discovered the divergence theorem in 1762’; Gauss ‘proved special cases of the divergence theorem’ in 1813 while working on the gravitational attraction of an elliptical spheroid and ‘proved additional special cases in 1833 and 1839’; ‘it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow’; and ‘special cases were proven by George Green in 1828’ in the Essay on electricity and magnetism.", "fetched": [ "2026-09-11" ] }, { "title": "Own reading", "url": null, "own_work": true, "note": "Lagrange’s 1773 memoir, read end to end for this book: it supplies the change of variables the divergence proof runs on, but has no closed surface, no flux and no 4π; the 1773 date on Gauss’s law is the literature’s attribution, not the memoir’s content.", "fetched": [] }, { "title": "Encyclopedia of Mathematics, Ostrogradski formula", "url": "https://encyclopediaofmath.org/wiki/Ostrogradski_formula", "own_work": false, "note": "records that the identity is ‘often referred to as: Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula’, that the three-dimensional formula ‘is attributed to Gauss, who proved a particular case in 1813, and to Ostrogradski, who later generalized it to general dimension’ (1831, generalized 1838, both Mem. Acad. Sci. St Petersbourg), and that ‘the formula is already present in the works of Euler and other mathematicians of the 18th century’.", "fetched": [ "2026-09-11" ] }, { "title": "MacTutor, Mikhail Ostrogradski", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Ostrogradski/", "own_work": false, "note": "Ostrogradski submitted Demonstration d’un theoreme du calcul integral to the Paris Academy on 13 February 1826, in which he ‘states and proves the general divergence theorem’; quoting Victor Katz, he ‘presented this theorem again in a paper in Paris on 6 August 1827, and finally in St Petersburg on 5 November 1828. The latter presentation was the only one published by Ostrogradski, appearing in 1831 in Note sur la Theorie de la Chaleur’; the page adds that Gauss proved special cases in 1833 and 1839 and that ‘the theorem is now often named after Gauss’.", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Added for Q11.3, ruled on by the author on 11 September 2026: the prose takes the earliest date, and the unclarity gets a card of its own rather than a silent choice. The card is #divattrib. DEEP RESEARCH DONE, 11 September 2026, and it changed the book. Katz 1979 was read in full and Lagrange’s two memoirs were read in the original. Lagrange 1762 is not a statement of the divergence theorem — it integrates by parts and then disposes of the boundary terms — and the credit traces to a footnote in a general-reference article rather than to any historian. Lagrange 1773 is not a statement of Gauss’s law: the memoir was read end to end and contains no closed surface, no flux and no 4π; what it supplies is the change of variables Gauss’s proof runs on, which is a real priority for a method and not for the law. Nothing supports 1835; the citation attached to that year in the article that prints it is the 1813 paper. And Gauss’s paper was not posthumous: it was published in 1813, and the text in Werke v. 5 is a reprint. All four claims have been removed from the prose and the cards. What remains genuinely contested is the divergence theorem itself — Gauss’s three special cases of 1813, Ostrogradsky’s general proof of 1826, Green’s special cases of 1828 — so the entry stays a judgement, and under D1 the prose prints the earliest it can source, ~1813, and names Gauss." }, { "id": "r-11.26", "url": "https://www.teclamairme.com/#r-11.26", "chapter": 11, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: pi is transcendental by Lindemann’s theorem of 1882; Gibbs and Heaviside separated nabla from the quaternions in the 1880s.", "sources": [ { "title": "Lindemann–Weierstrass theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "read raw: “Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below)..”", "fetched": [ "2026-09-18" ] }, { "title": "Euclidean vector, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euclidean_vector", "own_work": false, "note": "read raw: “The first half of Gibbs’s Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-11.31", "url": "https://www.teclamairme.com/#r-11.31", "chapter": 11, "status": "confirmed", "claim": "The loop on an integral sign, ∮, marks an integral taken all the way round something closed: a line integral round a closed curve, or a surface integral over a closed surface.", "sources": [ { "title": "Maxwell’s equations, Wikipedia", "url": "https://en.wikipedia.org/wiki/Maxwell%27s_equations", "own_work": false, "note": "read via the REST endpoint (revision 1375267474), in its key to the integral signs: the closed surface integral over ∂Ω “is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed,” and ∮ over ∂Σ “is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed..”", "fetched": [ "2026-09-29" ] }, { "title": "MacTutor, Earliest Uses of Symbols of Calculus", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/calculus/", "own_work": false, "note": "under “Integration around a closed path”: “a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868 - 1951) in 1917,” and E. B. Wilson “used a small circle below the standard integral symbol to denote integration around a closed curve.”", "fetched": [ "2026-09-29" ] } ], "editorial_note": "" }, { "id": "r-12.1", "url": "https://www.teclamairme.com/#r-12.1", "chapter": 12, "status": "confirmed", "claim": "Euler characteristic is V − E + F for a polyhedral decomposition, equals 2 − 2g for a closed orientable surface of genus g, and gives 2 for the sphere, 0 for the torus and Klein bottle, 1 for the projective plane, and −2 for the double torus. Euler announced the polyhedron formula in a letter to Christian Goldbach in 1750 and published the details in two papers. Those papers, E230 and E231, were written in 1750 and 1751 and printed in Novi Commentarii Academiae Scientiarum Petropolitanae vol. 4, the volume for 1752–53, which itself appeared only in 1758; the literature cites both 1752 and 1758. Under D1 the page prints ~1752, the earliest, and the card explains the volume year. ℝP² is the sphere with antipodal points identified, or a Möbius band with a disk glued on, and cannot be embedded in space without crossing itself; for nonorientable surfaces χ = 2 − g. Cauchy’s proof dates from 1811. The cube has V, E, F = 8, 12, 6 and the icosahedron 12, 30, 20, and V − E + F = 2 leaves exactly five regular solids. Closed surfaces of genus greater than 1 are hyperbolic.", "sources": [ { "title": "Wolfram MathWorld, Euler Characteristic", "url": "https://mathworld.wolfram.com/EulerCharacteristic.html", "own_work": false, "note": "gives χ = V − E + F, χ = 2 − 2g, and the table values for sphere, torus, Klein bottle, projective plane and double torus; its table of Euler characteristics gives the Klein bottle and the Möbius strip both 0.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Topology history", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/", "own_work": false, "note": "records that in 1750 Euler wrote to Christian Goldbach giving the polyhedron formula v − e + f = 2, and that he published the details in 1752 in two papers.", "fetched": [ "2026-09-09" ] }, { "title": "Euler Archive, E230 Elementa doctrinae solidorum", "url": "https://scholarlycommons.pacific.edu/euler-works/230/", "own_work": false, "note": "records E230 in Novi Commentarii Academiae Scientarum Petropolitanae vol. 4, pp. 109–140, dated 1758, with its companion E231 Demonstratio nonnullarum insignium proprietatum at pp. 140–160 of the same volume (the Euler Archive’s E231 record; pp. 94–108 is the Opera Omnia reprint); this page gives E230 as written in 1750 and published in 1758; the 1752–53 volume year and E231’s 1751 come from the E231 record and other sources, not this page.", "fetched": [ "2026-09-20" ] }, { "title": "Real projective plane, Wikipedia", "url": "https://en.wikipedia.org/wiki/Real_projective_plane", "own_work": false, "note": "constructed “by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk”; “It cannot be embedded in three-dimensional Euclidean space without intersecting itself.”", "fetched": [ "2026-09-22" ] }, { "title": "Genus, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/Genus.html", "own_work": false, "note": "“For a nonorientable surface such as a real projective plane (genus 1) or Klein bottle (genus 2), the relationship is χ=2-g”", "fetched": [ "2026-09-22" ] }, { "title": "Euler characteristic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler_characteristic", "own_work": false, "note": "“The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly half that of the corresponding sphere”; “Any contractible space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore, its Euler characteristic is 1.”; gives Cauchy’s proof as 1811: “One was given by Cauchy in 1811.”", "fetched": [ "2026-09-22" ] }, { "title": "Platonic solid, Wikipedia", "url": "https://en.wikipedia.org/wiki/Platonic_solid", "own_work": false, "note": "its table gives the cube 8 vertices, 12 edges, 6 faces and the regular icosahedron 12, 30, 20; the topological proof: “The key is Euler’s observation that V − E + F = 2,” whence “1/q + 1/p > 1/2,” and “there are only five possibilities for {p, q}.”", "fetched": [ "2026-09-22" ] }, { "title": "Uniformization theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Uniformization_theorem", "own_work": false, "note": "compact Riemann surfaces “with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1”", "fetched": [ "2026-09-22" ] }, { "title": "E231, Demonstratio nonnullarum insignium proprietatum…, Euler Archive", "url": "https://scholarlycommons.pacific.edu/euler-works/231/", "own_work": false, "note": "“Written Date 1751,” “Published Date 1758,” “Novi Commentarii academiae scientiarum Petropolitanae, Volume 4, pp. 140-160.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.2", "url": "https://www.teclamairme.com/#r-12.2", "chapter": 12, "status": "confirmed", "claim": "Poincaré’s 1895 Analysis situs made topology a systematic study of invariants such as homology, Betti numbers and the fundamental group. Enrico Betti introduced the counts in a paper of 1871 and Poincaré named them after him; Emmy Noether’s observation of 1925 recast them as the ranks of abelian homology groups. The word topology is older than any of this: Johann Benedict Listing used it first. A topologist cannot tell a coffee mug from a doughnut, and homology, hence the Euler characteristic, is a homotopy invariant. Sources differ on the fundamental group’s year: Wikipedia and MacTutor’s history of topology give 1895 (Analysis situs), MacTutor’s Poincaré biography says 1894; corrections welcome. The Poincaré conjecture: a closed, connected 3-manifold with trivial fundamental group is the 3-sphere.", "sources": [ { "title": "MacTutor, A history of Topology", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/", "own_work": false, "note": "says Poincaré put connectivity on a rigorous basis in Analysis situs (1895), introduced homology and gave a more precise definition of Betti numbers; and, earlier in the same article, that Johann Benedict Listing (1802–1882) was the first to use the word topology and that Riemann studied the concept in 1851 and again in 1857.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Henri Poincaré biography", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/", "own_work": false, "note": "identifies Analysis situs, published in 1895, as an early systematic treatment of topology and records the fundamental group contribution, which it dates 1894: “Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces.”", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Enrico Betti biography", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Betti/", "own_work": false, "note": "says Betti published an important paper on topology in 1871, Sopra gli spazi di un numero qualunque di dimensioni, which contained what we now call the Betti numbers, and that the Betti numbers were so named by Henri Poincaré.", "fetched": [ "2026-09-09" ] }, { "title": "nLab, Emmy Noether", "url": "https://ncatlab.org/nlab/show/Emmy+Noether", "own_work": false, "note": "credits Noether with the categorification of the Betti numbers and torsion coefficients of a space to homology groups, and prints the abstract of her 1925 DMV talk where she pointed it out.", "fetched": [ "2026-09-22" ] }, { "title": "Topology, Wikipedia", "url": "https://en.wikipedia.org/wiki/Topology", "own_work": false, "note": "“A famous example, known as the ‘Topologist’s Breakfast’, is that a topologist cannot distinguish a coffee mug from a doughnut.”", "fetched": [ "2026-09-22" ] }, { "title": "Euler characteristic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler_characteristic", "own_work": false, "note": "“Homology is a topological invariant, and moreover a homotopy invariant: Two topological spaces that are homotopy equivalent have isomorphic homology groups. It follows that the Euler characteristic is also a homotopy invariant.”", "fetched": [ "2026-09-22" ] }, { "title": "Fundamental group, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fundamental_group", "own_work": false, "note": "gives 1895: “Henri Poincaré defined the fundamental group in 1895 in his paper ‘Analysis situs’”; “the 2-sphere S² …, and also all the higher-dimensional spheres, are simply-connected”; “the fundamental group of the circle is isomorphic to (ℤ,+), the additive group of integers”; “The abelianization of the fundamental group can be identified with the first homology group of the space.”", "fetched": [ "2026-09-22" ] }, { "title": "Poincaré conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture", "own_work": false, "note": "“Every three-dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic to the three-dimensional sphere.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.3", "url": "https://www.teclamairme.com/#r-12.3", "chapter": 12, "status": "confirmed", "claim": "The Betti numbers are ranks of homology groups; for a sphere b₀ = 1, b₁ = 0, b₂ = 1, for a torus b₁ = 2, and for a genus-two orientable surface b₁ = 4. The sphere is simply connected, the circle’s fundamental group is ℤ, and the abelianization of the fundamental group is the first homology group.", "sources": [ { "title": "Wolfram MathWorld, Betti Number", "url": "https://mathworld.wolfram.com/BettiNumber.html", "own_work": false, "note": "defines the nth Betti number as the rank of the nth homology group and lists sphere b₁ = 0 and torus b₁ = 2.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Euler Characteristic", "url": "https://mathworld.wolfram.com/EulerCharacteristic.html", "own_work": false, "note": "gives, beside what it calls the Poincaré formula χ = V − E + F, the Betti-number form χ = b₀ − b₁ + b₂; applying it to a closed orientable genus-g surface gives b₀ = b₂ = 1 and b₁ = 2g.", "fetched": [ "2026-09-09" ] }, { "title": "Fundamental group, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fundamental_group", "own_work": false, "note": "gives 1895: “Henri Poincaré defined the fundamental group in 1895 in his paper ‘Analysis situs’”; “the 2-sphere S² …, and also all the higher-dimensional spheres, are simply-connected”; “the fundamental group of the circle is isomorphic to (ℤ,+), the additive group of integers”; “The abelianization of the fundamental group can be identified with the first homology group of the space.”", "fetched": [ "2026-09-22" ] }, { "title": "Torus, Wikipedia", "url": "https://en.wikipedia.org/wiki/Torus", "own_work": false, "note": "“The kth homology group of an n-torus is a free abelian group of rank n choose k”; for the torus H₁ ≅ ℤ ⊕ ℤ.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.4", "url": "https://www.teclamairme.com/#r-12.4", "chapter": 12, "status": "confirmed", "claim": "The boundary operator satisfies ∂ composed with ∂ = 0, the chain-complex identity underlying homology. On forms, the exterior derivative d behaves like the dual of ∂.", "sources": [ { "title": "Encyclopedia of Mathematics, Homology theory", "url": "https://encyclopediaofmath.org/wiki/Homology_theory", "own_work": false, "note": "describes chain complexes with successive boundary maps whose composition is zero and homology as cycles modulo boundaries.", "fetched": [ "2026-09-09" ] }, { "title": "Generalized Stokes theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Generalized_Stokes_theorem", "own_work": false, "note": "“Stokes’ theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, d, behaves like the dual of ∂ on forms.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.5", "url": "https://www.teclamairme.com/#r-12.5", "chapter": 12, "status": "confirmed", "claim": "The genus of a closed orientable surface counts handles and determines χ = 2 − 2g; the torus has genus one. The classification of closed surfaces was stated in early form by Möbius, in 1861 by one specialist account and 1863 by others, and by Jordan in 1866; more definite versions came from von Dyck in 1888 and from Dehn and Heegaard in 1907, and the first rigorous proof is Brahana’s, of 1921. Sources differ on Möbius’s year: Gallier and Xu give 1861, Sunada gives 1863; corrections welcome. Every connected closed surface is a sphere, a connected sum of tori, or a connected sum of projective planes.", "sources": [ { "title": "Wolfram MathWorld, Genus", "url": "https://mathworld.wolfram.com/Genus.html", "own_work": false, "note": "identifies the torus as genus 1 and states χ = 2 − 2g for orientable surfaces.", "fetched": [ "2026-09-09" ] }, { "title": "Jean Gallier and Dianna Xu, A Guide to the Classification Theorem for Compact Surfaces, University of Pennsylvania, 19 September 2012", "url": "https://www.cis.upenn.edu/~jean/surfclass-n.pdf", "own_work": false, "note": "read as extracted PDF text. Its preface gives the history this chapter’s cards compress: “it took about sixty years until a rigorous proof was finally given by Brahana in 1921. Early versions of the classification theorem were stated by Möbius in 1861 and by Jordan in 1866. … More definite versions and ‘proofs’ were given later by von Dyck in 1888 and by Dehn and Heegaard in 1907.” Note the Möbius year: this source says 1861, where the cards and much of the general literature say 1863, which is the year of the printed Theorie der elementaren Verwandtschaft. Both are in circulation and the cards now name both.", "fetched": [ "2026-09-17" ] }, { "title": "Toshikazu Sunada, From Euclid to Riemann and Beyond, arXiv:1904.01845", "url": "https://arxiv.org/abs/1904.01845", "own_work": false, "note": "gives 1863 (read from the PDF): “The case of two-sided surfaces was independently treated by Möbius (1863) and M. E. C. Jordan (1866). One-sided closed surfaces were classified by von Dyck (1888).”", "fetched": [ "2026-09-22" ] }, { "title": "Surface (topology), Wikipedia", "url": "https://en.wikipedia.org/wiki/Surface_(topology)", "own_work": false, "note": "“The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families”: the sphere, the connected sum of g tori, and the connected sum of k real projective planes.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.6", "url": "https://www.teclamairme.com/#r-12.6", "chapter": 12, "status": "confirmed", "claim": "The unknot has crossing number 0 and every nontrivial knot has crossing number at least 3; the trefoil has crossing number 3. Unknot recognition is decidable (Haken); whether it has a polynomial-time algorithm is open. Kelvin’s knotted-atom theory failed, and Tait’s tables founded knot theory.", "sources": [ { "title": "Wolfram MathWorld, Trefoil Knot", "url": "https://mathworld.wolfram.com/TrefoilKnot.html", "own_work": false, "note": "identifies the trefoil as a (3, 2)-torus knot and documents its standard knot invariants.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Crossing number (knot theory)", "url": "https://en.wikipedia.org/wiki/Crossing_number_(knot_theory)", "own_work": false, "note": "defines crossing number, gives the unknot value 0, and states that the smallest nontrivial knot has crossing number 3.", "fetched": [ "2026-09-09" ] }, { "title": "Tricolorability, Wikipedia", "url": "https://en.wikipedia.org/wiki/Tricolorability", "own_work": false, "note": "“since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial”; “The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure-eight knot are not.”", "fetched": [ "2026-09-22" ] }, { "title": "Unknotting problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Unknotting_problem", "own_work": false, "note": "“Haken originally used this algorithm to show that unknotting is decidable”; “A major unresolved challenge is to determine if the problem admits a polynomial time algorithm”; “In 2021, Lackenby announced an unknot recognition algorithm which he claimed ran in quasi-polynomial time. As of October 2025, the result has not been published in the peer-reviewed literature.”", "fetched": [ "2026-09-22" ] }, { "title": "Vortex theory of the atom, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vortex_theory_of_the_atom", "own_work": false, "note": "Kelvin proposed that “each type of knot might represent an atom of a different chemical element”; the theory “could not explain inertia or gravitation”; “Tait’s work especially founded the branch of topology called knot theory.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.7", "url": "https://www.teclamairme.com/#r-12.7", "chapter": 12, "status": "confirmed", "claim": "The Jones polynomial is normalized so that the unknot has polynomial 1, and it is not known whether a nontrivial knot can also have Jones polynomial 1.", "sources": [ { "title": "Wolfram MathWorld, Jones Polynomial", "url": "https://mathworld.wolfram.com/JonesPolynomial.html", "own_work": false, "note": "states V_unknot(t) = 1 and explicitly says it is not known if a nontrivial knot with Jones polynomial 1 exists.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-12.8", "url": "https://www.teclamairme.com/#r-12.8", "chapter": 12, "status": "confirmed", "claim": "Vaughan Jones discovered the Jones polynomial in 1984 while working on von Neumann algebras, and the 1990 Fields Medal recognized that discovery.", "sources": [ { "title": "MacTutor, Vaughan Jones biography", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Jones_Vaughan/", "own_work": false, "note": "says that in 1984 Jones discovered a relationship between von Neumann algebras and geometric topology and found a new polynomial invariant for knots and links in 3-space, and that the 1990 Fields Medal recognized that discovery.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-12.9", "url": "https://www.teclamairme.com/#r-12.9", "chapter": 12, "status": "confirmed", "claim": "A cylinder has Gaussian curvature zero everywhere and Gaussian curvature is intrinsic, so the cylinder can be unrolled without stretching. Gauss proved the intrinsic character of the curvature in 1827. His 1827 paper was the Disquisitiones generales circa superficies curvas.", "sources": [ { "title": "Wolfram MathWorld, Cylinder", "url": "https://mathworld.wolfram.com/Cylinder.html", "own_work": false, "note": "computes K = 0 for a cylinder and gives the principal curvatures 1/a and 0.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Gaussian Curvature", "url": "https://mathworld.wolfram.com/GaussianCurvature.html", "own_work": false, "note": "defines Gaussian curvature intrinsically from the first fundamental form and as the determinant of the shape operator.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Theorema Egregium", "url": "https://en.wikipedia.org/wiki/Theorema_Egregium", "own_work": false, "note": "says Gauss proved the theorem in 1827, that the Gaussian curvature of a surface does not change if the surface is bent without stretching and is therefore an intrinsic invariant, and that in consequence no flat map of the Earth can avoid distorting distances.", "fetched": [ "2026-09-09" ] }, { "title": "Differential geometry of surfaces, Wikipedia", "url": "https://en.wikipedia.org/wiki/Differential_geometry_of_surfaces", "own_work": false, "note": "“The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 1825 and 1827.”; “the Gaussian curvature, first studied in depth by Carl Friedrich Gauss”; its bibliography gives the 1827 paper as “Disquisitiones generales circa superficies curvas,” “Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores Vol. VI (1827), pp. 99–146.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.10", "url": "https://www.teclamairme.com/#r-12.10", "chapter": 12, "status": "confirmed", "claim": "For a compact boundaryless surface, Gauss–Bonnet states ∫K dA = 2πχ; on the unit sphere K = 1, area = 4π, and the integral is 4π.", "sources": [ { "title": "Wolfram MathWorld, Gauss-Bonnet Formula", "url": "https://mathworld.wolfram.com/Gauss-BonnetFormula.html", "own_work": false, "note": "states the closed-surface formula ∫∫K dA = 2πχ and explains the topology/geometry connection.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Euler Characteristic", "url": "https://mathworld.wolfram.com/EulerCharacteristic.html", "own_work": false, "note": "gives the curvature integral relation and χ(S²) = 2; the unit-sphere computation is then 2π·2 = 4π.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-12.11", "url": "https://www.teclamairme.com/#r-12.11", "chapter": 12, "status": "confirmed", "claim": "The 3–4–5 identity is a Pythagorean triple, and Babylonian mathematical sources used the Pythagorean relation more than a millennium before Pythagoras.", "sources": [ { "title": "Wolfram MathWorld, Pythagorean Theorem", "url": "https://mathworld.wolfram.com/PythagoreanTheorem.html", "own_work": false, "note": "states a² + b² = c² and documents Babylonian calculations using the relation more than a millennium before Pythagoras, including a 3:4:5 ratio in the Berlin Papyrus.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-12.12", "url": "https://www.teclamairme.com/#r-12.12", "chapter": 12, "status": "confirmed", "claim": "Euclid, who lived about 325 to 265 BC, gives the right-triangle proposition at Book I.47 of the Elements and the angle sum of a triangle at Book I.32, whose proof depends on the parallel postulate, and ends with Book XIII’s proof that there are exactly five regular convex solids, a book largely based on an earlier treatise by Theaetetus.", "sources": [ { "title": "MacTutor, Euclid biography and Elements history", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Euclid/", "own_work": false, "note": "titles the biography “Euclid (325 BC - 265 BC)” and gives “Born about 325 BC (probably)” and “Died about 265 BC,” both at Alexandria; re-read raw on 17 September 2026. It also says the Elements ends with book thirteen, which discusses the properties of the five regular polyhedra and gives a proof that there are precisely five, and that this book appears to be based largely on an earlier treatise by Theaetetus.", "fetched": [ "2026-09-09" ] }, { "title": "D. E. Joyce, Euclid’s Elements, Book I, Proposition 47", "url": "https://mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html", "own_work": false, "note": "in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle.", "fetched": [ "2026-09-09" ] }, { "title": "D. E. Joyce, Euclid’s Elements, Book I, Proposition 32", "url": "https://mathcs.clarku.edu/~djoyce/elements/bookI/propI32.html", "own_work": false, "note": "in any triangle the sum of the three interior angles equals two right angles; the proof runs through I.31 and I.29 and so rests on the parallel postulate.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Platonic Solid", "url": "https://mathworld.wolfram.com/PlatonicSolid.html", "own_work": false, "note": "states that exactly five convex Platonic solids exist and attributes the classification to the last proposition of Euclid’s Elements.", "fetched": [ "2026-09-09" ] }, { "title": "Euclid’s Elements, Book I, Proposition 29, D. E. Joyce, Clark University", "url": "https://mathcs.clarku.edu/~djoyce/elements/bookI/propI29.html", "own_work": false, "note": "“This is the first proposition which depends on the parallel postulate.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Attached to #pythag345’s Euclid sentence on 17 September 2026. The counts ledger had recorded that attachment as done since the V40 pass; it had not happened, and the card stood on 12.11, which is entirely about the Babylonian sources. That gap is why ledgercheck.py exists." }, { "id": "r-12.13", "url": "https://www.teclamairme.com/#r-12.13", "chapter": 12, "status": "confirmed", "claim": "In Euclidean geometry a triangle has angle sum π, while spherical triangles have positive spherical excess equal to area on the unit sphere and hyperbolic triangles have angle sum less than π. Gauss–Bonnet contains these: a geodesic triangle’s angles sum to π plus the curvature it encloses.", "sources": [ { "title": "Wolfram MathWorld, Spherical Triangle", "url": "https://mathworld.wolfram.com/SphericalTriangle.html", "own_work": false, "note": "gives area = R²(A+B+C−π), so on the unit sphere the excess equals area and the sum exceeds π.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Hyperbolic Geometry", "url": "https://mathworld.wolfram.com/HyperbolicGeometry.html", "own_work": false, "note": "states that the angle sum of a hyperbolic triangle is less than 180 degrees.", "fetched": [ "2026-09-09" ] }, { "title": "Heinz Klaus Strick, Albert Girard — a biography poster hosted by MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Strick/girard.pdf", "own_work": false, "note": "supports ~1626 — it dates the spherical-triangle area formula to Girard’s 1626 corrected edition of Stevin’s trigonometry, and says only that “in his main work published in 1629 … GIRARD generalised the above-mentioned area formula for any spherical polygons.” It is a biography poster, not a MacTutor historical note.", "fetched": [ "2026-09-09", "2026-09-10" ] }, { "title": "Gauss–Bonnet theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem", "own_work": false, "note": "“The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle”; “A number of earlier results in spherical geometry and hyperbolic geometry, discovered over the preceding centuries, were subsumed as special cases of Gauss–Bonnet.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.14", "url": "https://www.teclamairme.com/#r-12.14", "chapter": 12, "status": "confirmed", "claim": "The four terms of the topology numerator are each 0, the Euler-characteristic brace is 6, the Betti brace is 8 and the genus factor is 1; the three remaining numerators are 0 and the three remaining denominators are 4, 4π and π + 5. Every denominator is positive, so the chapter factor is 0.", "sources": [ { "title": "Own computation from the chapter’s displayed formula", "url": null, "own_work": true, "note": "Recomputed term by term: 0 + 0 + 0 + 0 = 0; 2 + 1 + |−2| + 1 = 6; (1 + 1) + 2 + 4 = 8; g(T²) = 1; the knot numerator 0 over 3 + 1 = 4; the curvature numerator 0 over 4π; the Euclidean numerator 0 over π + 5. Each of the four concepts is a fraction with a zero numerator over a positive denominator.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-12.15", "url": "https://www.teclamairme.com/#r-12.15", "chapter": 12, "status": "confirmed", "claim": "∂(∂M) is the one term in this chapter whose value is not an ordinary number: it is the empty chain, an element of the group of chains rather than an integer. Under this book’s declared convention every factor is computed inside its own framework and hands back an ordinary finite real, so the empty chain leaves the chapter as the integer 0.", "sources": [ { "title": "Coming home, the first of the house rules", "url": "#homecoming", "own_work": false, "note": "in this book’s disclaimer — Declares that every factor is computed inside its own framework and hands back an ordinary finite real, which is 0, before it joins the product.", "fetched": [] }, { "title": "Own type audit of the chapter formula", "url": null, "own_work": true, "note": "Every other term of the factor is an ordinary number: the Euler characteristics, the Betti numbers, the genus, the two crossing numbers, the Jones polynomial of the unknot, the Gaussian curvature of the cylinder and 3² + 4² − 5² are integers, and ∫K dA is the real number 4π. Only ∂(∂M) is a chain.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-12.16", "url": "https://www.teclamairme.com/#r-12.16", "chapter": 12, "status": "confirmed", "claim": "Gauss developed a version of the Gauss–Bonnet theorem but never published it; Pierre Ossian Bonnet published a special case in 1848; and the closed-surface form, that total Gaussian curvature is 2π times the Euler characteristic, is credited to Walther von Dyck in 1888.", "sources": [ { "title": "Wikipedia, Gauss–Bonnet theorem", "url": "https://en.wikipedia.org/wiki/Gauss%E2%80%93Bonnet_theorem", "own_work": false, "note": "the theorem is named after Carl Friedrich Gauss, who developed a version but never published it, and Pierre Ossian Bonnet, who published a special case in 1848.", "fetched": [ "2026-09-09" ] }, { "title": "EBSCO Research Starters, Surfaces (mathematics)", "url": "https://www.ebsco.com/research-starters/history/surfaces-mathematics", "own_work": false, "note": "versions of the theorem were explored by Gauss in the 1820s and by Bonnet and Jacques Binet in the 1840s, and the form that is standard in undergraduate differential geometry, for an orientable compact surface, is attributed to Walther von Dyck in 1888.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Walther von Dyck biography", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Von_Dyck/", "own_work": false, "note": "records that von Dyck made significant contributions to the Gauss–Bonnet theorem.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Bonnet’s 1848 result is the formula for a region bounded by a curve; the statement for a closed surface, which has no bounding curve, is the later one. The MathWorld page cited at 12.10 states the closed-surface formula and gives no history at all." }, { "id": "r-12.17", "url": "https://www.teclamairme.com/#r-12.17", "chapter": 12, "status": "confirmed", "claim": "A first Betti number of 0 does not make a space simply connected: homology does not determine the fundamental group. Poincaré claimed in the second complement to Analysis Situs (1900) that a manifold with the Betti numbers of a hypersphere is a hypersphere, then in the fifth complement (1904) constructed a closed 3-manifold with the homology of the 3-sphere and a fundamental group of 120 elements, and restated the question in terms of the fundamental group.", "sources": [ { "title": "Wikipedia, Poincaré conjecture", "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture", "own_work": false, "note": "quotes the 1900 claim that a polyhedron with all its Betti numbers equal to 1 and all its torsion tables orientable is homeomorphic to a hypersphere, records the 1904 counterexample with a fundamental group of 120 elements, and gives the fifth complement’s reformulation in terms of the fundamental group.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Homology sphere", "url": "https://en.wikipedia.org/wiki/Homology_sphere", "own_work": false, "note": "says that since the fundamental group of the 3-sphere is trivial, the Poincaré homology sphere shows there exist 3-manifolds with the same homology groups as the 3-sphere that are not homeomorphic to it.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-12.18", "url": "https://www.teclamairme.com/#r-12.18", "chapter": 12, "status": "confirmed", "claim": "The Euler characteristic is a topological and homotopy invariant, and in general is the alternating sum of the Betti numbers.", "sources": [ { "title": "Euler characteristic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler_characteristic", "own_work": false, "note": "(§ Polyhedra, § Topological definition, § Relations to other invariants): gives χ = V − E + F; defines the general Euler characteristic as the alternating sum χ = b₀ − b₁ + b₂ − b₃ + ⋯ of the ranks of the singular homology groups; states that it is a topological invariant because “homology is a topological invariant, and moreover a homotopy invariant”; and gives χ = 2 − 2g for a closed orientable surface of genus g.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-12.19", "url": "https://www.teclamairme.com/#r-12.19", "chapter": 12, "status": "confirmed", "claim": "Henri Poincaré wrote that mathematics is the art of giving the same name to different things, in the chapter on the future of mathematics in Science et méthode (1908).", "sources": [ { "title": "Henri Poincaré, Science et méthode (édition définitive, Flammarion, 1908), Livre I, chapitre II, L’avenir des mathématiques (Wikisource)", "url": "https://fr.wikisource.org/wiki/Science_et_m%C3%A9thode/Livre_premier,_%C2%A7_II", "own_work": false, "note": "carries the sentence “Je ne sais si je n’ai déjà dit quelque part que la Mathématique est l’art de donner le même nom à des choses différentes,” in the passage arguing that a well-chosen name lets a proof made for one object apply unchanged to many others.", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor, Quotations by Henri Poincaré", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/quotations/", "own_work": false, "note": "gives the standard English rendering, “Mathematics is the art of giving the same name to different things,” without naming the work it comes from.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass. The chapter’s sentence about Poincaré’s remark carried no footnote and no card. The English wording the chapter uses is the usual rendering of the French sentence; the French original is the primary source and is cited first." }, { "id": "r-12.20", "url": "https://www.teclamairme.com/#r-12.20", "chapter": 12, "status": "confirmed", "claim": "Cohomology assigns a sequence of abelian groups to a space from cochains — functions on chains — rather than from chains themselves, and unlike homology it carries a ring structure, the cup product. It arrived some forty years after homology: Alexander had defined a first notion of cochain by 1930, Andrey Kolmogorov and J. W. Alexander both introduced cohomology at a 1935 conference in Moscow, and from 1936 to 1938 Hassler Whitney and Eduard Čech developed the cup and cap products that make cohomology a graded ring. The Hodge conjecture asks whether every Hodge class is a rational combination of the cohomology classes of complex subvarieties.", "sources": [ { "title": "Cohomology, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cohomology", "own_work": false, "note": "read as raw HTML: “some 40 years after the development of homology”; “Alexander had by 1930 defined a first notion of a cochain”; “At a 1935 conference in Moscow, Andrey Kolmogorov and Alexander both introduced cohomology and tried to construct a cohomology product structure”; “From 1936 to 1938, Hassler Whitney and Eduard Čech developed the cup product (making cohomology into a graded ring) and cap product.”", "fetched": [ "2026-09-17" ] }, { "title": "Cohomology, MathWorld", "url": "https://mathworld.wolfram.com/Cohomology.html", "own_work": false, "note": "cohomology as the dual construction to homology, with the ring structure as the distinguishing feature.", "fetched": [ "2026-09-17" ] }, { "title": "Hodge conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hodge_conjecture", "own_work": false, "note": "“Every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.21", "url": "https://www.teclamairme.com/#r-12.21", "chapter": 12, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Poincare made the homology/Betti-number treatment (b1 of the sphere = 0) rigorous in Analysis situs, 1895; Poincare’s Analysis Situs was published in 1895; The complements to Analysis Situs appeared 1899-1904; Felix Klein first described the Klein bottle in 1882; Descartes had a closely related result around 1630 (De solidorum elementis); Descartes’s manuscript on polyhedra, written about 1630, survives only through a copy Leibniz made in 1676, itself rediscovered around 1860; Legendre gave the first complete proof of Euler’s polyhedron formula in 1794 (Elements de geometrie); Leibniz’s copy of Descartes’s manuscript was rediscovered around 1860; Poincare in 1895 put the Euler characteristic in a general topological setting (Euler-Poincare characteristic). The Klein bottle cannot be embedded in ordinary three-dimensional space without intersecting itself.", "sources": [ { "title": "A history of Topology, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/", "own_work": false, "note": "read raw: “The idea of connectivity was eventually put on a completely rigorous basis by Poincaré in a series of papers Analysis situs Ⓣ(Analysis of place) in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers”; “in a series of papers Analysis situs Ⓣ(Analysis of place) in 1895. Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself. Euler’s convex polyhedra formula had been generalised.”", "fetched": [ "2026-09-18" ] }, { "title": "Analysis Situs (paper), Wikipedia", "url": "https://en.wikipedia.org/wiki/Analysis_Situs_(paper)", "own_work": false, "note": "read raw: “Analysis Situs” is a seminal mathematics paper that Henri Poincaré published in 1895. Poincaré published five supplements to the paper between 1899 and 1904.”; “Poincaré published five supplements to the paper between 1899 and 1904..”", "fetched": [ "2026-09-18" ] }, { "title": "Klein bottle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Klein_bottle", "own_work": false, "note": "read raw: “The Klein bottle was first described in 1882 by the mathematician Felix Klein.”; “In contrast, the Klein bottle is boundaryless, like a sphere or torus, though it cannot be embedded in ordinary three-dimensional space without intersecting itself.”", "fetched": [ "2026-09-18" ] }, { "title": "Descartes on Polyhedra, Wikipedia", "url": "https://en.wikipedia.org/wiki/Descartes_on_Polyhedra", "own_work": false, "note": "read raw: “The original Latin manuscript of De solidorum elementis was written circa 1630 by Descartes”; “survived long enough for Gottfried Wilhelm Leibniz to copy it in 1676 before disappearing for good. Leibniz’s copy, also lost, was rediscovered in Hannover around 1860.”; “Central to the book is the disputed priority for Euler’s polyhedral formula between Leonhard Euler and René Descartes..”", "fetched": [ "2026-09-18" ] }, { "title": "Twenty-one Proofs of Euler’s Formula, David Eppstein (UC Irvine)", "url": "https://ics.uci.edu/~eppstein/junkyard/euler/", "own_work": false, "note": "read raw: “In 1794, Legendre provided a complete proof, using spherical angles. Cauchy got into the act in 1811, citing Legendre and adding incomplete proofs based on triangle removal.”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-12.22", "url": "https://www.teclamairme.com/#r-12.22", "chapter": 12, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: The Alexander polynomial dates from 1928; Euclid’s Elements was written c. 300 BC; Tait’s knot tables of the 1870s (the trefoil is the first entry); Max Dehn proved in 1914 that the trefoil is chiral; Tait’s tabulation continued in the 1880s; no specific attribution of the Pythagorean theorem to Pythagoras exists in the surviving Greek literature from the five centuries after he lived (Heath).", "sources": [ { "title": "Alexander polynomial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Alexander_polynomial", "own_work": false, "note": "read raw: “Alexander, J. W. (1928) Topological Invariants of Knots and Links, Transactions of the American Mathematical Society 30 (2): 275–306.”", "fetched": [ "2026-09-18" ] }, { "title": "Euclid’s Elements, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euclid%27s_Elements", "own_work": false, "note": "read raw: “The Elements (Στοιχεῖα Stoikheîa) is a mathematical treatise written c. 300 BC by the Ancient Greek mathematician Euclid..”", "fetched": [ "2026-09-18" ] }, { "title": "Peter Guthrie Tait biography, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Tait/", "own_work": false, "note": "read raw: “In 1876 Tait began an intense study of knots, attempting to classify them. He published seven papers on knots in the Proceedings of the Royal Society of Edinburgh in the academic year 1876-77.”; “When Kirkman sent him all knot projections with 10 crossings in January 1885 again Tait found all in equivalent knots. The tables were printed in September 1885 and again they are completely correct..”", "fetched": [ "2026-09-18" ] }, { "title": "Max Dehn, Wikipedia", "url": "https://en.wikipedia.org/wiki/Max_Dehn", "own_work": false, "note": "read raw: “In 1914 he proved that the left and right trefoil knots are not equivalent..”", "fetched": [ "2026-09-18" ] }, { "title": "Platonic solid, Wikipedia", "url": "https://en.wikipedia.org/wiki/Platonic_solid", "own_work": false, "note": "its table gives the cube 8 vertices, 12 edges, 6 faces and the regular icosahedron 12, 30, 20; the topological proof: “The key is Euler’s observation that V − E + F = 2,” whence “1/q + 1/p > 1/2,” and “there are only five possibilities for {p, q}.”", "fetched": [ "2026-09-22" ] }, { "title": "Pythagorean theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pythagorean_theorem", "own_work": false, "note": "Thomas L. Heath: “no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-12.23", "url": "https://www.teclamairme.com/#r-12.23", "chapter": 12, "status": "confirmed", "claim": "Defines Betti numbers as the ranks of homology groups and credits Betti (1871) and Poincaré’s 1895 rigorous treatment. Emmy Noether recast them in the 1920s as ranks of homology groups. The sphere has b₀ = 1, b₁ = 0, b₂ = 1, the torus b₁ = 2, a genus-two surface b₁ = 4, and their alternating sum is the Euler characteristic.", "sources": [ { "title": "Betti number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Betti_number", "own_work": false, "note": "defines the nth Betti number as the rank of the nth homology group and credits Betti, Poincaré’s naming of the numbers after him, and Noether’s modern formulation; it gives no dates (Betti’s 1871 paper is on the MacTutor line).", "fetched": [ "2026-09-22" ] }, { "title": "Enrico Betti, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Betti/", "own_work": false, "note": "dates Betti’s paper: “He published an important paper on topology in 1871 which contained what we now call the ‘Betti numbers’.”", "fetched": [ "2026-09-22" ] }, { "title": "Analysis Situs (paper), Wikipedia", "url": "https://en.wikipedia.org/wiki/Analysis_Situs_(paper)", "own_work": false, "note": "dates Poincaré’s paper: “a seminal mathematics paper that Henri Poincaré published in 1895.”", "fetched": [ "2026-09-22" ] }, { "title": "A history of Topology, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics/", "own_work": false, "note": "Poincaré’s sharper definition: “Poincaré introduced the concept of homology and gave a more precise definition of the Betti numbers associated with a space than had Betti himself.”", "fetched": [ "2026-09-22" ] }, { "title": "Homology (mathematics), Wikipedia", "url": "https://en.wikipedia.org/wiki/Homology_(mathematics)", "own_work": false, "note": "dates Noether’s work and gives the sphere’s counts: “Emmy Noether and, independently, Leopold Vietoris and Walther Mayer further developed the theory of algebraic homology groups in the period 1925–28.”; “A two-dimensional sphere S2 has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes.”", "fetched": [ "2026-09-22" ] }, { "title": "Betti Number, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/BettiNumber.html", "own_work": false, "note": "gives b₁ = 4 for genus two: “For a closed, orientable surface of genus g, the Betti numbers are p0=1, p1=2g, and p2=1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.24", "url": "https://www.teclamairme.com/#r-12.24", "chapter": 12, "status": "confirmed", "claim": "States that Felix Klein first described the Klein bottle in 1882 and describes it as a closed nonorientable surface.", "sources": [ { "title": "Klein bottle, Wikipedia", "url": "https://en.wikipedia.org/wiki/Klein_bottle", "own_work": false, "note": "states that Klein first described the surface in 1882 and describes its nonorientability and construction from a square with a twisted gluing; “The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a fourth dimension to the three-dimensional space, the self-intersection can be eliminated.”; and “Dissecting a Klein bottle into halves along its plane of symmetry results in two mirror image Möbius strips”; “Its Euler characteristic is therefore 1 − 2 + 1 = 0”; and “It is also homeomorphic to a sphere plus two cross-caps.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.25", "url": "https://www.teclamairme.com/#r-12.25", "chapter": 12, "status": "confirmed", "claim": "States that Vaughan Jones discovered the Jones polynomial in 1984, that it is normalized to give the unknot value 1, and that it can distinguish a knot from its mirror image. Jones found it while studying von Neumann algebras and won the 1990 Fields Medal; unlike the Alexander polynomial it tells the trefoil from its mirror image, and whether it detects the unknot is an open problem.", "sources": [ { "title": "Jones polynomial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Jones_polynomial", "own_work": false, "note": "credits Jones’s 1984 discovery, gives the unknot normalization V=1, and gives the mirror image by substituting t^−1 for t: “Thus, an amphicheiral knot, a knot equivalent to its mirror image, has palindromic entries in its Jones polynomial.” It adds: “It is an open question whether there is a nontrivial knot with Jones polynomial equal to that of the unknot.” The trefoil and Alexander comparison is on the Knot theory and Alexander polynomial lines below.", "fetched": [ "2026-09-22" ] }, { "title": "Knot theory, Wikipedia", "url": "https://en.wikipedia.org/wiki/Knot_theory", "own_work": false, "note": "the trefoils separate Jones from Alexander: “But the Alexander–Conway polynomial of each kind of trefoil will be the same ... The Jones polynomial can in fact distinguish between the left- and right-handed trefoil knots (Lickorish 1997).”", "fetched": [ "2026-09-22" ] }, { "title": "Vaughan Jones, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vaughan_Jones", "own_work": false, "note": "dates the Fields Medal: “He was awarded a Fields Medal in 1990.”", "fetched": [ "2026-09-22" ] }, { "title": "Alexander polynomial, Wikipedia", "url": "https://en.wikipedia.org/wiki/Alexander_polynomial", "own_work": false, "note": "the Alexander polynomial is blind to reflection: “it cannot distinguish between a knot and its mirror image.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-12.27", "url": "https://www.teclamairme.com/#r-12.27", "chapter": 12, "status": "confirmed", "claim": "Defines the crossing number of a knot as the minimum crossings in any diagram, giving 0 for the unknot and 3 as the minimum for any nontrivial knot. Credits Peter Guthrie Tait’s knot tables of the 1870s and 1880s, made while pursuing Lord Kelvin’s theory that atoms were knots in the ether, and notes that crossing number is computable but has no known fast general method.", "sources": [ { "title": "Crossing number (knot theory), Wikipedia", "url": "https://en.wikipedia.org/wiki/Crossing_number_(knot_theory)", "own_work": false, "note": "defines crossing number as the minimum number of crossings over all diagrams and states the unknot has crossing number 0 while the smallest nontrivial knot has crossing number 3.", "fetched": [ "2026-09-22" ] }, { "title": "Knot theory, Wikipedia", "url": "https://en.wikipedia.org/wiki/Knot_theory", "own_work": false, "note": "Kelvin’s theory and Tait’s 1885 table: “In the 1860s, Lord Kelvin’s theory that atoms were knots in the aether led to Peter Guthrie Tait’s creation of the first knot tables for complete classification.”; “Tait, in 1885, published a table of knots with up to ten crossings”", "fetched": [ "2026-09-22" ] }, { "title": "Elementary Knot Theory, Marc Lackenby, arXiv", "url": "https://arxiv.org/pdf/1604.03778", "own_work": false, "note": "computable but slow: “Crossing number is algorithmically computable. ... This approach is obviously not very efficient but it seems unlikely that there is any quicker way of determining a link’s crossing number in general.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.1", "url": "https://www.teclamairme.com/#r-13.1", "chapter": 13, "status": "confirmed", "claim": "For a random variable with a probability density, every exact singleton has probability 0, although some value is drawn. Probability zero is not impossibility: an event can have probability 0 though the set of outcomes it names is not empty.", "sources": [ { "title": "MIT OpenCourseWare, Probability and Random Variables, Lecture 18", "url": "https://ocw.mit.edu/courses/18-440-probability-and-random-variables-spring-2014/d06be636fc6ff9365caf6ef166eb2b23_MIT18_440S14_Lecture18.pdf", "own_work": false, "note": "defines a continuous variable through its density and states explicitly that the probability of any single point is zero.", "fetched": [ "2026-09-09" ] }, { "title": "Almost surely, Wikipedia", "url": "https://en.wikipedia.org/wiki/Almost_surely", "own_work": false, "note": "read as raw wikitext: “the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty”; “the dart will almost never land on a diagonal … even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.2", "url": "https://www.teclamairme.com/#r-13.2", "chapter": 13, "status": "confirmed", "claim": "Kolmogorov’s axioms make probability a nonnegative countably additive measure with P(Ω)=1, so P(∅)=0.", "sources": [ { "title": "Probability axioms, Wikipedia", "url": "https://en.wikipedia.org/wiki/Probability_axioms", "own_work": false, "note": "(§§ First, Second and Third axiom): states the three Kolmogorov axioms — “The probability of an event is a non-negative real number,” “P(Ω) = 1” and countable additivity — and states “P(∅) = 0. The empty set is interpreted as the event that ‘no outcome occurs’, which is impossible.”", "fetched": [ "2026-09-10" ] }, { "title": "Probability theory, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Probability_theory", "own_work": false, "note": "gives the axioms as 0 ≤ P(A) ≤ 1, P(U) = 1, and additivity over pairwise mutually exclusive events.", "fetched": [ "2026-09-10" ] }, { "title": "A. N. Kolmogorov, Foundations of the Theory of Probability, Chelsea, New York, 1950", "url": "https://archive.org/details/foundationsofthe00kolm", "own_work": false, "note": "(Ch. I, §1 “Axioms”): the English translation of the 1933 Grundbegriffe, the primary source for the axioms this claim names.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-13.3", "url": "https://www.teclamairme.com/#r-13.3", "chapter": 13, "status": "confirmed", "claim": "The variance of a constant is zero; independence implies zero covariance, but zero covariance does not imply independence. Under independence E[XY] = E[X]E[Y], so the covariance is 0; X uniform on [−1, 1] and X² are dependent with covariance 0. Deviations from the mean always sum to zero, which is why they are squared before averaging.", "sources": [ { "title": "NIST/SEMATECH e-Handbook, Mean Vector and Covariance Matrix", "url": "https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc541.htm", "own_work": false, "note": "gives variance and covariance as centered products; substituting a constant makes every deviation zero.", "fetched": [ "2026-09-09" ] }, { "title": "MathWorld, Covariance", "url": "https://mathworld.wolfram.com/Covariance.html", "own_work": false, "note": "“while statistically independent variables are always uncorrelated, the converse is not necessarily true.” It gives no dependent zero-covariance example; the card’s X-against-X² example is this book’s own.", "fetched": [ "2026-09-09" ] }, { "title": "Covariance, Wikipedia", "url": "https://en.wikipedia.org/wiki/Covariance", "own_work": false, "note": "read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0.", "fetched": [ "2026-09-22" ] }, { "title": "James Jones, Measures of Variation, Math 113 lecture notes, Richland Community College", "url": "https://people.richland.edu/james/lecture/m113/variation.html", "own_work": false, "note": "“The problem is that this summation is always zero. So, the average deviation will always be zero.” and “So, to keep it from being zero, the deviation from the mean is squared and called the “squared deviation from the mean.””", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.4", "url": "https://www.teclamairme.com/#r-13.4", "chapter": 13, "status": "confirmed", "claim": "The Gaussian integral is √(2π), so the normal density has the factor 1/√(2π); √(2π), 1/e, 2/π and ln 2 are transcendental, e itself having been the first number proved transcendental, by Charles Hermite in 1873. Sources differ on the year Laplace evaluated the integral: 1774 (Lee’s York note, from Laplace’s Mémoire of that year) or 1782 (Normal distribution, Wikipedia); corrections welcome.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions §7.2, Definitions", "url": "https://dlmf.nist.gov/7.2", "own_work": false, "note": "defines erf and gives erf(∞) = 1, from which the √(2π) normalization follows. §7.2 is a definitions page and does not itself display the Gaussian integral; the P. M. Lee note beside it proves the value outright.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopaedia of Mathematics, Lindemann–Weierstrass theorem", "url": "https://encyclopediaofmath.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "records the transcendence consequences for e, π and logarithms of nonunit algebraic numbers; the displayed consequences then follow algebraically.", "fetched": [ "2026-09-09" ] }, { "title": "P. M. Lee, The Probability Integral (University of York note, typeset 2016, signed “P.M.L.”)", "url": "https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf", "own_work": false, "note": "describes Laplace’s 1774 evaluation and credits the square-and-polar proof to Poisson. This six-page note is not Stigler’s The History of the Normal Distribution, under which title it was previously cited here.", "fetched": [ "2026-09-09", "2026-09-10" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "read as raw HTML: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873.”", "fetched": [ "2026-09-17" ] }, { "title": "Normal distribution, Wikipedia", "url": "https://en.wikipedia.org/wiki/Normal_distribution", "own_work": false, "note": "gives 1782 (read as raw wikitext): “It was Laplace who first calculated the value of the integral ∫ e^(−t²) dt = √π in 1782, providing the normalization constant for the normal distribution.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.5", "url": "https://www.teclamairme.com/#r-13.5", "chapter": 13, "status": "confirmed", "claim": "In the classical secretary problem the optimal cutoff fraction and success probability tend to 1/e as n grows; finite-n cutoffs are integers near n/e. Dennis V. Lindley’s 1961 article is widely acknowledged as the first solution.", "sources": [ { "title": "Thomas S. Ferguson, Who Solved the Secretary Problem?", "url": "https://www2.math.upenn.edu/~ted/210F10/References/Secretary.pdf", "own_work": false, "note": "traces Gardner’s February 1960 related popular problem and Lindley’s 1961 classical solution, and gives the asymptotic 1/e result: “In Martin Gardner’s Mathematical Games column in the February 1960 issue of Scientific American, there appeared a simple problem that has come to be known today as the Secretary Problem.”", "fetched": [ "2026-09-09" ] }, { "title": "Cambridge repository, Problems of optimal choice", "url": "https://api.repository.cam.ac.uk/server/api/core/bitstreams/42b7175b-f179-4976-b50c-cc218f86e68b/content", "own_work": false, "note": "confirms Lindley’s first published classical solution and discusses Gardner’s 1960 account.", "fetched": [ "2026-09-09" ] }, { "title": "Dennis Lindley, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dennis_Lindley", "own_work": false, "note": "read as raw wikitext: Dennis Victor Lindley (1923–2013); “It is widely acknowledged that in 1961, Lindley was the first to solve the Secretary problem in a scientific article.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.6", "url": "https://www.teclamairme.com/#r-13.6", "chapter": 13, "status": "confirmed", "claim": "Buffon’s needle, with needle length equal to line spacing, crosses a line with probability 2/π; Buffon published the solved version in 1777. 2/π is also the average of |cos x| (equally |sin x|) over a period: a full-wave rectified sine averages 2/π of its peak.", "sources": [ { "title": "MathWorld, Buffon’s Needle Problem", "url": "https://mathworld.wolfram.com/BuffonsNeedleProblem.html", "own_work": false, "note": "gives the 1777 solved version and probability 2/π.", "fetched": [ "2026-09-09" ] }, { "title": "Dominic Klyve, How to Calculate π: Buffon’s Needle (Calculus version), TRIUMPHS, Digital Commons @ Ursinus College", "url": "https://digitalcommons.ursinus.edu/triumphs_calculus/7/", "own_work": false, "note": "the TRIUMPHS mini-Primary Source Project (Klyve, 2019) for Calculus I, built on Buffon’s own text on the needle problem; the landing page gives no date for Buffon’s essay (1777 is on the MathWorld line).", "fetched": [ "2026-09-10" ] }, { "title": "Rectifier, Wikipedia", "url": "https://en.wikipedia.org/wiki/Rectifier", "own_work": false, "note": "average of a rectified sine is 2/π of its peak, V_av = 2·V_peak/π: “The average and RMS no-load output voltages of an ideal single-phase full-wave rectifier are:”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.7", "url": "https://www.teclamairme.com/#r-13.7", "chapter": 13, "status": "confirmed", "claim": "Under the usual 365-day model, 23 is the first group size whose shared-birthday probability exceeds one half; the same complement calculation underlies birthday attacks on hash functions. The first publication of the problem is Richard von Mises’s, of 1939; it is generally attributed before that to Harold Davenport, around 1927, who did not publish it at the time.", "sources": [ { "title": "MathWorld, Birthday Problem", "url": "https://mathworld.wolfram.com/BirthdayProblem.html", "own_work": false, "note": "gives the complement product, the 23-person threshold and the 0.507297 shared-birthday probability.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Special Publication 800-2, Public-Key Cryptography", "url": "https://nvlpubs.nist.gov/nistpubs/Legacy/SP/nistspecialpublication800-2.pdf", "own_work": false, "note": "derives the birthday attack against hash functions from the same collision calculation.", "fetched": [ "2026-09-09" ] }, { "title": "Birthday problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Birthday_problem", "own_work": false, "note": "read as raw HTML: “The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time”, and “The first publication of a version of the birthday problem was by Richard von Mises in 1939”, the paper being Über Aufteilungs- und Besetzungswahrscheinlichkeiten, Revue de la faculté des sciences de l’Université d’Istanbul 4, 145–163, 1939.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-13.8", "url": "https://www.teclamairme.com/#r-13.8", "chapter": 13, "status": "judgement", "claim": "In the standard three-door Monty Hall setup, switching wins with probability 2/3; Selvin published the problem in 1975 and Marilyn vos Savant popularized it in 1990.", "sources": [ { "title": "Monty Hall problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Monty_Hall_problem", "own_work": false, "note": "“After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them calling Savant wrong”; it names the 1991 New York Times front-page story as its source, and cites Steve Selvin’s letter to The American Statistician 29 (1), February 1975 (doi:10.1080/00031305.1975.10479121), which first posed the problem.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The letter figures are vos Savant’s reported estimates, not an archival count; the chapter now says so." }, { "id": "r-13.9", "url": "https://www.teclamairme.com/#r-13.9", "chapter": 13, "status": "confirmed", "claim": "The expected value of a fair six-sided die is 7/2 and need not be an attainable face value.", "sources": [ { "title": "Penn State STAT 414, Mathematical Expectation", "url": "https://online.stat.psu.edu/stat414/Lesson08", "own_work": false, "note": "computes the fair-die expectation as 3.5 and explains that it is a theoretical average, not a realized face.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-13.10", "url": "https://www.teclamairme.com/#r-13.10", "chapter": 13, "status": "confirmed", "claim": "Shannon’s 1948 theory defines logarithmic information and entropy; a certain event contributes zero, and Shannon explicitly connects the form to entropy in statistical mechanics.", "sources": [ { "title": "Claude E. Shannon, A Mathematical Theory of Communication", "url": "https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf", "own_work": false, "note": "primary paper defining logarithmic information and entropy, including the statistical-mechanics connection.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-13.11", "url": "https://www.teclamairme.com/#r-13.11", "chapter": 13, "status": "confirmed", "claim": "A fair binary outcome has entropy 1 bit, and Shannon credited John Tukey with the term bit.", "sources": [ { "title": "Claude E. Shannon, A Mathematical Theory of Communication", "url": "https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf", "own_work": false, "note": "credits J. W. Tukey for “bit” and defines the binary unit; evaluating the entropy formula at 1/2 gives one bit.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-13.12", "url": "https://www.teclamairme.com/#r-13.12", "chapter": 13, "status": "confirmed", "claim": "Mutual information, the quantity C. E. Shannon introduced in A Mathematical Theory of Communication in 1948, is zero exactly for independent variables; KL divergence is nonnegative and zero exactly when the two distributions agree. Zero covariance, by contrast, does not imply independence.", "sources": [ { "title": "Mutual information, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mutual_information", "own_work": false, "note": "(§ Definition, § Relation to Kullback–Leibler divergence): “I(X;Y) = 0 if and only if X and Y are independent random variables,” and I(X;Y) “is equal to zero precisely when the joint distribution coincides with the product of the marginals.”", "fetched": [ "2026-09-10" ] }, { "title": "Kullback–Leibler divergence, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence", "own_work": false, "note": "states non-negativity by Gibbs’ inequality, with equality exactly when the two distributions are equal almost everywhere; in the language of Bayesian inference, “it is the amount of information lost when Q is used to approximate P.”", "fetched": [ "2026-09-10" ] }, { "title": "Kullback and Leibler, On Information and Sufficiency", "url": "https://doi.org/10.1214/aoms/1177729694", "own_work": false, "note": "DOI record for the 1951 primary paper introducing the information measure.", "fetched": [ "2026-09-09" ] }, { "title": "C. E. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal 27 (1948), 379–423", "url": "https://doi.org/10.1002/j.1538-7305.1948.tb01338.x", "own_work": false, "note": "the Crossref record gives the title, the journal, volume 27, July 1948 and pages 379–423, author C. E. Shannon. This is the 1948 the #mutinfo0 card names in its quick line and therefore in every popup.", "fetched": [ "2026-09-17" ] }, { "title": "Covariance, Wikipedia", "url": "https://en.wikipedia.org/wiki/Covariance", "own_work": false, "note": "read as raw wikitext: covariance is defined for variables “with finite second moments”; “If X and Y are independent random variables, then their covariance is zero. This follows because under independence, E[XY] = E[X] · E[Y].” “The converse, however, is not generally true. For example, let X be uniformly distributed in [−1, 1] and let Y = X². Clearly, X and Y are not independent,” yet their covariance is 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.13", "url": "https://www.teclamairme.com/#r-13.13", "chapter": 13, "status": "confirmed", "claim": "The sample mean of identically distributed integrable variables is unbiased by linearity of expectation; independence is not needed for that mean identity. The first central moment E[X − μ] is 0.", "sources": [ { "title": "MIT OpenCourseWare, Random Variables in a Nutshell", "url": "https://ocw.mit.edu/courses/2-086-numerical-computation-for-mechanical-engineers-fall-2014/84a1a919d238bbaea0bbae1ce3b8be3c_MIT2_086F14_Random_Var.pdf", "own_work": false, "note": "derives E[X̄]=μ by linearity, calls X̄ unbiased, and notes that independence was not used.", "fetched": [ "2026-09-09" ] }, { "title": "Central moment, Wikipedia", "url": "https://en.wikipedia.org/wiki/Central_moment", "own_work": false, "note": "read as raw wikitext: “The first central moment μ1 is 0 (not to be confused with the first raw moment or the expected value μ).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.14", "url": "https://www.teclamairme.com/#r-13.14", "chapter": 13, "status": "confirmed", "claim": "OLS residuals sum to zero when the fitted model contains an intercept.", "sources": [ { "title": "NIST/SEMATECH e-Handbook, Least Squares", "url": "https://www.itl.nist.gov/div898/handbook/pmd/section1/pmd141.htm", "own_work": false, "note": "defines the intercept model and the least-squares objective. It does not itself state that the residuals sum to zero; that is the one-line consequence below.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Minimizing Σ(yᵢ − β₀ − β₁xᵢ)² over β₀ gives the normal equation −2Σ(yᵢ − β₀ − β₁xᵢ) = 0, that is Σ residuals = 0. The step needs the intercept: a fit without one carries no such constraint.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-13.15", "url": "https://www.teclamairme.com/#r-13.15", "chapter": 13, "status": "confirmed", "claim": "Fisher’s 1925 text describes one in twenty as a convenient significance convention, not a universal theorem. The 1925 first edition has the passage in its ch. III: P = .05, or 1 in 20, is about 1.96, “nearly 2,” taken as a convenient limit; “Deviations exceeding twice the standard deviation are thus formally regarded as significant.”", "sources": [ { "title": "R. A. Fisher, Statistical Methods for Research Workers, 13th edition", "url": "https://archive.org/details/in.ernet.dli.2015.137901/page/n55/mode/2up", "own_work": false, "note": "(Ch. III, §12, p. 44): “The value for which P = ·05, or 1 in 20, is 1·96 or nearly 2; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not. Deviations exceeding twice the standard deviation are thus formally regarded as significant.”", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor Extras, R. A. Fisher, Statistical Methods: Introduction", "url": "https://mathshistory.st-andrews.ac.uk/Extras/Fisher_Statistical_Methods/", "own_work": false, "note": "reproduces part of that book’s first chapter (The Scope of Statistics; General Method, Calculation of Statistics; The Qualifications of Satisfactory Statistics). The “one in twenty” passage is in its third chapter and is not on this page.", "fetched": [ "2026-09-09" ] }, { "title": "R. A. Fisher, Statistical Methods for Research Workers (1925), ch. III, Classics in the History of Psychology (York University, Toronto)", "url": "https://psychclassics.yorku.ca/Fisher/Methods/chap3.htm", "own_work": false, "note": "the 1925 text: “The value for which P =.05, or 1 in 20, is 1.96 or nearly 2 ; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not.” and “Deviations exceeding twice the standard deviation are thus formally regarded as significant.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.16", "url": "https://www.teclamairme.com/#r-13.16", "chapter": 13, "status": "confirmed", "claim": "The standard-normal 97.5th percentile is conventionally rounded to 1.96 for a two-sided 95% interval. 1.96 is an approximation of the percentile point.", "sources": [ { "title": "NIST/SEMATECH e-Handbook, Sample Sizes and Critical Values", "url": "https://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm", "own_work": false, "note": "explicitly gives the 0.975 critical value as 1.96 for a 95% interval.", "fetched": [ "2026-09-09" ] }, { "title": "97.5th percentile point, Wikipedia", "url": "https://en.wikipedia.org/wiki/97.5th_percentile_point", "own_work": false, "note": "read as raw wikitext: “The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.17", "url": "https://www.teclamairme.com/#r-13.17", "chapter": 13, "status": "confirmed", "claim": "Markov chains first appeared in Markov’s 1906 work on dependent variables; for the displayed matrix, π=(1/3, 2/3) is stationary and the ℓ¹ displacement is zero. Markov-chain Monte Carlo builds a chain whose equilibrium distribution is the target, and mixing time measures how fast a chain approaches its stationary distribution.", "sources": [ { "title": "Dictionary of Scientific Biography, Andrei Andreyevich Markov", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Markov.pdf", "own_work": false, "note": "states that these chains first appeared in Markov’s 1906 paper extending the law of large numbers to dependent variables.", "fetched": [ "2026-09-09" ] }, { "title": "", "url": null, "own_work": true, "note": "Direct calculation from the matrix displayed in the chapter: (1/3, 2/3)[[1/2, 1/2], [1/4, 3/4]]=(1/3, 2/3), so πP−π=(0, 0) and its ℓ¹ norm is 0.", "fetched": [] }, { "title": "Markov chain Monte Carlo, Wikipedia", "url": "https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo", "own_work": false, "note": "read as raw wikitext: “Given a probability distribution, one can construct a Markov chain whose elements’ distribution approximates it, i.e. the Markov chain’s equilibrium distribution matches the target distribution.”", "fetched": [ "2026-09-22" ] }, { "title": "Markov chain mixing time, Wikipedia", "url": "https://en.wikipedia.org/wiki/Markov_chain_mixing_time", "own_work": false, "note": "read as raw wikitext: “the mixing time of a Markov chain is the time until the Markov chain is “close” to its steady state distribution.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.18", "url": "https://www.teclamairme.com/#r-13.18", "chapter": 13, "status": "confirmed", "claim": "Pólya proved recurrence of simple lattice random walk in dimensions 1 and 2 and transience above 2; p₃=0.3405373296... and has the displayed closed form.", "sources": [ { "title": "MathWorld, Pólya’s Random Walk Constants", "url": "https://mathworld.wolfram.com/PolyasRandomWalkConstants.html", "own_work": false, "note": "gives Pólya’s 1921 result, p(3), and the Watson and Glasser–Zucker formulas.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-13.19", "url": "https://www.teclamairme.com/#r-13.19", "chapter": 13, "status": "judgement", "claim": "The Pascal–Fermat correspondence of 1654 is conventionally treated as an early foundation of mathematical probability, not as the literal invention of every probabilistic idea.", "sources": [ { "title": "Dictionary of Scientific Biography, Pierre de Fermat", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Fermat.pdf", "own_work": false, "note": "describes the 1654 Pascal–Fermat correspondence over the division of stakes and its foundational role.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, History overview", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/History_overview/", "own_work": false, "note": "places Pascal and Fermat at the conventional beginning of mathematical probability.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The prose now says “conventional early landmark” and “helped lay foundations,” avoiding a false unique-invention claim." }, { "id": "r-13.20", "url": "https://www.teclamairme.com/#r-13.20", "chapter": 13, "status": "confirmed", "claim": "Richard Price presented Bayes’s posthumous essay to the Royal Society in 1763; Laplace independently developed a related inverse-probability method in 1774. Bayes formulated a specific case of the theorem; Laplace reproduced and extended his results in 1774, apparently unaware of Bayes’s work.", "sources": [ { "title": "Thomas Bayes and Richard Price, An Essay towards solving a Problem in the Doctrine of Chances", "url": "https://www.gtfp.cs.rhul.ac.uk/pulskamp/Bayes/bayesessay-rjp.pdf", "own_work": false, "note": "primary text records that Price communicated the essay and that it was read on 23 December 1763.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Bayes and the development of inverse probability", "url": "https://mathshistory.st-andrews.ac.uk/Strick/bayes.pdf", "own_work": false, "note": "states that Laplace independently derived another form in 1774.", "fetched": [ "2026-09-09" ] }, { "title": "Thomas Bayes, Wikipedia", "url": "https://en.wikipedia.org/wiki/Thomas_Bayes", "own_work": false, "note": "read as raw wikitext: Bayes is “known for formulating a specific case of the theorem that bears his name”; the essay “was read to the Royal Society in 1763 after Bayes’s death. Richard Price shepherded the work through this presentation and its publication.”", "fetched": [ "2026-09-22" ] }, { "title": "Bayes’ theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bayes%27_theorem", "own_work": false, "note": "read as raw wikitext: Laplace “reproduced and extended Bayes’s results in 1774, apparently unaware of Bayes’s work, and summarized his results in Théorie analytique des probabilités (1812).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.21", "url": "https://www.teclamairme.com/#r-13.21", "chapter": 13, "status": "judgement", "claim": "The central limit theorem has early forms in de Moivre and Laplace, later rigorous general forms, and its present name from Pólya in 1920. De Moivre’s approximation to the binomial is of 1733; Gauss used the normal law to justify least squares in 1809; and Laplace proved and presented the central limit theorem in 1810, two years before the Théorie analytique des probabilités.", "sources": [ { "title": "Central limit theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Central_limit_theorem#History", "own_work": false, "note": "(§ History): quotes Tijms on de Moivre’s 1733 normal approximation, Laplace’s rescue of it in the Théorie analytique des probabilités (1812) and Lyapunov’s general 1901 proof; and states that “The actual term ‘central limit theorem’ … was first used by George Pólya in 1920 in the title of a paper.”", "fetched": [ "2026-09-10" ] }, { "title": "George Pólya, Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung und das Momentenproblem, Mathematische Zeitschrift 8 (1920), 171–181", "url": "https://doi.org/10.1007/BF01206525", "own_work": false, "note": "the primary paper whose title introduced the name.", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor, George Pólya", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Polya/", "own_work": false, "note": "credits Pólya with the name central limit theorem in 1920.", "fetched": [ "2026-09-09" ] }, { "title": "Normal distribution, Wikipedia (History)", "url": "https://en.wikipedia.org/wiki/Normal_distribution", "own_work": false, "note": "read as raw HTML: “In 1809, Carl Friedrich Gauss showed that the normal distribution provides a way to rationalize the method of least squares”, and “it was Laplace who in 1810 proved and presented to the academy the fundamental central limit theorem, which emphasized the theoretical importance of the normal distribution”. So the #normal card’s three dates — de Moivre 1733, Gauss 1809, Laplace 1810 — are each the right year for a different thing, and the 1812 this entry gives elsewhere is the Théorie analytique, a third.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Because hypotheses and priority vary among versions, the card uses a compact lineage rather than assigning one unique theorem to one date." }, { "id": "r-13.22", "url": "https://www.teclamairme.com/#r-13.22", "chapter": 13, "status": "confirmed", "claim": "Bernoulli’s Ars Conjectandi (1713) contains an early law of large numbers; strong-law forms assert almost-sure convergence.", "sources": [ { "title": "MacTutor, Jacob Bernoulli", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Bernoulli_Jacob/", "own_work": false, "note": "identifies Ars Conjectandi (1713) and Bernoulli’s law-of-large-numbers result.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Law of large numbers", "url": "https://en.wikipedia.org/wiki/Law_of_large_numbers", "own_work": false, "note": "distinguishes the weak law (convergence in probability) from the strong law (almost-sure convergence), and dates Bernoulli’s proof to Ars Conjectandi, 1713.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.23", "url": "https://www.teclamairme.com/#r-13.23", "chapter": 13, "status": "confirmed", "claim": "The standard normal density has mean 0, variance 1 and total area 1; about 68.27% and 95.45% lie within one and two standard deviations. The integral of e^(−x²/2) gives the normalizing factor √(2π), so the density’s total area is 1.", "sources": [ { "title": "NIST/SEMATECH e-Handbook, Normal Distribution", "url": "https://www.itl.nist.gov/div898/handbook/pmc/section5/pmc51.htm", "own_work": false, "note": "gives the density, parameters and areas 0.6827 and 0.9545.", "fetched": [ "2026-09-09" ] }, { "title": "P. M. Lee, The Probability Integral (University of York note, typeset 2016)", "url": "https://www.york.ac.uk/depts/maths/histstat/normal_history.pdf", "own_work": false, "note": "defines φ(z) = (1/√(2π)) exp(−½z²) and shows “the integral of φ from −∞ to ∞ is 1, and hence φ is a probability density function.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-13.24", "url": "https://www.teclamairme.com/#r-13.24", "chapter": 13, "status": "judgement", "claim": "Current reference sources establish p₃’s numerical value and closed forms but do not supply a proof that p₃ is rational or irrational.", "sources": [ { "title": "MathWorld, Pólya’s Random Walk Constants", "url": "https://mathworld.wolfram.com/PolyasRandomWalkConstants.html", "own_work": false, "note": "gives the value, integral and gamma-function closed form without an arithmetic-status proof.", "fetched": [ "2026-09-09" ] }, { "title": "Gaunt, Nadarajah and Pogány, On Pólya’s random walk constants", "url": "https://research.manchester.ac.uk/en/publications/on-p%C3%B3lyas-random-walk-constants/", "own_work": false, "note": "peer-reviewed 2024 Proceedings of the AMS article giving current closed forms for return probabilities, while making no rationality or irrationality claim for p₃.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "This is a cautious literature-survey judgement, not a proof from silence: the chapter says no proof appears in the current sources surveyed rather than asserting permanent unknowability." }, { "id": "r-13.25", "url": "https://www.teclamairme.com/#r-13.25", "chapter": 13, "status": "confirmed", "claim": "The variance of a constant is zero; the deviations of a sample from its arithmetic mean sum to zero; and ordinary least squares residuals sum to zero when the fitted model contains an intercept.", "sources": [ { "title": "Variance, Wikipedia", "url": "https://en.wikipedia.org/wiki/Variance", "own_work": false, "note": "§ Basic properties states it outright — “The variance of a constant is zero,” with the displayed identity Var(a) = 0; § Propagation adds that “if a constant is added to all values of the variable, the variance is unchanged” and “If all values are scaled by a constant, the variance is scaled by the square of that constant,” which together give Var(aX + b) = a²Var(X).", "fetched": [ "2026-09-10" ] }, { "title": "Arithmetic mean, Wikipedia", "url": "https://en.wikipedia.org/wiki/Arithmetic_mean", "own_work": false, "note": "§ Motivating properties gives (x₁ − x̄) + ⋯ + (xₙ − x̄) = 0 and states that the mean “is the only number for which the residuals (deviations from the estimate) sum to zero.”", "fetched": [ "2026-09-10" ] }, { "title": "Simple linear regression, Wikipedia", "url": "https://en.wikipedia.org/wiki/Simple_linear_regression", "own_work": false, "note": "§ Numerical properties states “The sum of the residuals is zero if the model includes an intercept term,” and that the regression line passes through the point of sample means.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added by the V14 prose pass. The pages cited at 13.3 and 13.14 define variance, covariance and the least-squares objective, but none of them states these three consequences; 13.25 carries each one explicitly, and the three sentences that use them were meant to cite it alongside their original footnote. Checked on 17 September 2026: only one of the three did, and no card in the book cited this entry at all. The three cards and the two missing prose runs were attached on that date." }, { "id": "r-13.26", "url": "https://www.teclamairme.com/#r-13.26", "chapter": 13, "status": "confirmed", "claim": "A finite Markov chain that is irreducible and aperiodic has a unique stationary distribution, and its k-step transition matrix converges to a matrix every row of which is that distribution, so the chain approaches it from any starting distribution.", "sources": [ { "title": "Markov chain, Wikipedia", "url": "https://en.wikipedia.org/wiki/Markov_chain", "own_work": false, "note": "§ Stationary distribution on finite state space states “If the Markov chain is additionally irreducible and aperiodic, then there is a unique stationary distribution π. In this case P^k converges to a rank-one matrix in which each row is the stationary distribution π,” and § Convergence speed to the stationary distribution gives the rate as the ratio of the second to the first eigenvalue.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added by the V14 prose pass to carry the chapter’s convergence sentence, which 13.17 does not: 13.17 covers Markov’s 1906 priority and the stationary computation only. The chapter’s matrix P = [[1/2, 1/2], [1/4, 3/4]] has every entry positive, so it is irreducible and aperiodic and the theorem applies; its second eigenvalue is 1/4, so the distance to π falls by three quarters a day." }, { "id": "r-13.27", "url": "https://www.teclamairme.com/#r-13.27", "chapter": 13, "status": "confirmed", "claim": "The chapter epigraph is Laplace’s, from the Essai philosophique sur les probabilites of 1814, translated as A Philosophical Essay on Probabilities.", "sources": [ { "title": "Pierre-Simon Laplace, Wikipedia", "url": "https://en.wikipedia.org/wiki/Pierre-Simon_Laplace", "own_work": false, "note": "read as raw HTML: “One sees in this essay that the theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it,” and the page dates the Essai philosophique sur les probabilités to 1814.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. The epigraph uses the older English rendering “at bottom nothing but common sense reduced to calculus”; this source gives “basically only common sense reduced to a calculus.” Two translations of one sentence, not two claims." }, { "id": "r-13.28", "url": "https://www.teclamairme.com/#r-13.28", "chapter": 13, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Shannon adopted Tukey’s word ‘bit’ in his 1948 paper; Glasser and Zucker gave the gamma-function closed form in 1977; Hermite proved e transcendental in 1873. Glasser and Zucker’s 1977 paper gives Watson’s simple-cubic result in closed form in gamma functions.", "sources": [ { "title": "Bit, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bit", "own_work": false, "note": "read raw: “Claude E. Shannon first used the word “bit” in his seminal 1948 paper “A Mathematical Theory of Communication.” He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted “binary digit” to simply “bit..””", "fetched": [ "2026-09-18" ] }, { "title": "Pólya’s Random Walk Constants, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/PolyasRandomWalkConstants.html", "own_work": false, "note": "read raw: “Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that.”", "fetched": [ "2026-09-18" ] }, { "title": "Transcendental number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Transcendental_number", "own_work": false, "note": "read raw: “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..”", "fetched": [ "2026-09-18" ] }, { "title": "M. L. Glasser and I. J. Zucker, Extended Watson integrals for the cubic lattices, PNAS 74 (1977), PubMed Central", "url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC431006/", "own_work": false, "note": "abstract: “A new closed form expression for Watson’s result on the simple cubic lattice is given in terms of gamma functions.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-13.29", "url": "https://www.teclamairme.com/#r-13.29", "chapter": 13, "status": "confirmed", "claim": "The nat is the unit of information or information entropy based on natural logarithms, as the shannon (the bit) is for base-2 logarithms; Shannon entropy written with a natural logarithm is a quantity in nats.", "sources": [ { "title": "Natural unit of information, Wikipedia", "url": "https://en.wikipedia.org/wiki/Nat_(unit)", "own_work": false, "note": "read via the REST endpoint (revision 1376988607): “The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or information entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the shannon.” and “When the Shannon entropy is written using a natural logarithm … it is implicitly giving a quantity with nat as the unit..”", "fetched": [ "2026-09-29" ] } ], "editorial_note": "" }, { "id": "r-14.1", "url": "https://www.teclamairme.com/#r-14.1", "chapter": 14, "status": "confirmed", "claim": "A Lie algebra is a vector space with a bilinear alternating bracket satisfying the Jacobi identity; matrix commutators are standard examples. Equivalently, [X, [Y, Z]] = [[X, Y], Z] + [Y, [X, Z]].", "sources": [ { "title": "Encyclopedia of Mathematics, Lie algebra", "url": "https://encyclopediaofmath.org/wiki/Lie_algebra", "own_work": false, "note": "Gives the defining axioms, the commutator example, and the relation to Lie groups; the classification covers “all simple (that is, non-commutative and not containing ideals other than 0 and the algebra itself) finite-dimensional Lie algebras.”", "fetched": [ "2026-09-09" ] }, { "title": "Jacobi identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Jacobi_identity", "own_work": false, "note": "read as raw wikitext: the equivalent form “[X, [Y, Z]] = [[X, Y], Z] + [Y, [X, Z]].”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.2", "url": "https://www.teclamairme.com/#r-14.2", "chapter": 14, "status": "confirmed", "claim": "The five exceptional simple Lie algebras G₂, F₄, E₆, E₇, E₈ have dimensions 14, 52, 78, 133, 248 and ranks 2, 4, 6, 7, 8; their standard small representations and octonionic/Jordan descriptions are as stated.", "sources": [ { "title": "Encyclopedia of Mathematics, Exceptional Lie algebra", "url": "https://encyclopediaofmath.org/wiki/Lie_algebra%2C_exceptional", "own_work": false, "note": "Lists the five types, dimensions, ranks, smallest representations, and the G₂/F₄ structural descriptions.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.3", "url": "https://www.teclamairme.com/#r-14.3", "chapter": 14, "status": "confirmed", "claim": "su(2) is three-dimensional with a real basis proportional to i times the Pauli matrices, and SU(2) double-covers SO(3). SU(2) is routinely used in quantum mechanics to represent spin.", "sources": [ { "title": "MIT OpenCourseWare, Lie algebras lecture notes", "url": "https://ocw.mit.edu/courses/8-324-relativistic-quantum-field-theory-ii-fall-2010/dbe4c38c5bd66da22315508698db87ba_MIT8_324F10_lie_algebras3.pdf", "own_work": false, "note": "Defines su(2) by traceless anti-Hermitian 2 × 2 matrices and gives the Pauli-matrix basis.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Cayley–Klein parameters", "url": "https://encyclopediaofmath.org/wiki/Cayley-Klein_parameters", "own_work": false, "note": "Describes the two-to-one homomorphism from SU(2) to SO(3).", "fetched": [ "2026-09-09" ] }, { "title": "Special unitary group, Wikipedia", "url": "https://en.wikipedia.org/wiki/Special_unitary_group", "own_work": false, "note": "read as raw wikitext: “This representation is routinely used in quantum mechanics to represent the spin of fundamental particles such as electrons.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.4", "url": "https://www.teclamairme.com/#r-14.4", "chapter": 14, "status": "confirmed", "claim": "Grassmann introduced the exterior algebra in 1844; Clifford’s algebra imposes the quadratic relation that makes orthogonal vectors anticommute.", "sources": [ { "title": "MacTutor, Hermann Grassmann", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Grassmann/", "own_work": false, "note": "Dates the 1844 Ausdehnungslehre and describes Grassmann’s exterior-algebra work and its reception.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Clifford algebra", "url": "https://encyclopediaofmath.org/wiki/Clifford_algebra", "own_work": false, "note": "gives the Clifford relations and the exterior-algebra basis construction, and supports 1876 — the algebra was “first investigated by W. Clifford in 1876.” Other accounts date it to the 1878 paper Applications of Grassmann’s Extensive Algebra; under the tilde convention the card shows the earlier year and this note carries both.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.5", "url": "https://www.teclamairme.com/#r-14.5", "chapter": 14, "status": "confirmed", "claim": "A Clifford algebra on an n-dimensional space has dimension 2^n; for n=3 the grade dimensions are 1,3,3,1 and the even subalgebra is quaternionic. The even subalgebra of Cl₃,₀(ℝ) is Hamilton’s quaternion algebra.", "sources": [ { "title": "Encyclopedia of Mathematics, Clifford algebra", "url": "https://encyclopediaofmath.org/wiki/Clifford_algebra", "own_work": false, "note": "Gives the subset-indexed basis and dimension 2^n.", "fetched": [ "2026-09-09" ] }, { "title": "John Baez, The Octonions", "url": "https://math.ucr.edu/home/baez/octonions/", "own_work": false, "note": "The contents page of John C. Baez’s The Octonions, which has a section on Clifford algebras; the page itself does not state the even-subalgebra result, which the Clifford algebra page carries.", "fetched": [ "2026-09-09" ] }, { "title": "Clifford algebra, Wikipedia", "url": "https://en.wikipedia.org/wiki/Clifford_algebra", "own_work": false, "note": "read as raw wikitext: “Hamilton’s quaternions are constructed as the even subalgebra of the Clifford algebra Cl3,0(R),” “which shows that the even subalgebra Cl[0]3,0(R) is Hamilton’s real quaternion algebra.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.6", "url": "https://www.teclamairme.com/#r-14.6", "chapter": 14, "status": "confirmed", "claim": "The n-th roots of unity are the vertices of a regular n-gon and sum to zero for n≥2; Gauss used cyclotomic/root sums in the 17-gon construction.", "sources": [ { "title": "Encyclopedia of Mathematics, Root", "url": "https://encyclopediaofmath.org/wiki/Root", "own_work": false, "note": "Describes roots of unity as regular-polygon vertices.", "fetched": [ "2026-09-09" ] }, { "title": "Root of unity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Root_of_unity#Summation", "own_work": false, "note": "(§ Summation): “Let SR(n) be the sum of all the nth roots of unity, primitive or not. Then SR(n) = 1 for n = 1 and 0 for n > 1.” This is the source for the vanishing sum itself. It also carries the link to characters: “The n th roots of unity form an irreducible representation of any cyclic group of order n. The orthogonality relationship also follows from group-theoretic principles as described in Character group.”", "fetched": [ "2026-09-10" ] }, { "title": "Stephan Ramon Garcia, Trevor Hyde and Bob Lutz, Gauss’s hidden menagerie: from cyclotomy to supercharacters, arXiv:1501.07507 (Notices AMS 62 (2015) 878–888)", "url": "https://arxiv.org/abs/1501.07507", "own_work": false, "note": "the same article, free on arXiv: Gauss’s study of the sums of complex exponentials known as Gaussian periods, behind the constructibility of the 17-gon. It does not state the vanishing sum.", "fetched": [ "2026-09-22" ] }, { "title": "Wikipedia, Gaussian period", "url": "https://en.wikipedia.org/wiki/Gaussian_period", "own_work": false, "note": "Gaussian periods were introduced by Gauss as the basis of his compass-and-straightedge theory; “the construction of the heptadecagon … depended on the algebra of such periods.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.7", "url": "https://www.teclamairme.com/#r-14.7", "chapter": 14, "status": "confirmed", "claim": "A nontrivial irreducible character of a finite group sums to zero by orthogonality with the trivial character; Frobenius introduced group characters in 1896.", "sources": [ { "title": "MIT, Introduction to Representation Theory", "url": "https://math.mit.edu/~etingof/representationtheorybook.pdf", "own_work": false, "note": "Gives irreducible-character orthogonality; pairing with the trivial character yields the stated sum.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Ferdinand Georg Frobenius", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Frobenius/", "own_work": false, "note": "Dates Frobenius’s first paper on group characters to 1896.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.8", "url": "https://www.teclamairme.com/#r-14.8", "chapter": 14, "status": "confirmed", "claim": "A₅ has order 60, is the smallest nonabelian simple group, and is the icosahedral rotation group; S₅ has order 120 and is not solvable. The icosahedral rotation group has order 60 and is isomorphic to A₅; since S₅ is not solvable, the general quintic has no solution in radicals.", "sources": [ { "title": "Encyclopedia of Mathematics, Symmetric group", "url": "https://encyclopediaofmath.org/wiki/Symmetric_group", "own_work": false, "note": "States simplicity of Aₙ for n≥5 and nonsolvability of Sₙ for n≥5.", "fetched": [ "2026-09-09" ] }, { "title": "Cambridge, An aspect of icosahedral symmetry", "url": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/DBBE2B6B1D886B677B5E498741916976/S0008439500017525a.pdf/an-aspect-of-icosahedral-symmetry.pdf", "own_work": false, "note": "“As an abstract group, the icosahedral group G is the smallest non-abelian simple group, isomorphic to the alternating group A5 on five letters.” The paper does not call it the rotation group or give its order; the Icosahedral symmetry page carries both.", "fetched": [ "2026-09-09" ] }, { "title": "Icosahedral symmetry, Wikipedia", "url": "https://en.wikipedia.org/wiki/Icosahedral_symmetry", "own_work": false, "note": "read as raw wikitext: “The icosahedral rotation group I is of order 60. The group I is isomorphic to A5, the alternating group of even permutations of five objects.”", "fetched": [ "2026-09-22" ] }, { "title": "Galois theory, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Galois_theory", "own_work": false, "note": "“Since a symmetric group is unsolvable for n≥5, there are, generally speaking, no solutions in radicals of quintic equations and equations of a higher degree (Abel’s theorem).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.9", "url": "https://www.teclamairme.com/#r-14.9", "chapter": 14, "status": "confirmed", "claim": "The classification of finite simple groups leaves exactly 26 sporadic groups outside the cyclic, alternating, and Lie-type families; the 2004 quasithin work closed the last recognized gap.", "sources": [ { "title": "Wikipedia, Classification of finite simple groups", "url": "https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups", "own_work": false, "note": "every finite simple group is cyclic, alternating, of Lie type, or one of 26 sporadic groups; “Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221-page proof for the missing quasithin case.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.10", "url": "https://www.teclamairme.com/#r-14.10", "chapter": 14, "status": "confirmed", "claim": "The Monster is the largest sporadic group, has the stated exact order, and its smallest nontrivial complex character degree is 196883.", "sources": [ { "title": "ATLAS of Finite Group Representations, Monster", "url": "https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M/", "own_work": false, "note": "Gives the Monster’s exact order, its maximal subgroups and conjugacy classes; its Representations section is marked “Porting incomplete,” so the 196883 degree rests on the Wikipedia line.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Monster group", "url": "https://en.wikipedia.org/wiki/Monster_group", "own_work": false, "note": "the Monster is the largest sporadic simple group, of order 808017424794512875886459904961710757005754368000000000, and its smallest faithful complex representation has dimension 196883.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.11", "url": "https://www.teclamairme.com/#r-14.11", "chapter": 14, "status": "confirmed", "claim": "McKay’s 196883+1=196884 observation led to the Conway–Norton moonshine conjectures, which Borcherds proved in 1992; he received a Fields Medal in 1998. 196884 is the coefficient of q in J(τ) = 1/q + 744 + 196884q + ⋯.", "sources": [ { "title": "Wikipedia, Monstrous moonshine", "url": "https://en.wikipedia.org/wiki/Monstrous_moonshine", "own_work": false, "note": "McKay’s observation that the coefficient 196884 is one more than 196883, the Conway–Norton moonshine conjecture, and Borcherds’s proof. (First cited here: M. Broué, In memoriam: John H. Conway, Bull. AMS 58 (2021), doi:10.1090/bull/1702).", "fetched": [ "2026-09-22" ] }, { "title": "International Mathematical Union, 1998 Fields Medals", "url": "https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/1998/index.html", "own_work": false, "note": "Official IMU record of Borcherds’s 1998 Fields Medal.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.12", "url": "https://www.teclamairme.com/#r-14.12", "chapter": 14, "status": "confirmed", "claim": "The discriminant of ax²+bx+c is b²−4ac and vanishes exactly when a polynomial has a repeated root.", "sources": [ { "title": "Encyclopedia of Mathematics, Discriminant", "url": "https://encyclopediaofmath.org/wiki/Discriminant", "own_work": false, "note": "Defines the discriminant by root differences, gives the quadratic case b²−4ac, and states the repeated-root criterion.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.13", "url": "https://www.teclamairme.com/#r-14.13", "chapter": 14, "status": "confirmed", "claim": "An extension degree is vector-space dimension; [ℚ(√2):ℚ]=2 and degrees multiply in towers. ℚ(√2) has basis {1, √2} over ℚ.", "sources": [ { "title": "Encyclopedia of Mathematics, Extension of a field", "url": "https://encyclopediaofmath.org/wiki/Extension_of_a_field", "own_work": false, "note": "Defines extension degree, relates it to minimal-polynomial degree, and gives the tower law.", "fetched": [ "2026-09-09" ] }, { "title": "Field extension, Wikipedia", "url": "https://en.wikipedia.org/wiki/Field_extension", "own_work": false, "note": "read as raw wikitext: “The degree is 2 because {1, √2} can serve as a basis.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.14", "url": "https://www.teclamairme.com/#r-14.14", "chapter": 14, "status": "confirmed", "claim": "Gal(ℚ(ζ₅)/ℚ) is isomorphic to (ℤ/5ℤ)^× and has order φ(5)=4, acting by ζ₅↦ζ₅^k. Each σ in the Galois group sends ζₙ to ζₙ^a for an integer a prime to n.", "sources": [ { "title": "Encyclopedia of Mathematics, Cyclotomic field", "url": "https://encyclopediaofmath.org/wiki/Cyclotomic_field", "own_work": false, "note": "Identifies the cyclotomic Galois group with the unit group modulo n and gives degree φ(n).", "fetched": [ "2026-09-09" ] }, { "title": "Cyclotomic field, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cyclotomic_field", "own_work": false, "note": "read as raw wikitext: “The isomorphism sends each σ ∈ Gal(Q(ζn)/Q) to a mod n, where a is an integer such that σ(ζn) = ζn^a.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.15", "url": "https://www.teclamairme.com/#r-14.15", "chapter": 14, "status": "confirmed", "claim": "Ruffini argued for the impossibility in 1799 and only Cauchy accorded it full credence, Abel proved the general quintic impossibility in 1824, and Galois theory explains it through nonsolvability of S₅. Sₙ is not solvable for n ≥ 5, and Aₙ is then simple and non-abelian.", "sources": [ { "title": "Encyclopedia of Mathematics, Galois theory", "url": "https://encyclopediaofmath.org/wiki/Galois_theory", "own_work": false, "note": "Connects solvability by radicals to solvable groups and the general quintic to nonsolvable symmetric groups.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor Dictionary of Scientific Biography, Paolo Ruffini", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Ruffini.pdf", "own_work": false, "note": "Documents Ruffini’s 1799 attempt and Abel’s independent 1824 proof.", "fetched": [ "2026-09-09" ] }, { "title": "Symmetric group, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Symmetric_group", "own_work": false, "note": "“For n ≤ 4 the group S_n is solvable, but for n ≥ 5 it is not solvable and A_n is a simple non-Abelian group.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.16", "url": "https://www.teclamairme.com/#r-14.16", "chapter": 14, "status": "confirmed", "claim": "E₈ has 240 roots, and the associated E₈ lattice gives the optimal sphere packing in eight dimensions, proved by Viazovska.", "sources": [ { "title": "Annals of Mathematics, The sphere packing problem in dimension 8", "url": "https://annals.math.princeton.edu/2017/185-3/p07", "own_work": false, "note": "Viazovska’s primary paper proves optimality of the E₈ lattice packing.", "fetched": [ "2026-09-09" ] }, { "title": "AMS Feature Column, Eight-dimensional spheres and E₈", "url": "https://mathvoices.ams.org/featurecolumn/2022/09/01/eight-dimensional-spheres-and-the-exceptional-e_8/", "own_work": false, "note": "Relates the 240 roots/shortest vectors to the E₈ lattice packing.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.17", "url": "https://www.teclamairme.com/#r-14.17", "chapter": 14, "status": "confirmed", "claim": "E₆ has a 27-dimensional fundamental representation and has been used in grand-unified models since 1976.", "sources": [ { "title": "Encyclopedia of Mathematics, Exceptional Lie algebra", "url": "https://encyclopediaofmath.org/wiki/Lie_algebra%2C_exceptional", "own_work": false, "note": "Lists 27 as the simplest nontrivial representation dimension for E₆.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Feza Gürsey", "url": "https://en.wikipedia.org/wiki/Feza_G%C3%BCrsey", "own_work": false, "note": "credits Gürsey with the E6 grand unified theory and cites the 1976 paper: “A universal gauge theory model based on E6.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-14.18", "url": "https://www.teclamairme.com/#r-14.18", "chapter": 14, "status": "confirmed", "claim": "The Jacobi identity appears in Jacobi’s Nova methodus, published by Clebsch in 1862 — Jacobi died on 18 February 1851, so the year is the posthumous edition’s and not his — and Weyl introduced the term Lie algebra in the 1930s. Corrected 11 September 2026: this entry previously read “Jacobi’s 1862 Poisson-bracket paper,” which the cited source does not say. The source cites the 1862 paper; it does not date Jacobi’s authorship to it.", "sources": [ { "title": "Encyclopedia of Mathematics, Poisson brackets", "url": "https://encyclopediaofmath.org/wiki/Poisson_brackets", "own_work": false, "note": "States the Jacobi identity and cites Jacobi’s 1862 paper.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Lie algebra", "url": "https://encyclopediaofmath.org/wiki/Lie_algebra", "own_work": false, "note": "Dates Weyl’s introduction of the term Lie algebra to 1934.", "fetched": [ "2026-09-09" ] }, { "title": "Carl Jacobi, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Jacobi/", "own_work": false, "note": "gives Jacobi’s death as 18 February 1851, in Berlin.", "fetched": [ "2026-09-22" ] }, { "title": "David Eugene Smith, History of Modern Mathematics (1896), Project Gutenberg", "url": "https://www.gutenberg.org/files/8746/8746-pdf.pdf", "own_work": false, "note": "read from the PDF: “Jacobi (1827) also gave an analysis of Pfaff’s method, besides developing an original one (1836) which Clebsch published (1862).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.19", "url": "https://www.teclamairme.com/#r-14.19", "chapter": 14, "status": "confirmed", "claim": "Gauss was born 30 April 1777, and his diary’s first entry is dated 30 March 1796; he was therefore eighteen, not nineteen, when he recorded the regular-17-gon discovery.", "sources": [ { "title": "Gauss’s diary, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gauss%27s_diary", "own_work": false, "note": "(§ Sample entries): “Entry 1, dated 1796, March 30, states Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc.”", "fetched": [ "2026-09-10" ] }, { "title": "Göttingen State and University Library, catalogue record for Gauss’s Mathematisches Tagebuch, 30.3.1796 – 9.7.1814", "url": "https://gdz.sub.uni-goettingen.de/id/DE-611-HS-3382323", "own_work": false, "note": "the digitisation record for the manuscript (Cod. Ms. Gauß Math. 48 Cim.), dated 30.3.1796 – 9.7.1814; re-read 24 September 2026, the scanned pages are now viewable in its viewer (they were not in September’s first reading).", "fetched": [ "2026-09-10" ] }, { "title": "MacTutor/Royal Astronomical Society obituary of Gauss", "url": "https://mathshistory.st-andrews.ac.uk/Obituaries/Gauss_RAS/", "own_work": false, "note": "Gives Gauss’s birth date as 30 April 1777.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-14.20", "url": "https://www.teclamairme.com/#r-14.20", "chapter": 14, "status": "confirmed", "claim": "Over an algebraically closed field of characteristic 0 the finite-dimensional simple Lie algebras correspond to irreducible reduced root systems: the four classical series of types A–D and the five exceptional types E₆, E₇, E₈, F₄, G₂.", "sources": [ { "title": "Encyclopedia of Mathematics, Lie algebra, semi-simple", "url": "https://encyclopediaofmath.org/wiki/Lie_algebra,_semi-simple", "own_work": false, "note": "frames the whole classification with “Henceforth finite-dimensional semi-simple Lie algebras over a field k of characteristic 0 are considered,” states that over an algebraically closed field the classification of simple non-Abelian Lie algebras “essentially coincides with the classification of … irreducible reduced root systems,” calls the algebras of types A–D classical, and names E₆, E₇, E₈, F₄, G₂ as the exceptional ones.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass. The chapter’s sentence about the classification carried no footnote, and r-14.2 — the Encyclopedia of Mathematics article on exceptional Lie algebras — states only the five exceptional types, not the four classical series; that article opens “A simple Lie algebra (see Lie algebra, semi-simple) that is not classical” and so defers the classification itself to this page. The words finite-dimensional were added to the chapter’s sentence in the same pass: that is the hypothesis this source states, and the classification is false without it." }, { "id": "r-14.21", "url": "https://www.teclamairme.com/#r-14.21", "chapter": 14, "status": "confirmed", "claim": "Quadratic, cubic and quartic equations are solvable by radicals — the cubic by del Ferro, Tartaglia and Cardano, the quartic by Ferrari in 1540 — and Ruffini’s 1799 proof that the general equation of degree above four is not solvable by radicals was complete except for one gap.", "sources": [ { "title": "MacTutor, Paolo Ruffini", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Ruffini/", "own_work": false, "note": "carries “Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians,” “The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan” and “Ferrari had solved the quartic by radicals in 1540,” and, of Ruffini’s 1799 book on the impossibility above degree four, “It is remarkable work and, except for one gap, proves the theorem as Ruffini claimed.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass. The chapter’s sentence listing formulas by radicals for quadratics, cubics and quartics carried no footnote of any kind. The word incomplete in “Ruffini’s 1799 argument was incomplete” is asserted by r-14.15’s claim line, but r-14.15’s readable source — the MacTutor Dictionary of Scientific Biography article on Ruffini — says only “Only Cauchy accorded them full credence” and never that the argument had a gap; r-14.15’s other source, the Encyclopedia of Mathematics article on Galois theory, could not be re-read during this pass — encyclopediaofmath.org answered HTTP 502 to every request from this client on 10 September 2026, intermittently for other articles too, so this is a reachability problem and not a dead link. This page carries both halves in so many words." }, { "id": "r-14.22", "url": "https://www.teclamairme.com/#r-14.22", "chapter": 14, "status": "confirmed", "claim": "The chapter epigraph is Weyl’s, from Symmetry, Princeton University Press, 1952.", "sources": [ { "title": "Hermann Weyl, Wikiquote", "url": "https://en.wikiquote.org/wiki/Hermann_Weyl", "own_work": false, "note": "read as raw HTML: it quotes Symmetry and cites it as “Symmetry, Princeton Univ. Press, 1952,” which is the book and the year the footer gives. The page number the footer gives, p. 5, is the book’s own and is not in this source.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book and the year are sourced; the page reference is the author’s own reading of the printed book and is offered as such." }, { "id": "r-14.23", "url": "https://www.teclamairme.com/#r-14.23", "chapter": 14, "status": "confirmed", "claim": "Cites the order of the Tits group and the convention that files it among the Lie-type families rather than counting it as a twenty-seventh sporadic simple group. The Tits group, named for Jacques Tits, has order 17,971,200 and is the index-2 derived subgroup of the Ree group ²F₄(2); it belongs to the family ²F₄(2²ⁿ⁺¹)′ but is not strictly of Lie type.", "sources": [ { "title": "Tits group, Wikipedia", "url": "https://en.wikipedia.org/wiki/Tits_group", "own_work": false, "note": "gives the group’s order and its status as an index-2 subgroup of the Ree group 2F4(2), classified with the Lie-type groups.", "fetched": [ "2026-09-22" ] }, { "title": "Classification of finite simple groups, Wikipedia", "url": "https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups", "own_work": false, "note": "carries why it is a borderline case: “the Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.25", "url": "https://www.teclamairme.com/#r-14.25", "chapter": 14, "status": "confirmed", "claim": "Cites the definition and q-expansion of the modular j-invariant, including its coefficient 196884. The expansion continues 21493760q²; 196884 is one more than 196883, the smallest representation of the Monster, McKay’s observation behind monstrous moonshine; j((1+√−163)/2) = −640320³, so e^(π√163) lies within about 10⁻¹² of 640320³ + 744.", "sources": [ { "title": "j-invariant, Wikipedia", "url": "https://en.wikipedia.org/wiki/J-invariant", "own_work": false, "note": "gives the q-expansion 1/q + 744 + 196884q + ... and the function’s role classifying elliptic curves.", "fetched": [ "2026-09-22" ] }, { "title": "Monstrous moonshine, Wikipedia", "url": "https://en.wikipedia.org/wiki/Monstrous_moonshine", "own_work": false, "note": "carries McKay’s observation: “Specifically, McKay noticed that the coefficient 196884 was precisely one more than 196883, the degree of the smallest faithful complex representation of the monster group.”", "fetched": [ "2026-09-22" ] }, { "title": "Heegner number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Heegner_number", "own_work": false, "note": "carries how close e^(π√163) comes: “e^(π√163) = 262,537,412,640,768,743.999,999,999,999 25…” and “e^(π√163) ≈ 640,320³ + 744,” with “the linear term of the error is, −196,884/e^(π√163) ≈ −0.000,000,000,000 75”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.26", "url": "https://www.teclamairme.com/#r-14.26", "chapter": 14, "status": "confirmed", "claim": "Cites the Conway-Norton monstrous moonshine conjecture and Borcherds’s 1992 proof. McKay’s observation that 196884 is one more than 196883 prompted the conjectures, and Borcherds received the Fields Medal in 1998 in part for the proof.", "sources": [ { "title": "Monstrous moonshine, Wikipedia", "url": "https://en.wikipedia.org/wiki/Monstrous_moonshine", "own_work": false, "note": "states McKay’s observation, the Conway-Norton conjectures of 1979, and Borcherds’s 1992 proof; the moonshine module is a vertex operator algebra “which has the monster group as its group of symmetries,” and “The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras.”; McKay found that the first few terms of the J-invariant “could be expressed in terms of linear combinations of the dimensions of the irreducible representations” of the monster group, and Conway and Norton “conjectured the existence of an infinite-dimensional graded representation of M, whose graded traces T_{g} are the expansions of precisely the functions on their list.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-14.27", "url": "https://www.teclamairme.com/#r-14.27", "chapter": 14, "status": "confirmed", "claim": "A solvable group is one that can be built from abelian groups; a polynomial equation is solvable in radicals exactly when its Galois group is solvable, in characteristic 0; S₅ has the composition factors A₅ and C₂, and A₅ is not abelian, so S₅ is not solvable; the symmetries of ℚ(√2) over ℚ form a group of order 2 that swaps √2 and −√2.", "sources": [ { "title": "Solvable group, Wikipedia", "url": "https://en.wikipedia.org/wiki/Solvable_group", "own_work": false, "note": "“a solvable group or soluble group is a group that can be constructed from abelian groups using extensions”; “a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0)”; “The group S5 is not solvable — it has a composition series {E, A5, S5} …, giving factor groups isomorphic to A5 and C2; and A5 is not abelian”; and, in its example, “Aut(Q(√2)/Q) ≅ Z/2 with group action f(±√2) = ∓√2” (formulas as typeset on the page).", "fetched": [ "2026-09-29" ] } ], "editorial_note": "" }, { "id": "r-15.1", "url": "https://www.teclamairme.com/#r-15.1", "chapter": 15, "status": "confirmed", "claim": "A Hilbert space is a complete inner-product space; l² and L² are standard examples, and the inner product defines norm and orthogonality.", "sources": [ { "title": "Encyclopedia of Mathematics, Hilbert space", "url": "https://encyclopediaofmath.org/wiki/Hilbert_space", "own_work": false, "note": "defines Hilbert space, gives l² and L² as examples, and defines orthogonality by vanishing inner product. Its numbered axioms require the space to be infinite-dimensional, but the article adds that this condition is often omitted, so the source carries BOTH conventions. Recorded because this entry is cited in support of the wide definition while its headline axioms give the narrow one; see 15.24.", "fetched": [ "2026-09-09", "2026-09-11" ] } ], "editorial_note": "" }, { "id": "r-15.2", "url": "https://www.teclamairme.com/#r-15.2", "chapter": 15, "status": "confirmed", "claim": "Hilbert’s integral-equation work around 1909 established the basis for the infinite-dimensional space bearing his name; von Neumann made abstract Hilbert space the rigorous common setting for quantum mechanics and published his foundational book in 1932.", "sources": [ { "title": "MacTutor, David Hilbert", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Hilbert/", "own_work": false, "note": "Dates Hilbert’s integral-equation work to about 1909 and connects it directly to infinite-dimensional Hilbert space and quantum mechanics.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Hilbert space", "url": "https://encyclopediaofmath.org/wiki/Hilbert_space", "own_work": false, "note": "Credits Hilbert with l² and L² in his integral-equation work and von Neumann, Riesz and Stone with the abstract definition.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, The quantum age begins", "url": "https://mathshistory.st-andrews.ac.uk/HistTopics/The_Quantum_age_begins/", "own_work": false, "note": "Records von Neumann’s 1932 rigorous operator-algebra foundation for quantum theory.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The inherited 1906–1912/1927/1929 compressed chronology was replaced by the dates these sources actually support." }, { "id": "r-15.3", "url": "https://www.teclamairme.com/#r-15.3", "chapter": 15, "status": "confirmed", "claim": "Bra-ket notation represents inner products as ⟨a|b⟩; orthogonal vectors have zero inner product, and the computational basis states of a qubit are orthonormal. A bra times a ket is a scalar, and the standard basis vectors form an orthonormal basis.", "sources": [ { "title": "Dirac, A new notation for quantum mechanics", "url": "https://doi.org/10.1017/S0305004100021162", "own_work": false, "note": "Primary 1939 paper introducing Dirac’s notation.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Hilbert space", "url": "https://encyclopediaofmath.org/wiki/Hilbert_space", "own_work": false, "note": "Defines orthogonality as a zero inner product.", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Learning, Quantum information", "url": "https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information", "own_work": false, "note": "Gives the standard qubit bases, inner products and measurement probabilities.", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Learning, Basics of quantum information: Inner products and projections", "url": "https://quantum.cloud.ibm.com/learning/courses/basics-of-quantum-information/quantum-circuits/inner-products-and-projections", "own_work": false, "note": "“We also have that the product of a bra vector and a ket vector, viewed as matrices either having a single row or a single column, results in a scalar.”; “for any classical state set Σ, the set of all standard basis vectors {|a⟩ : a∈Σ} is an orthonormal basis.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.4", "url": "https://www.teclamairme.com/#r-15.4", "chapter": 15, "status": "confirmed", "claim": "Orthogonal pure quantum states can be perfectly distinguished, whereas non-orthogonal states cannot; the same framework yields the no-cloning limitation. Any two orthogonal states can be discriminated perfectly, non-orthogonal ones cannot; a cloning unitary can work only for states that are identical up to phase or orthogonal.", "sources": [ { "title": "IBM Quantum Learning, Quantum circuits", "url": "https://learning.quantum.ibm.com/course/basics-of-quantum-information/quantum-circuits", "own_work": false, "note": "the lesson introduction: it introduces inner products, orthogonality and projective measurements, and says “we’ll derive fundamental limitations on quantum information, including the no-cloning theorem and the impossibility to perfectly discriminate non-orthogonal quantum states” (the derivations are quoted on the Limitations line below).", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Learning, Basics of quantum information: Limitations on quantum information", "url": "https://quantum.cloud.ibm.com/learning/courses/basics-of-quantum-information/quantum-circuits/limitations-on-quantum-information", "own_work": false, "note": "“if we have two quantum states |ψ⟩ and |φ⟩ that are not orthogonal … then it’s impossible to discriminate them perfectly”; “It is possible, by the way, to perfectly discriminate any two states that are orthogonal”; “The no-cloning theorem shows it is impossible to create a perfect copy of an unknown quantum state.”", "fetched": [ "2026-09-22" ] }, { "title": "No-cloning theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/No-cloning_theorem", "own_work": false, "note": "read as raw wikitext: “Hence by the Cauchy–Schwarz inequality either |φ⟩ = e^{iβ}|ψ⟩ or |φ⟩ is orthogonal to |ψ⟩. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.5", "url": "https://www.teclamairme.com/#r-15.5", "chapter": 15, "status": "confirmed", "claim": "The Born rule assigns probabilities from squared amplitudes; Born proposed the statistical interpretation in 1926 and received half of the 1954 Nobel Prize for it.", "sources": [ { "title": "Nobel Prize, Physics 1954 summary", "url": "https://www.nobelprize.org/prizes/physics/1954/summary/", "own_work": false, "note": "Official award citation credits Born for the statistical interpretation of the wavefunction.", "fetched": [ "2026-09-09" ] }, { "title": "Max Born, Nobel Lecture", "url": "https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf", "own_work": false, "note": "Born’s primary lecture states that squared amplitudes give state probabilities and recounts his 1926 statistical interpretation.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.6", "url": "https://www.teclamairme.com/#r-15.6", "chapter": 15, "status": "confirmed", "claim": "The Hadamard gate sends |0⟩ to |+⟩=(|0⟩+|1⟩)/√2, giving overlap 1/√2 with |0⟩ and measurement probability 1/2.", "sources": [ { "title": "IBM Quantum Learning, Quantum information", "url": "https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information", "own_work": false, "note": "Gives H|0⟩=|+⟩ and the equal computational-basis probabilities.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.7", "url": "https://www.teclamairme.com/#r-15.7", "chapter": 15, "status": "confirmed", "claim": "The Pauli X matrix flips |0⟩ and |1⟩, serves as the quantum NOT gate, and squares to the 2×2 identity; Pauli introduced the matrices in 1927.", "sources": [ { "title": "Encyclopedia of Mathematics, Pauli matrices", "url": "https://encyclopediaofmath.org/wiki/Pauli_matrices", "own_work": false, "note": "Gives the matrices, their 1927 attribution, and their algebra, including σᵢ²=I.", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Composer reference", "url": "https://quantum.cloud.ibm.com/docs/en/guides/composer", "own_work": false, "note": "Identifies Pauli X as the standard NOT gate.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.8", "url": "https://www.teclamairme.com/#r-15.8", "chapter": 15, "status": "confirmed", "claim": "Grassmann variables anticommute and square to zero and are used for fermionic path integrals; Pauli formulated the exclusion principle in 1925, and it applies to fermions. Grassmann variables anticommute and square to zero, and their anticommutation is that of the Pauli exclusion principle.", "sources": [ { "title": "MIT OpenCourseWare, Path Integrals of Fermions", "url": "https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/resources/ocw_8323_lecture19_2023apr19_mp4/", "own_work": false, "note": "Graduate lecture covers Grassmann anticommutation and fermionic path integrals.", "fetched": [ "2026-09-09" ] }, { "title": "Nobel Prize, Wolfgang Pauli facts", "url": "https://www.nobelprize.org/prizes/physics/1945/pauli/facts/", "own_work": false, "note": "Dates Pauli’s exclusion principle to 1925 and states its same-state prohibition.", "fetched": [ "2026-09-09" ] }, { "title": "Nobel Prize, Physics 1901–2000", "url": "https://www.nobelprize.org/nobel_prizes/themes/physics/karlsson/", "own_work": false, "note": "Connects the exclusion principle with half-integer-spin fermions; “This principle was later found to be associated with the symmetry of wave functions for particles of half-integer spins in general, distinguishing what is now called fermions from the bosonic particles whose spins are integer multiples of” (the unit is an image on the page, not in its text).", "fetched": [ "2026-09-09" ] }, { "title": "Grassmann number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Grassmann_number", "own_work": false, "note": "read as raw wikitext: “the generators and thus the algebra are anti-commuting,” θᵢθⱼ = −θⱼθᵢ; “The squares of the elements vanish”; “the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.9", "url": "https://www.teclamairme.com/#r-15.9", "chapter": 15, "status": "confirmed", "claim": "The canonical relation between a position and its own momentum is [x̂, p̂] = iℏI; Born and Jordan supplied the canonical matrix-mechanics relation in 1925. For the three-dimensional form with the Kronecker delta, and therefore for x-position commuting with y-momentum, see 15.23: neither source below prints it. The sign follows [x̂, p̂ₓ] = x̂p̂ₓ − p̂ₓx̂ = iℏI.", "sources": [ { "title": "Encyclopedia of Mathematics, Uncertainty principle", "url": "https://encyclopediaofmath.org/wiki/Uncertainty_principle", "own_work": false, "note": "prints the same-component canonical commutators with the identity operator, [p̂ₓ, x̂] = [p̂_y, ŷ] = [p̂_z, ẑ] = iℏE, with no cross terms and no Kronecker delta. The annotation here used to read “Gives the Cartesian canonical commutators,” which 15.23 contradicts in the same reference list; corrected 17 September 2026. Its sign is the reverse of this book’s: [x̂, p̂] = iℏI gives [p̂ₓ, x̂] = −iℏI. The book does not follow this page’s sign; the Canonical commutation relation line carries it.", "fetched": [ "2026-09-09" ] }, { "title": "Born–Jordan Quantization and the Equivalence of Matrix and Wave Mechanics", "url": "https://arxiv.org/abs/1405.2519", "own_work": false, "note": "the abstract dates the Born and Jordan paper to 1925: “The aim of the famous Born and Jordan 1925 paper was to put Heisenberg’s matrix mechanics on a firm mathematical basis.” It does not print the commutation relation itself.", "fetched": [ "2026-09-09" ] }, { "title": "Canonical commutation relation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Canonical_commutation_relation", "own_work": false, "note": "read as raw wikitext: “[x̂, p̂ₓ] = iℏI … where [x, pₓ] = x pₓ − pₓ x is the commutator of x and pₓ.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.10", "url": "https://www.teclamairme.com/#r-15.10", "chapter": 15, "status": "confirmed", "claim": "For |+⟩, σz yields +1 and −1 with equal probability and expectation zero; equivalently σz|+⟩=|−⟩ and ⟨+|−⟩=0. {|+⟩, |−⟩} is an orthonormal basis, so ⟨+|−⟩ = 0.", "sources": [ { "title": "Encyclopedia of Mathematics, Pauli matrices", "url": "https://encyclopediaofmath.org/wiki/Pauli_matrices", "own_work": false, "note": "Gives σz=diag(1, −1) and its eigenvalues.", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Learning, Quantum information", "url": "https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information", "own_work": false, "note": "defines the plus state and the minus state, and of a standard basis measurement says “Measuring either state produces the same output distribution as the other,” 0 and 1 each with probability 1/2 (their orthogonality is on the Inner products line below).", "fetched": [ "2026-09-09" ] }, { "title": "IBM Quantum Learning, Basics of quantum information: Inner products and projections", "url": "https://quantum.cloud.ibm.com/learning/courses/basics-of-quantum-information/quantum-circuits/inner-products-and-projections", "own_work": false, "note": "“{|+⟩,|−⟩} is an orthonormal basis for the 2-dimensional space corresponding to a single qubit.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.11", "url": "https://www.teclamairme.com/#r-15.11", "chapter": 15, "status": "confirmed", "claim": "Self-adjoint observables have real spectrum and orthogonal eigenspaces for distinct eigenvalues; Schrödinger published wave mechanics in 1926 and showed its equivalence to matrix mechanics. The first 1926 paper, Quantisierung als Eigenwertproblem, treated time-independent systems; a third, in May, showed the equivalence with matrix mechanics.", "sources": [ { "title": "Encyclopedia of Mathematics, Linear operator", "url": "https://encyclopediaofmath.org/wiki/Linear_operator", "own_work": false, "note": "States that self-adjoint spectra are real, distinct eigenspaces are orthogonal, and self-adjoint operators represent quantum observables.", "fetched": [ "2026-09-09" ] }, { "title": "MIT OpenCourseWare, Mathematical formalism of quantum mechanics", "url": "https://ocw.mit.edu/courses/18-700-linear-algebra-fall-2013/cc993d0c7953d1aa7d57997b1e43b7da_MIT18_700F13_qntm_mechnc.pdf", "own_work": false, "note": "Explains self-adjoint observables, spectral values and the finite-dimensional spectral decomposition.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Erwin Schrödinger", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Schrodinger/", "own_work": false, "note": "Documents Schrödinger’s 1926 papers and his equivalence argument for wave and matrix mechanics; “Schrödinger published his revolutionary work relating to wave mechanics and the general theory of relativity in a series of six papers in 1926”; “In its most general form, the time-dependent Schrödinger equation describes the time evolution of closed quantum physical systems.”", "fetched": [ "2026-09-09" ] }, { "title": "Erwin Schrödinger, Wikipedia", "url": "https://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger", "own_work": false, "note": "read as raw wikitext: “In January 1926, Schrödinger published in Annalen der Physik the paper “Quantisierung als Eigenwertproblem” (Quantization as an Eigenvalue Problem)… In this paper, he gave a “derivation” of the wave equation for time-independent systems”; “A third paper, published in May, showed the equivalence of his approach to that of Werner Heisenberg’s matrix mechanics.”; “A fourth paper in this series showed how to treat problems in which the system changes with time, as in scattering problems.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.12", "url": "https://www.teclamairme.com/#r-15.12", "chapter": 15, "status": "confirmed", "claim": "Kennard proved ΔxΔp≥ℏ/2 in 1927, Robertson generalized it in 1929, and Gaussian wavepackets attain equality, so min ΔxΔp/ℏ=1/2.", "sources": [ { "title": "Encyclopedia of Mathematics, Uncertainty principle", "url": "https://encyclopediaofmath.org/wiki/Uncertainty_principle", "own_work": false, "note": "Gives the inequality and the Kennard 1927 and Robertson 1929 chronology; its comments: “W. Heisenberg [a1] presented this uncertainty principle in 1927. In the same year E.H. Kennard [a2] found relation (2), while the general relation (1) was proved by H.P. Robertson [a3] in 1929..”", "fetched": [ "2026-09-09" ] }, { "title": "University of Texas at Austin, Quantum Mechanics", "url": "https://farside.ph.utexas.edu/teaching/qmech/qmech.pdf", "own_work": false, "note": "States that equality is achieved by Gaussian wavepackets.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.13", "url": "https://www.teclamairme.com/#r-15.13", "chapter": 15, "status": "confirmed", "claim": "For the CHSH expression, local hidden-variable correlations are bounded by 2 and quantum correlations by 2√2; quantum experiments can approach the latter value.", "sources": [ { "title": "Poh, Joshi, Cerè, Cabello and Kurtsiefer, Approaching Tsirelson’s Bound in a Photon Pair Experiment", "url": "https://harvest.aps.org/v2/journals/articles/10.1103/PhysRevLett.115.180408/fulltext", "own_work": false, "note": "Primary Physical Review Letters article states both bounds and reports an experiment approaching 2√2.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.14", "url": "https://www.teclamairme.com/#r-15.14", "chapter": 15, "status": "confirmed", "claim": "Clauser, Aspect and Zeilinger received the 2022 Nobel Prize for entangled-photon and Bell-inequality experiments; loophole-free Bell tests were reported in 2015. The prize followed the Bell-test experiments; the first loophole-free tests were published in 2015.", "sources": [ { "title": "Nobel Prize, Physics 2022 summary", "url": "https://www.nobelprize.org/prizes/physics/2022/summary/", "own_work": false, "note": "Official award citation.", "fetched": [ "2026-09-09" ] }, { "title": "NIST, A significant-loophole-free test of Bell’s theorem with entangled photons", "url": "https://www.nist.gov/publications/significant-loophole-free-test-bells-theorem-entangled-photons", "own_work": false, "note": "Records the 2015 test and observed violation.", "fetched": [ "2026-09-09" ] }, { "title": "Bell test, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bell_test", "own_work": false, "note": "read as raw wikitext: “These efforts to experimentally validate violations of the Bell inequalities resulted in John Clauser, Alain Aspect, and Anton Zeilinger being awarded the 2022 Nobel Prize in Physics”; “In 2015 the first three significant-loophole-free Bell-tests were published within three months by independent groups in Delft, Vienna and Boulder.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.15", "url": "https://www.teclamairme.com/#r-15.15", "chapter": 15, "status": "confirmed", "claim": "Fourier presented his heat memoir in 1807 and published the book in 1822; over a full period, sines and cosines are orthogonal, including ∫₀²π sin x cos x dx=0.", "sources": [ { "title": "MacTutor, Joseph Fourier", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Fourier/", "own_work": false, "note": "Dates the 1807 presentation, the objections by Lagrange and Laplace, and the 1822 publication.", "fetched": [ "2026-09-09" ] }, { "title": "MIT OpenCourseWare, Orthogonality Relations", "url": "https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/618691a1edaa81b0a75303e09b4cb4b5_MIT18_03SCF11_s21_7text.pdf", "own_work": false, "note": "Derives the sine-cosine orthogonality integrals over a full period.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "MIT’s derivation supports the displayed integral and its vanishing." }, { "id": "r-15.16", "url": "https://www.teclamairme.com/#r-15.16", "chapter": 15, "status": "confirmed", "claim": "Under a consistent normalization, Plancherel’s theorem makes the Fourier transform unitary on L² and preserves its norm; Plancherel established it in 1910. With f̂(ξ) = ∫f(x)e^(−2πixξ)dx the extension to L² is unitary.", "sources": [ { "title": "Encyclopedia of Mathematics, Plancherel theorem", "url": "https://encyclopediaofmath.org/index.php?title=Plancherel_theorem", "own_work": false, "note": "States the unitary L² extension, norm identity and 1910 date.", "fetched": [ "2026-09-09" ] }, { "title": "Plancherel theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Plancherel_theorem", "own_work": false, "note": "read as raw wikitext: proven “by Michel Plancherel in 1910”; the transform of an L¹ function “is defined as the Lebesgue integral” f̂(ξ) = ∫ f(x)e^(−2πixξ) dx; its L² extension is an isometry, and “This isometry is actually a unitary map.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.17", "url": "https://www.teclamairme.com/#r-15.17", "chapter": 15, "status": "judgement", "claim": "The limiting Fourier overshoot at a jump is Si(π)/π−1/2≈0.089489872, and its spike narrows without losing relative height; the chapter leaves the number’s arithmetic classification open.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §6.16(i)", "url": "https://dlmf.nist.gov/6.16.i", "own_work": false, "note": "Gives the sine-integral formula, limiting overshoot and Gibbs’s 1899 report.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Wilbraham–Gibbs Constant", "url": "https://mathworld.wolfram.com/Wilbraham-GibbsConstant.html", "own_work": false, "note": "Gives the constant, decimal value and Wilbraham 1848/Gibbs 1899 history; page last updated 2 September 2026.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Neither standard reference assigns a rational, irrational, algebraic or transcendental class, and targeted searches found no theorem settling it. The page therefore states only that this review leaves the classification open, rather than claiming an exhaustive proof of the literature’s absence." }, { "id": "r-15.18", "url": "https://www.teclamairme.com/#r-15.18", "chapter": 15, "status": "confirmed", "claim": "L² is the complete Hilbert space of square-integrable functions modulo equality almost everywhere; Riesz and Fischer independently proved the relevant L²/ℓ² Fourier-coefficient theorem in 1907. Riesz and Fischer proved the theorem independently in 1907; an element of L² is a class of functions equal almost everywhere.", "sources": [ { "title": "Encyclopedia of Mathematics, Hilbert space", "url": "https://encyclopediaofmath.org/wiki/Hilbert_space", "own_work": false, "note": "Defines the L² example and completeness framework.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Orthogonal series", "url": "https://encyclopediaofmath.org/wiki/Orthogonal_series", "own_work": false, "note": "Dates the Riesz and Fischer theorem to 1907 and states the L²/ℓ² correspondence.", "fetched": [ "2026-09-09" ] }, { "title": "Square-integrable function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Square-integrable_function", "own_work": false, "note": "read as raw wikitext: “The square integrable functions (in the sense mentioned in which a “function” actually means an equivalence class of functions that are equal almost everywhere) form an inner product space.”", "fetched": [ "2026-09-22" ] }, { "title": "Riesz–Fischer theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riesz%E2%80%93Fischer_theorem", "own_work": false, "note": "read as raw wikitext: “The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.19", "url": "https://www.teclamairme.com/#r-15.19", "chapter": 15, "status": "confirmed", "claim": "Position- and momentum-space wavefunctions are Fourier transforms of one another, and Gaussian wavepackets transform to Gaussians and attain the position-momentum uncertainty minimum. Canonical conjugate quantities are those related by definition such that one is the Fourier transform of the other.", "sources": [ { "title": "Durham University, Momentum-space wave function", "url": "https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/momentum_space.html", "own_work": false, "note": "Gives the position/momentum Fourier-transform pair and operator representations.", "fetched": [ "2026-09-09" ] }, { "title": "University of Texas at Austin, Quantum Mechanics", "url": "https://farside.ph.utexas.edu/teaching/qmech/qmech.pdf", "own_work": false, "note": "Gives Gaussian position/momentum wavepackets and the minimum uncertainty product.", "fetched": [ "2026-09-09" ] }, { "title": "Canonical commutation relation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Canonical_commutation_relation", "own_work": false, "note": "read as raw wikitext: “the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-15.20", "url": "https://www.teclamairme.com/#r-15.20", "chapter": 15, "status": "confirmed", "claim": "The chapter’s displayed inner products, Pauli expectations, trace and Grassmann anticommutator follow directly from the cited definitions and matrices; for the cross-coordinate commutator see 15.23. Grassmann variables anticommute, θ₁θ₂ = −θ₂θ₁, and square to zero.", "sources": [ { "title": "IBM Quantum Learning, Quantum information", "url": "https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information", "own_work": false, "note": "Gives |0⟩, |1⟩, |+⟩, |−⟩ and their measurement and inner-product relations.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Pauli matrices", "url": "https://encyclopediaofmath.org/wiki/Pauli_matrices", "own_work": false, "note": "Gives the explicit matrices and σᵢ²=I.", "fetched": [ "2026-09-09" ] }, { "title": "MIT OpenCourseWare, Path Integrals of Fermions", "url": "https://ocw.mit.edu/courses/8-323-relativistic-quantum-field-theory-i-spring-2023/resources/ocw_8323_lecture19_2023apr19_mp4/", "own_work": false, "note": "A video lecture on fermionic path integrals; the page itself prints no text of the anticommutation rule, which the Grassmann number line carries.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Uncertainty principle", "url": "https://encyclopediaofmath.org/wiki/Uncertainty_principle", "own_work": false, "note": "Prints the same-component canonical commutators, in the order [p̂ₓ, x̂] and with the opposite sign to this book’s (see 15.9); for the cross-coordinate case see 15.23.", "fetched": [ "2026-09-09" ] }, { "title": "Grassmann number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Grassmann_number", "own_work": false, "note": "read as raw wikitext: “the generators and thus the algebra are anti-commuting,” θᵢθⱼ = −θⱼθᵢ; “The squares of the elements vanish”; “the anti-commuting behavior can be strongly identified with the quantum-mechanical behavior of fermions: the anti-commutation is that of the Pauli exclusion principle.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The cited external definitions and matrices support each displayed value." }, { "id": "r-15.21", "url": "https://www.teclamairme.com/#r-15.21", "chapter": 15, "status": "confirmed", "claim": "The canonical commutator is operator-valued and has no finite-dimensional matrix realization, since a finite commutator has trace zero while a nonzero scalar multiple of the identity does not.", "sources": [ { "title": "Canonical commutation relation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Canonical_commutation_relation", "own_work": false, "note": "(§ Weyl relations): gives the trace obstruction — the relation Tr(AB) = Tr(BA) “gives a nonzero number on the right and zero on the left” — so [x̂, p̂] = iℏ has no finite-dimensional matrix realization.", "fetched": [ "2026-09-10" ] }, { "title": "Fritz, On infinite-dimensional state spaces", "url": "https://arxiv.org/abs/1202.3817", "own_work": false, "note": "States that [x, p]=i can be realized exactly only on an infinite-dimensional Hilbert space.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-15.22", "url": "https://www.teclamairme.com/#r-15.22", "chapter": 15, "status": "confirmed", "claim": "The chapter epigraph is Feynman’s statement that nobody understands quantum mechanics, from the sixth Messenger Lecture, Probability and Uncertainty, delivered at Cornell in 1964 and collected as the sixth chapter of The Character of Physical Law, first published by the BBC in 1965.", "sources": [ { "title": "Wikiquote, Richard Feynman", "url": "https://en.wikiquote.org/wiki/Richard_Feynman", "own_work": false, "note": "gives “I think I can safely say that nobody understands quantum mechanics” from The Character of Physical Law, in that book’s sixth lecture, “Probability and Uncertainty — the Quantum Mechanical View of Nature,” the Messenger Lectures at Cornell, November 1964.", "fetched": [ "2026-09-22" ] }, { "title": "The Character of Physical Law, Wikipedia", "url": "https://en.wikipedia.org/wiki/The_Character_of_Physical_Law", "own_work": false, "note": "read as raw HTML: the infobox gives “Published 1965 (BBC)”; the lectures were delivered “at Cornell University, as part of the Messenger Lectures series” and “The BBC recorded the lectures and published a book under the same title the following year”; and the list of the seven topics has as its sixth “Probability and uncertainty – the quantum mechanical view of nature,” which is the sixth chapter the epigraph cites.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-15.23", "url": "https://www.teclamairme.com/#r-15.23", "chapter": 15, "status": "confirmed", "claim": "The three-dimensional canonical commutation relations carry a Kronecker delta, [x̂ᵢ, p̂ⱼ] = iℏδᵢⱼI, so a position coordinate commutes with a different component of the momentum and the two can be measured simultaneously.", "sources": [ { "title": "University of Texas at Austin, Quantum Mechanics", "url": "https://farside.ph.utexas.edu/teaching/qmech/qmech.pdf", "own_work": false, "note": "Chapter 7, Three-Dimensional Quantum Mechanics, p. 95: Eqs. (7.15)–(7.17) display [xᵢ, xⱼ] = 0, [pᵢ, pⱼ] = 0 and [xᵢ, pⱼ] = iℏδᵢⱼ, and the text immediately beneath them says it is “perfectly possible to simultaneously measure two different positions coordinates, or two different components of the momentum,” the only restriction being a coordinate against its own conjugate momentum. Eq. (9.12), p. 116, repeats it as “the fundamental commutation relation for the xᵢ and the pᵢ.”", "fetched": [ "2026-09-10" ] }, { "title": "Canonical commutation relation, Wikipedia", "url": "https://en.wikipedia.org/wiki/Canonical_commutation_relation", "own_work": false, "note": "The lead section gives the generalized relation [x̂ᵢ, p̂ⱼ] = iℏδᵢⱼ, whose Kronecker delta vanishes whenever the indices differ.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added by the V14 prose pass to close the chapter’s open WRONG-REF row. 15.9’s two pages do not carry the Kronecker-delta form: the Encyclopedia of Mathematics Uncertainty principle entry prints only the same-component commutators [p̂ₓ, x̂] = [p̂_y, ŷ] = [p̂_z, ẑ] = iℏE, with no cross terms and no delta, and de Gosson (arXiv:1405.2519) gives only the one-dimensional pq − qp = −iℏ1. The claim itself is standard; 15.23 is the source that states it. The University of Texas text was already cited in this chapter at 15.12 and 15.19, and the Wikipedia article at 15.21." }, { "id": "r-15.24", "url": "https://www.teclamairme.com/#r-15.24", "chapter": 15, "status": "confirmed", "claim": "Von Neumann required infinite-dimensionality as a named axiom of abstract Hilbert space, deliberately, because that choice makes the space unique up to isomorphism and so establishes the equivalence of matrix and wave mechanics. Modern usage generally drops the requirement, so a finite-dimensional complete inner-product space such as C² counts as a Hilbert space.", "sources": [ { "title": "Anthony Duncan, annotated translation of von Neumann, Mathematische Begründung der Quantenmechanik (1927)", "url": "https://arxiv.org/pdf/2406.02149", "own_work": false, "note": "gives the 1927 axioms A to E verbatim. Axiom C reads “H has infinitely many dimensions: i.e., there are an arbitrarily large number of linearly independent elements of H,” and von Neumann states that the five properties fully determine the space. This is the direct evidence that the infinite-dimensionality was required by name and on purpose.", "fetched": [ "2026-09-11" ] }, { "title": "Robert B. Griffiths, Hilbert Space Quantum Mechanics, Carnegie Mellon qitd113, section 1.1", "url": "https://quantum.phys.cmu.edu/QCQI/qitd113.pdf", "own_work": false, "note": "names the split explicitly: the term is “often reserved for an infinite-dimensional inner product space” but “is often used nowadays ... in a way that includes finite-dimensional spaces,” and it calls a two-dimensional Hilbert space a qubit.", "fetched": [ "2026-09-11" ] }, { "title": "Federico Zalamea, Chasing Individuation: Mathematical Description of Physical Systems, PhD thesis, Université Paris Diderot, 2016, p. 117", "url": "https://arxiv.org/pdf/1612.02959", "own_work": false, "note": "quotes von Neumann’s 1932 book, in the Beyer translation at p. 45, giving the dimension condition in two forms — C.(n), exactly n linearly independent vectors, and C.(infinity), arbitrarily many — followed by “We obtain a different space R, depending on which we decide upon.” READ THE CHAIN OF CUSTODY BEFORE RELYING ON THIS: it is Zalamea quoting Beyer quoting von Neumann, and Zalamea is a doctoral thesis rather than a peer-reviewed article. The Beyer text itself is lending-restricted and could not be obtained here, and Zalamea’s PDF would not extract for an independent check — two attempts recovered 48 KB of a 2.3 MB file, and the ar5iv HTML redirects to the abstract — so this source is carried on the reviewing agent’s reading and not on the coordinator’s. The title and document type above WERE checked against the arXiv record on 11 September 2026, because the first draft of this entry carried a wrong title. Nothing in the book’s prose or cards rests on it — the card says only what the first source above supports.", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Added 11 September 2026, closing the deep-research item DR2. The book had been speaking two conventions: chapter T10’s prose said a Hilbert space has infinitely many dimensions, while chapter T15 computed its whole quantum brace in C² and the hilbert card gave the general definition. A survey of eleven sources found ten allowing finite dimensions, four naming Cⁿ or Rⁿ as a Hilbert space outright (Rudin, Conway, Kreyszig, MathWorld), and Nielsen and Chuang section 2.1.4 using the terms interchangeably in finite dimension. The two apparent holdouts do not define against it: Sakurai avoids the phrase rather than restricting it, and Watrous says he prefers “complex Euclidean space” precisely because “Hilbert space” is the more general notion. The book now speaks the wide convention everywhere. Do not claim that von Neumann called a finite-dimensional space a Hilbert space; he did not." }, { "id": "r-15.25", "url": "https://www.teclamairme.com/#r-15.25", "chapter": 15, "status": "confirmed", "claim": "Cites the CHSH inequality’s bound of 2 for local hidden-variable correlations and the derivation of the quantum bound 2√2. Clauser, Horne, Shimony and Holt published the inequality in 1969, and experiments have come close to the quantum limit 2√2. A local hidden-variable model computes each side’s outcome from that side’s own setting and a hidden state λ of the common source, never from the other side’s setting.", "sources": [ { "title": "CHSH inequality, Wikipedia", "url": "https://en.wikipedia.org/wiki/CHSH_inequality", "own_work": false, "note": "gives the 1969 Clauser-Horne-Shimony-Holt inequality, its local-hidden-variable bound of 2, and Tsirelson’s quantum bound of 2√2; it is “a constraint—on the statistical occurrence of “coincidences” in a Bell test” and “In practice, the inequality is routinely violated by modern experiments in quantum mechanics.” Its derivation “start[s] with the standard assumption of independence of the two sides,” the hidden variable λ being “drawn from a fixed distribution of possible states of the source,” and writes E(a, b) = ∫ A(a, λ)B(b, λ)ρ(λ) dλ, “where A and B are the outcomes,” so each outcome depends only on its own side’s setting and λ; “The maximum violation predicted by quantum mechanics is 2√2 (Tsirelson’s bound).”", "fetched": [ "2026-09-22" ] }, { "title": "Approaching Tsirelson’s bound in a photon pair experiment, arXiv", "url": "https://arxiv.org/abs/1506.01865", "own_work": false, "note": "Poh, Joshi, Ceré, Cabello and Kurtsiefer (2015): “a value 2.82759+-0.00051 is observed” and “the smallest distance with respect to Tsirelson’s bound ever reported, namely, 0.00084+-0.00051”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.1", "url": "https://www.teclamairme.com/#r-16.1", "chapter": 16, "status": "confirmed", "claim": "A contour integral is an integral along an oriented curve; the positive orientation of a circle is counter-clockwise. Cauchy’s 1825 memoir defined integration along arbitrary paths in the complex plane and established path independence in a regular domain. The memoir is the Mémoire sur les intégrales définies prises entre des limites imaginaires; Cauchy reached arbitrary closed paths only in 1846, proving the theorem anew by what is now called Green’s formula, and did not recall the 1825 memoir until 1851.", "sources": [ { "title": "MIT 18.783 Elliptic Curves (Fall 2025), Lecture 14: Elliptic curves over C (part I)", "url": "https://math.mit.edu/classes/18.783/2025/LectureNotes14.pdf", "own_work": false, "note": "defines contour integrals and states that positive orientation of a circle is counter-clockwise.", "fetched": [ "2026-09-09" ] }, { "title": "Dictionary of Scientific Biography, Augustin-Louis Cauchy", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf", "own_work": false, "note": "describes the 1825 memoir’s definition of integrals over arbitrary complex paths and its path-independence result; its bibliography lists “Cauchy, Mémoire sur les intègrales définies prises entre des limites imaginaires (Paris, 1825),” and “Cauchy finally introduced arbitrary closed integration paths, although not as an immediate consequence of his 1825 mémoire, which he did not remember until 1851 [1, 1st ser., XI, 328]; instead, he proved his integral theorem anew by means of what is now called Green’s formula.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.2", "url": "https://www.teclamairme.com/#r-16.2", "chapter": 16, "status": "confirmed", "claim": "Cauchy’s integral theorem gives zero around a closed curve bounding a region on which the function is holomorphic; because e^z is entire, its unit-circle integral is zero. Cauchy did not write the theorem over an arbitrary closed path until a sequence of Comptes rendus notes of 1846, and he reached it there by what is now called Green’s formula rather than as a consequence of his 1825 memoir, which he did not recall until 1851. Goursat’s 1884 and 1900 papers established the theorem without assuming continuity of the derivative. Before 1846 Cauchy restricted himself to rectangular and circular paths; the first of the 1846 notes is Sur les intégrales qui s’étendent à tous les points d’une courbe fermée (C. R. t. XXIII, 3 August 1846).", "sources": [ { "title": "Encyclopedia of Mathematics, Analytic function", "url": "https://encyclopediaofmath.org/wiki/Holomorphic_function", "own_work": false, "note": "states Cauchy’s theorem with the required domain condition and the equivalence of holomorphy and complex analyticity.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Édouard Goursat", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Goursat/", "own_work": false, "note": "identifies Goursat’s removal of the derivative-continuity condition and dates his first paper to 1884.", "fetched": [ "2026-09-09" ] }, { "title": "Dictionary of Scientific Biography, Édouard Goursat", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Goursat.pdf", "own_work": false, "note": "records the 1884 Acta Mathematica paper and the corresponding 1900 Transactions of the AMS paper.", "fetched": [ "2026-09-09" ] }, { "title": "Œuvres complètes d’Augustin Cauchy, 1st series, vol. X, table of contents (Mathdoc)", "url": "http://sites.mathdoc.fr/cgi-bin/oetoc?id=OE_CAUCHY_1_10", "own_work": false, "note": "lists “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée p.70-74 C. R., t. XXIII, p. 251 (3 août 1846).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.3", "url": "https://www.teclamairme.com/#r-16.3", "chapter": 16, "status": "confirmed", "claim": "The residue theorem makes a positively oriented contour integral 2πi times the sum of enclosed residues; applying it to 1/z on the unit circle gives 2πi. Each positive circuit of the origin adds 2πi to the logarithm.", "sources": [ { "title": "NIST DLMF §1.10(iv), Residue Theorem", "url": "https://dlmf.nist.gov/1.10.iv", "own_work": false, "note": "equation 1.10.8 states the residue theorem with the 2πi normalization and positive orientation.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §4.2(i), Logarithms", "url": "https://dlmf.nist.gov/4.2.i", "own_work": false, "note": "“the general logarithm is given by Ln z = ln z + 2kπi, where k is the excess of the number of times the path … crosses the negative real axis in the positive sense over the number of times in the negative sense.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.4", "url": "https://www.teclamairme.com/#r-16.4", "chapter": 16, "status": "confirmed", "claim": "A residue is the coefficient of (z−a)^−1 in a Laurent expansion. The expansion of e^(1/z) gives residue 1 and an essential singularity at zero; Great Picard’s theorem says every complex value, with at most one exception, occurs infinitely often in every neighborhood of such a point. Since exp has no zeros, e^(1/z) omits the value 0. The residue theorem: (1/2πi)∫_C f(z) dz is the sum of the residues within C.", "sources": [ { "title": "NIST DLMF §1.10(iii), Laurent Series", "url": "https://dlmf.nist.gov/1.10.iii", "own_work": false, "note": "defines residues as Laurent coefficients, classifies essential singularities (“if aₙ ≠ 0 for infinitely many negative n, then z₀ is an isolated essential singularity”); the exponential series used after substituting 1/z is DLMF 4.2.19, on the §4.2(iii) line.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Essential singular point", "url": "https://encyclopediaofmath.org/wiki/Essential_singular_point", "own_work": false, "note": "states the infinitely-often form of Great Picard’s theorem with at most one exceptional finite value.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §4.2(iii), The exponential function", "url": "https://dlmf.nist.gov/4.2.iii", "own_work": false, "note": "4.2.19: exp z = 1 + z/1! + z²/2! + z³/3! + ⋯; “The function exp is an entire function of z, with no real or complex zeros.”", "fetched": [ "2026-09-22" ] }, { "title": "NIST DLMF §1.10(iv), Residue theorem", "url": "https://dlmf.nist.gov/1.10.iv", "own_work": false, "note": "“(1/2πi)∫_C f(z) dz = sum of the residues of f(z) within C.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.5", "url": "https://www.teclamairme.com/#r-16.5", "chapter": 16, "status": "confirmed", "claim": "The Cauchy–Riemann equations are equivalent, in Wirtinger notation, to ∂f/∂z̄ = 0. Wirtinger introduced the operators in a ~1926 paper; the history and the publisher’s record date it differently and both are given below.", "sources": [ { "title": "Encyclopedia of Mathematics, Cauchy–Riemann equations", "url": "https://encyclopediaofmath.org/wiki/Cauchy-Riemann_equations", "own_work": false, "note": "gives both real equations and their ∂/∂z̄ form.", "fetched": [ "2026-09-09" ] }, { "title": "R. Michael Range, Some landmarks in the history of the tangential Cauchy–Riemann equations", "url": "https://www.esi.ac.at/preprints/esi2225.pdf", "own_work": false, "note": "traces the operators to Wirtinger’s 1926 paper and displays their definitions and the holomorphy condition — supports 1926.", "fetched": [ "2026-09-09" ] }, { "title": "W. Wirtinger, Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen, Mathematische Annalen 97 (1927), 357–375 — Springer’s record for the printed volume", "url": "https://doi.org/10.1007/BF01447872", "own_work": false, "note": "supports 1927 — the publisher’s own record dates the printed volume to 1927.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-16.6", "url": "https://www.teclamairme.com/#r-16.6", "chapter": 16, "status": "confirmed", "claim": "The winding number n(γ, a) = (1/(2πi))∮γ dz/(z−a) is an integer counting signed turns; the counter-clockwise unit circle has winding number 1. Counterclockwise turns count as positive; the residue theorem counts each residue as many times as the curve winds round it; deforming the contour without crossing a singularity leaves the integral unchanged.", "sources": [ { "title": "MIT 18.783 Elliptic Curves (Spring 2013), Lecture 17", "url": "https://math.mit.edu/classes/18.783/2013/LectureNotes17.pdf", "own_work": false, "note": "gives the complex-integral formula, calls it an integer, and identifies it as the number of turns around the point.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §1.10(iv), Residue Theorem", "url": "https://dlmf.nist.gov/1.10.iv", "own_work": false, "note": "supplies the 1/(2πi) normalization and the residue theorem for a simple closed contour (1.10.8), where “the path C is described in the positive sense”; the winding-number weighting of residues is on the Residue theorem line below.", "fetched": [ "2026-09-09" ] }, { "title": "Winding number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Winding_number", "own_work": false, "note": "read as raw wikitext: “When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative.”", "fetched": [ "2026-09-22" ] }, { "title": "Residue theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Residue_theorem", "own_work": false, "note": "read as raw wikitext: “the line integral of f around γ is equal to 2πi times the sum of residues, each counted as many times as γ winds around the respective point.”", "fetched": [ "2026-09-22" ] }, { "title": "Contour integration, Wikipedia", "url": "https://en.wikipedia.org/wiki/Contour_integration", "own_work": false, "note": "read as raw wikitext: “the integrals of holmorphic functions are invariant under deforming the contour, provided the deformation does not cross a singularity or branch cut.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.7", "url": "https://www.teclamairme.com/#r-16.7", "chapter": 16, "status": "confirmed", "claim": "The argument principle gives (1/(2πi))∮ f’/f dz = N−P around a positively oriented contour, counting zeros and poles with multiplicity. For z²+1 on |z|=2, the roots ±i lie inside and there are no poles, so the count is 2; winding-number root counting also yields a proof of the fundamental theorem of algebra.", "sources": [ { "title": "NIST DLMF §1.10(iv), Phase Principle", "url": "https://dlmf.nist.gov/1.10.iv", "own_work": false, "note": "equation 1.10.9 states the argument principle as N−P and specifies multiplicities.", "fetched": [ "2026-09-09" ] }, { "title": "MIT 18.095, The notion of topological degree", "url": "https://math.mit.edu/classes/18.095/2021IAP/", "own_work": false, "note": "the course record identifies winding number as the method used for a proof of the fundamental theorem of algebra.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.8", "url": "https://www.teclamairme.com/#r-16.8", "chapter": 16, "status": "confirmed", "claim": "A holomorphic function is complex-differentiable on an open set and is locally equal to its Taylor series; a meromorphic function is holomorphic except at isolated poles.", "sources": [ { "title": "Encyclopedia of Mathematics, Analytic function", "url": "https://encyclopediaofmath.org/wiki/Holomorphic_function", "own_work": false, "note": "states the equivalence of holomorphy and complex analyticity and gives the Cauchy integral framework.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §1.10(iii), Laurent Series", "url": "https://dlmf.nist.gov/1.10.iii", "own_work": false, "note": "defines meromorphic functions through isolated poles and analytic functions through Taylor expansions.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.9", "url": "https://www.teclamairme.com/#r-16.9", "chapter": 16, "status": "confirmed", "claim": "A Laurent series is a two-sided power series on an annulus and its a_−1 coefficient is the residue. Laurent submitted the result in 1843; Weierstrass had proved it in 1841 but did not publish it then.", "sources": [ { "title": "NIST DLMF §1.10(iii), Laurent Series", "url": "https://dlmf.nist.gov/1.10.iii", "own_work": false, "note": "states the annular two-sided expansion and identifies the residue coefficient.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Pierre Alphonse Laurent", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Laurent_Pierre/", "own_work": false, "note": "dates Laurent’s submission and Cauchy’s report to 1843 and documents Weierstrass’s unpublished 1841 proof: “He submitted a memoir for the Grand Prix of the Academy of Sciences of 1842, unfortunately after the final date for submission in 1843”; “Cauchy reported on Laurent’s entry Mémoire sur le calcul des variations, which contains the Laurent series for a complex function, on 20 May 1843.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.10", "url": "https://www.teclamairme.com/#r-16.10", "chapter": 16, "status": "confirmed", "claim": "The residue theorem reduces contour integrals to residue sums and supports the evaluation of real integrals. Cauchy began the calculus of residues in 1826, in Sur un nouveau genre de calcul analogue au calcul infinitésimal, which is where the word residue enters mathematics; his Turin memoirs date to 1831–32, were summarized in 1837, and were republished in 1840–41. Boas notes that only in 1826 does Cauchy introduce the word residue. Contour integration gives ∫e^(itx)/(x² + 1) dx = πe^(−t) for t ≥ 0, which is π at t = 0.", "sources": [ { "title": "Dictionary of Scientific Biography, Augustin-Louis Cauchy", "url": "https://mathshistory.st-andrews.ac.uk/DSB/Cauchy.pdf", "own_work": false, "note": "documents the Turin memoirs, their later publication history, and Cauchy’s use of residues for definite integrals.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §1.10(iv), Residue Theorem", "url": "https://dlmf.nist.gov/1.10.iv", "own_work": false, "note": "gives the modern theorem that converts a contour integral to a finite residue sum: “If f(z) is analytic within a simple closed contour C, and continuous within and on C—except in both instances for a finite number of singularities within C—then” (1/2πi)∫_C f(z) dz is the sum of the residues within C (1.10.8).", "fetched": [ "2026-09-09" ] }, { "title": "Augustin-Louis Cauchy, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Cauchy/", "own_work": false, "note": "“He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal.”", "fetched": [ "2026-09-22" ] }, { "title": "Harold P. Boas, Cauchy’s residue sore thumb, arXiv:1701.04887", "url": "https://arxiv.org/abs/1701.04887", "own_work": false, "note": "read from the PDF: “Cauchy does not have the present terminology of line integrals (path integrals), and only in 1826 does he introduce the word “residue.””", "fetched": [ "2026-09-22" ] }, { "title": "Contour integration, Wikipedia", "url": "https://en.wikipedia.org/wiki/Contour_integration", "own_work": false, "note": "read as raw wikitext: the worked example ∫ e^(itx)/(x² + 1) dx over the real line = πe^(−t) (t ≥ 0).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.11", "url": "https://www.teclamairme.com/#r-16.11", "chapter": 16, "status": "confirmed", "claim": "A bounded entire function is constant. Cauchy apparently first published the result in 1844 and Liouville presented it in lectures in 1847; it gives a standard short proof of the fundamental theorem of algebra.", "sources": [ { "title": "Encyclopedia of Mathematics, Liouville theorems", "url": "https://encyclopediaofmath.org/wiki/Liouville_theorems", "own_work": false, "note": "states the theorem and records the Cauchy 1844 and Liouville 1847 history.", "fetched": [ "2026-09-09" ] }, { "title": "MIT 18.04, Complex Variables with Applications", "url": "https://math.mit.edu/~dunkel/Teach/18.04_2019S/notes/1804_Main.pdf", "own_work": false, "note": "states Liouville’s theorem and derives the fundamental theorem of algebra from 1/p.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.12", "url": "https://www.teclamairme.com/#r-16.12", "chapter": 16, "status": "confirmed", "claim": "Riemann’s 1851 thesis developed complex-function theory using what are now called Riemann surfaces and conformal mappings; such surfaces make multivalued relations single-valued, and compact surfaces are classified in part by genus. √z has two branches and log z needs countably many sheets; a Riemann surface is a connected one-dimensional complex manifold, and the Riemann surfaces on the torus are the elliptic curves.", "sources": [ { "title": "MacTutor, Bernhard Riemann", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Riemann/", "own_work": false, "note": "dates the thesis to 1851 and describes its Riemann surfaces, conformal mappings, and connectivity.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Riemann surface", "url": "https://encyclopediaofmath.org/wiki/Riemann_surface", "own_work": false, "note": "explains single-valued uniformization, the 1851 result, and classification data by genus: “Closed Riemann surfaces are topologically completely determined by one number — the genus.”", "fetched": [ "2026-09-09" ] }, { "title": "Branch point, Wikipedia", "url": "https://en.wikipedia.org/wiki/Branch_point", "own_work": false, "note": "read as raw wikitext: “the function w = z^(1/2) has two branches”; “The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut.”", "fetched": [ "2026-09-22" ] }, { "title": "Riemann surface, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_surface", "own_work": false, "note": "read as raw wikitext: “a Riemann surface is a connected one-dimensional complex manifold”; “The 2-torus T² has many different Riemann surface structures, all of the form ℂ/(ℤ+τℤ) … These are called elliptic curves.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-16.13", "url": "https://www.teclamairme.com/#r-16.13", "chapter": 16, "status": "confirmed", "claim": "The holomorphic automorphisms of the Riemann sphere are Möbius transformations forming PSL(2, ℂ), and they preserve generalized circles. PSL(2, ℂ) corresponds to the connected, proper orthochronous Lorentz group, not the full Lorentz group.", "sources": [ { "title": "Harvard Math 55b course notes", "url": "https://people.math.harvard.edu/~ctm/home/text/class/harvard/55b/10/html/home/course/course.pdf", "own_work": false, "note": "states Aut(Ĉ) ≅ PSL₂(ℂ) and that these automorphisms send circles to circles.", "fetched": [ "2026-09-09" ] }, { "title": "Blagoje Oblak, From the Lorentz Group to the Celestial Sphere", "url": "https://arxiv.org/abs/1508.00920", "own_work": false, "note": "the published lecture notes identify the connected Lorentz group with the conformal transformations of the sphere.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.14", "url": "https://www.teclamairme.com/#r-16.14", "chapter": 16, "status": "confirmed", "claim": "A holomorphic map with nonzero derivative is conformal, and every simply connected proper plane domain is biholomorphic to the disc. Riemann formulated the mapping theorem in 1851; Mercator’s 1569 world map uses a conformal projection.", "sources": [ { "title": "Encyclopedia of Mathematics, Conformal mapping", "url": "https://encyclopediaofmath.org/wiki/Conformal_mapping", "own_work": false, "note": "defines conformality and connects the simply connected case to the Riemann mapping theorem.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Riemann theorem", "url": "https://encyclopediaofmath.org/wiki/Riemann_theorem", "own_work": false, "note": "states the mapping theorem and dates Riemann’s formulation to 1851.", "fetched": [ "2026-09-09" ] }, { "title": "Cartographica Helvetica, The maps of Europe by Gerard Mercator", "url": "https://www.kartengeschichte.ch/ch/summaries/e10a.html", "own_work": false, "note": "dates the world map to 1569 and identifies its cylindrical projection as conformal.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.15", "url": "https://www.teclamairme.com/#r-16.15", "chapter": 16, "status": "confirmed", "claim": "Hadamard printed the epigraph’s English sentence in his 1945 book; the underlying thought is adapted from a related sentence in Paul Painlevé’s 1900 account of his scientific work.", "sources": [ { "title": "Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field", "url": "https://archive.org/details/eassayonthepsych006281mbp", "own_work": false, "note": "the 1945 scan contains the quoted wording on page 123.", "fetched": [ "2026-09-09" ] }, { "title": "University of Iowa, source history for the Hadamard quotation", "url": "https://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html", "own_work": false, "note": "transcribes Hadamard’s sentence and traces the related original wording to Painlevé’s 1900 Analyse des travaux scientifiques.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-16.16", "url": "https://www.teclamairme.com/#r-16.16", "chapter": 16, "status": "confirmed", "claim": "Cauchy’s integral formula: if D is an open domain whose boundary consists of finitely many non-intersecting rectifiable curves oriented positively with respect to D, and f is holomorphic on D, then f(z) = (1/2πi)∫_∂D f(ζ)/(ζ − z) dζ for every z in D, so the values of a holomorphic function inside a loop are determined by its values on the loop. The formula makes a holomorphic function infinitely differentiable and, through the geometric series, equal to its power series; Cauchy’s integral theorem yields the formula.", "sources": [ { "title": "Encyclopedia of Mathematics, Analytic function", "url": "https://encyclopediaofmath.org/wiki/Holomorphic_function", "own_work": false, "note": "states the formula in so many words — “if $D$ is an open domain whose boundary consists of a finite number of non-intersecting rectifiable curves (oriented positively with respect to $D$) and $f: D\\to \\mathbb C$ is holomorphic, then $f(z) = \\frac{1}{2\\pi i} \\int_{\\partial D} \\frac{f (\\zeta)}{\\zeta-z}\\, d\\zeta \\quad \\forall z\\in D$” — and carries the hypotheses (open domain, finitely many non-intersecting rectifiable boundary curves, positive orientation) that the popular one-line version drops.", "fetched": [ "2026-09-10" ] }, { "title": "Cauchy’s integral formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula", "own_work": false, "note": "read as raw wikitext: “it implies that a function that is holomorphic in an open set is in fact infinitely differentiable there. Furthermore, it is an analytic function, meaning that it can be represented as a power series. The proof of this uses the dominated convergence theorem and the geometric series.”", "fetched": [ "2026-09-22" ] }, { "title": "Analytic function, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Analytic_function", "own_work": false, "note": "“Cauchy’s integral theorem yields Cauchy’s integral formula, which expresses the values of an analytic function inside a domain in terms of its values on the boundary.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added in the V14 prose pass for the new card `cauchyformula`. The chapter’s sentence “One complex derivative forces infinitely many, equality with the function’s own power series, and complete determination inside a loop by the values on it” is footnoted 16.8, and 16.8’s first source is this same Encyclopedia of Mathematics page, so the third clause was already carried by a source the chapter cites; what 16.8’s claim line states is only the holomorphic/analytic equivalence and the definition of meromorphic, which is why the integral formula is given its own entry here rather than being read into 16.8." }, { "id": "r-16.18", "url": "https://www.teclamairme.com/#r-16.18", "chapter": 16, "status": "confirmed", "claim": "States Green’s theorem, converting a line integral around a closed plane curve into a double integral over the enclosed region, from Green’s 1828 essay (a proof was finally given by Riemann in 1851), and its role as the two-dimensional case of Stokes’ theorem underlying Cauchy’s integral theorem.", "sources": [ { "title": "Green’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Green%27s_theorem", "own_work": false, "note": "states the theorem, dates it to George Green’s 1828 essay (Green “stated a similar result in an 1828 paper titled An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism”), says “A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his doctoral dissertation,” and describes its relation to Stokes’ theorem.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.1", "url": "https://www.teclamairme.com/#r-17.1", "chapter": 17, "status": "confirmed", "claim": "The chapter’s gamma, beta, hypergeometric, Bessel, Airy, error, Fresnel and orthogonal-polynomial functions are standard special-function families represented through integrals, series, differential equations and related constructions.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions", "url": "https://dlmf.nist.gov/", "own_work": false, "note": "the authoritative chapter index organizes the named families and their definitions.", "fetched": [ "2026-09-09" ] }, { "title": "Andrews, Askey and Roy, Special Functions", "url": "https://www.cambridge.org/core/books/special-functions/1F1C575CEA780EE774F5518C7963BF08/listing", "own_work": false, "note": "the publisher’s contents and preface describe the subject’s long use and the book’s treatment of these families.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.2", "url": "https://www.teclamairme.com/#r-17.2", "chapter": 17, "status": "confirmed", "claim": "Gamma extends factorials, has no zeros and simple poles at the nonpositive integers, the pole at −n with residue (−1)ⁿ/n!, was introduced by Euler in 1729 correspondence with Goldbach, and received the Γ notation in Legendre’s 1811 volume. Sources differ on the year of Legendre’s Γ: 1811 (Miller, transcribing Legendre’s Exercices de calcul intégral, p. 277) or 1814 (Encyclopedia of Mathematics); corrections welcome.", "sources": [ { "title": "NIST DLMF §5.2, Gamma Function", "url": "https://dlmf.nist.gov/5.2", "own_work": false, "note": "defines gamma, its meromorphic continuation, factorial relation and simple poles: “It is a meromorphic function with no zeros, and with simple poles of residue (−1)^{n}/n! at z = −n.”", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Gamma-function", "url": "https://encyclopediaofmath.org/wiki/Gamma-function", "own_work": false, "note": "records Euler’s 1729 introduction in a letter to Goldbach; it gives 1814 for Legendre’s symbol Γ and the name gamma-function (“proposed in 1814 by A.M. Legendre”).", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Earliest Uses of Function Symbols", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/functions/", "own_work": false, "note": "transcribes the Γ notation from page 277 of Legendre’s 1811 volume.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.3", "url": "https://www.teclamairme.com/#r-17.3", "chapter": 17, "status": "confirmed", "claim": "Γ(1/2)=√π and Γ(3/2)=√π/2; the recurrence identifies these with (−1/2)! and (1/2)! respectively.", "sources": [ { "title": "NIST DLMF §5.4, Special Values", "url": "https://dlmf.nist.gov/5.4", "own_work": false, "note": "gives the half-integer gamma values.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §5.5, Functional Relations", "url": "https://dlmf.nist.gov/5.5", "own_work": false, "note": "gives Γ(z+1)=zΓ(z).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.4", "url": "https://www.teclamairme.com/#r-17.4", "chapter": 17, "status": "confirmed", "claim": "The logarithmic integral uses a principal value across t=1, and the prime number theorem establishes its asymptotic relation to π(x); Hadamard and de la Vallée Poussin proved the theorem in 1896 after Gauss’s estimate. The prime number theorem can be written π(x) ∼ li(x).", "sources": [ { "title": "NIST DLMF §6.2, Exponential and Logarithmic Integrals", "url": "https://dlmf.nist.gov/6.2", "own_work": false, "note": "defines li by the Cauchy principal value.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §27.2, Functions of Number Theory", "url": "https://dlmf.nist.gov/27.2", "own_work": false, "note": "(eq. 27.2.3): states the prime number theorem in the form π(x) ∼ x/ln x, credits the 1896 proofs of Hadamard and de la Vallée Poussin, and names Gauss and Legendre for the conjecture. It does not use li or the words logarithmic integral.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §27.12, Asymptotic Formulas: Primes", "url": "https://dlmf.nist.gov/27.12", "own_work": false, "note": "(eq. 27.12.5): bounds |π(x) − li(x)| by O(x exp(−c(ln x)^½)), the sharper form from which π(x) ∼ li(x) follows. This is the DLMF section that ties π(x) to li(x).", "fetched": [ "2026-09-10" ] }, { "title": "Logarithmic integral function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Logarithmic_integral_function", "own_work": false, "note": "read as raw wikitext: “The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value”; “the prime number theorem states that: π(x) ∼ li(x).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.5", "url": "https://www.teclamairme.com/#r-17.5", "chapter": 17, "status": "confirmed", "claim": "The Ramanujan–Soldner constant is the unique positive zero μ≈1.451369234883; Soldner’s table appeared in 1809, Ramanujan later used the associated expansion, and a 2019 arithmetic survey reports that irrationality of μ and log μ is unknown. Ramanujan computed μ (as 1.45136380…, the correct value being 1.45136923488…).", "sources": [ { "title": "OEIS A070769, Ramanujan–Soldner constant", "url": "https://oeis.org/A070769", "own_work": false, "note": "gives the value and cites Soldner’s 1809 Théorie et tables d’une nouvelle fonction transcendante.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Logarithmic integral function", "url": "https://en.wikipedia.org/wiki/Logarithmic_integral_function", "own_work": false, "note": "defines the Ramanujan–Soldner constant μ ≈ 1.45136 92348 as the unique positive zero of li(x), and gives Ramanujan’s more rapidly convergent series.", "fetched": [ "2026-09-22" ] }, { "title": "Johann Georg von Soldner, Wikipedia", "url": "https://en.wikipedia.org/wiki/Johann_Georg_von_Soldner", "own_work": false, "note": "“He also published on the logarithmic integral function”; lists “Soldner, J. G. v. (1809). Théorie et tables d’une nouvelle fonction transcendante. München: Lindauer.”", "fetched": [ "2026-09-22" ] }, { "title": "Soldner’s Constant, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/SoldnersConstant.html", "own_work": false, "note": "Ramanujan calculated “μ = 1.45136380...,” while the correct value is 1.45136923488….", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.6", "url": "https://www.teclamairme.com/#r-17.6", "chapter": 17, "status": "confirmed", "claim": "The reciprocal gamma function 1/Γ(z) is entire and has zeros at the simple poles of Γ, including (1/Γ)(0)=0.", "sources": [ { "title": "NIST DLMF §5.2, Gamma Function", "url": "https://dlmf.nist.gov/5.2", "own_work": false, "note": "records that reciprocal gamma is entire with simple zeros at the nonpositive integers.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.7", "url": "https://www.teclamairme.com/#r-17.7", "chapter": 17, "status": "confirmed", "claim": "For the unnormalized convention sinc(x)=sin(x)/x, sinc(0)=1 by continuation and sinc(π)=0; sinc is the interpolation kernel in band-limited sampling. sinc(0) is defined as the limiting value 1.", "sources": [ { "title": "NIST DLMF §4.14, Trigonometric Functions", "url": "https://dlmf.nist.gov/4.14", "own_work": false, "note": "supplies the sine zeros used in the elementary evaluation: “In ℂ the zeros of sin z are z = kπ, k ∈ ℤ” (the limiting value sinc(0) = 1 is on the Sinc function line).", "fetched": [ "2026-09-09" ] }, { "title": "MIT OpenCourseWare, The Sampling Process", "url": "https://ocw.mit.edu/courses/2-693-principles-of-oceanographic-instrument-systems-sensors-and-measurements-13-998-spring-2004/91fd1c49b05bf805b0d7057cf45cf230_part2_smplingpro.pdf", "own_work": false, "note": "states that sinc is the ideal interpolation function in the sampling theorem.", "fetched": [ "2026-09-09" ] }, { "title": "Sinc function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Sinc_function", "own_work": false, "note": "read as raw wikitext: “In either case, the value at x = 0 is defined to be the limiting value” sinc(0) = lim sin(ax)/(ax) = 1.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.8", "url": "https://www.teclamairme.com/#r-17.8", "chapter": 17, "status": "confirmed", "claim": "For integer n≥1, J_n(0)=0 while J_0(0)=1, as follows from the defining Bessel series. Bessel’s equation comes from separating Laplace’s or Helmholtz’s equation in cylindrical coordinates; J is the solution finite at the origin; the zeros of J₀ set the axisymmetric modes of a circular membrane. For integer n, Jₙ(x) is the coefficient of tⁿ in the Laurent expansion of e^((x/2)(t − 1/t)).", "sources": [ { "title": "NIST DLMF §10.2, Bessel Functions", "url": "https://dlmf.nist.gov/10.2", "own_work": false, "note": "gives the defining series (10.2.2), from which J₀(0) = 1 and Jₙ(0) = 0 for n ≥ 1 follow; the page does not state those values itself.", "fetched": [ "2026-09-09" ] }, { "title": "Bessel function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bessel_function", "own_work": false, "note": "read as raw wikitext: “Bessel’s equation arises when finding separable solutions to Laplace’s equation and the Helmholtz equation in cylindrical or spherical coordinates”; applications include “Electromagnetic waves in a cylindrical waveguide,” “Heat conduction in a cylindrical object,” “Modes of vibration of a thin circular or annular acoustic membrane (such as a drumhead …)”; “For integer or positive α, Bessel functions of the first kind are finite at the origin”; “For integer order α = n, J_{n} is often defined via a Laurent series for a generating function: e^{(x/2)(t − 1/t)} = Σ_{n=−∞}^{∞} J_{n}(x) t^{n}.”", "fetched": [ "2026-09-22" ] }, { "title": "Vibrations of a circular membrane, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane", "own_work": false, "note": "the axisymmetric modes are set by the positive roots of J₀; the fundamental u₀₁ has α₀₁ = 2.40483.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.9", "url": "https://www.teclamairme.com/#r-17.9", "chapter": 17, "status": "confirmed", "claim": "The beta function satisfies B(a, b)=Γ(a)Γ(b)/Γ(a+b), hence B(1/2, 1/2)=π.", "sources": [ { "title": "NIST DLMF §5.12, Beta Function", "url": "https://dlmf.nist.gov/5.12", "own_work": false, "note": "gives the beta integral and gamma quotient.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.10", "url": "https://www.teclamairme.com/#r-17.10", "chapter": 17, "status": "confirmed", "claim": "The digamma function is ψ=Γ′/Γ and ψ(1)=−γ; current literature still does not establish whether the Euler–Mascheroni constant is irrational. ψ(z) = Γ′(z)/Γ(z); ψ(1) = −γ and ψ(n + 1) = Σ 1/k − γ.", "sources": [ { "title": "NIST DLMF §5.15, Polygamma Functions", "url": "https://dlmf.nist.gov/5.15", "own_work": false, "note": "defines the polygamma functions ψ⁽ⁿ⁾, n = 1, 2, …, whose properties “follow straightforwardly by differentiation of properties of the psi function,” and gives their special values (digamma’s definition and values are on the §5.2 and §5.4 lines).", "fetched": [ "2026-09-09" ] }, { "title": "Marek Wolf, 6+∞ New Expressions for the Euler–Mascheroni Constant (2025)", "url": "https://cmst.eu/wp-content/uploads/files/10.12921_cmst.2025.0000011_WOLF_acc.pdf", "own_work": false, "note": "the peer-reviewed paper explicitly states that irrationality of γ remains unknown.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §5.2, Gamma function: definitions", "url": "https://dlmf.nist.gov/5.2", "own_work": false, "note": "5.2.2: ψ(z) = Γ′(z)/Γ(z).", "fetched": [ "2026-09-22" ] }, { "title": "NIST DLMF §5.4, Gamma function: special values", "url": "https://dlmf.nist.gov/5.4", "own_work": false, "note": "5.4.12: ψ(1) = −γ; 5.4.14: ψ(n+1) = Σ_{k=1}^{n} 1/k − γ.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.11", "url": "https://www.teclamairme.com/#r-17.11", "chapter": 17, "status": "confirmed", "claim": "Lambert W is the multivalued inverse of w↦we^w; W(1)=Ω≈0.5671433, and its real branches meet at W(−1/e)=−1. Ω = W(1) = 0.56714 32904… Corless, Gonnet, Hare, Jeffrey and Knuth proposed the standard notation for W’s branches in their 1996 paper.", "sources": [ { "title": "NIST DLMF §4.13, Lambert W-Function", "url": "https://dlmf.nist.gov/4.13", "own_work": false, "note": "defines the inverse, its branches and the branch point W(−1/e) = −1 (the value Ω = W(1) is on the Omega constant line).", "fetched": [ "2026-09-09" ] }, { "title": "Corless et al., On the Lambert W Function (1996)", "url": "https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf", "own_work": false, "note": "the published paper’s author copy defines W and standardizes its branches and notation: “The text below is the same as that published in Advances in Computational Mathematics, Vol 5 (1996) 329 – 359”; authors R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth; “We use the letter W for this function, following early Maple usage.” and “An important part of this paper is our proposal of a standard notation for all the branches of W in the complex plane.”", "fetched": [ "2026-09-09" ] }, { "title": "Omega constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Omega_constant", "own_work": false, "note": "read as raw wikitext: “It is the value of W(1), where W is Lambert’s W function”; Ω = 0.56714 32904 09783 87299 …", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.12", "url": "https://www.teclamairme.com/#r-17.12", "chapter": 17, "status": "confirmed", "claim": "W(1) is transcendental: if algebraic nonzero Ω satisfied Ωe^Ω=1, Lindemann–Weierstrass would make e^Ω transcendental rather than algebraic 1/Ω. The same theorem makes π transcendental, and transcendence survives a nonzero rational scaling and a square root, so √π, π/2 and π itself are transcendental together.", "sources": [ { "title": "Encyclopedia of Mathematics, Lindemann–Weierstrass theorem", "url": "https://encyclopediaofmath.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "states the required transcendence of e^α for nonzero algebraic α.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §4.13, Lambert W-Function", "url": "https://dlmf.nist.gov/4.13", "own_work": false, "note": "supplies Ωe^Ω=1.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.13", "url": "https://www.teclamairme.com/#r-17.13", "chapter": 17, "status": "confirmed", "claim": "The first positive zero j_{0,1}≈2.4048256 determines the fundamental axisymmetric circular-drum mode, and Siegel’s 1929 work implies every nonzero J_{0} zero is transcendental.", "sources": [ { "title": "NIST DLMF §10.21, Zeros of Bessel Functions", "url": "https://dlmf.nist.gov/10.21", "own_work": false, "note": "defines the positive Bessel zeros j_{ν,m} and gives their properties and asymptotic expansions (numerical tables are referred elsewhere).", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Carl Ludwig Siegel", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Siegel/", "own_work": false, "note": "records Siegel’s 1929 result in these words: “for any non-zero algebraic integer r he showed that J0(r) is transcendental,” so no non-zero algebraic integer is a zero of J₀ (the statement for every nonzero root is on the Bessel function line).", "fetched": [ "2026-09-09" ] }, { "title": "Vibrations of a circular membrane, Wikipedia", "url": "https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane", "own_work": false, "note": "(§ The axisymmetric case, § Animations of several vibration modes): the axisymmetric modes are set by the positive roots of J₀, and the fundamental mode u₀₁ has α₀₁ = 2.40483.", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Bessel function", "url": "https://en.wikipedia.org/wiki/Bessel_function", "own_work": false, "note": "its section on the zeros states that “Siegel proved in 1929 that when ν is rational, all nonzero roots of Jν(x) and J′ν(x) are transcendental,” citing Lorch and Muldoon, Int. J. Math. Math. Sci. 18 (1995) 551–560 (doi:10.1155/S0161171295000706).", "fetched": [ "2026-09-22" ] }, { "title": "D. S. Karachalios, I. V. Gosea, Q. Zhang, A. C. Antoulas, Case study: Approximations of the Bessel function, arXiv:1801.03390", "url": "https://arxiv.org/abs/1801.03390", "own_work": false, "note": "read from the PDF: “The first 6 solutions of the equation J0(s) = 0, with 15 digits precision are: … 2.40482555769577.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.14", "url": "https://www.teclamairme.com/#r-17.14", "chapter": 17, "status": "confirmed", "claim": "erf(1)≈0.8427008 equals the standard-normal probability within √2 standard deviations under erf(x)=2/√π∫_0^x e^(−t²)dt. erf(a/(σ√2)) is the probability of an error between −a and a; at a = σ√2 this is erf(1) = 0.842700793…", "sources": [ { "title": "NIST DLMF §7.2, Error Functions", "url": "https://dlmf.nist.gov/7.2", "own_work": false, "note": "defines erf and gives its normalization.", "fetched": [ "2026-09-09" ] }, { "title": "NIST/SEMATECH e-Handbook, Normal Distribution", "url": "https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm", "own_work": false, "note": "gives the normal density and cumulative-probability standardization used for the probability interpretation.", "fetched": [ "2026-09-09" ] }, { "title": "Error function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Error_function", "own_work": false, "note": "read as raw wikitext: “The defining integral cannot be evaluated in closed form in terms of elementary functions”; erf(x) = 2Φ(x√2) − 1; for a normal distribution with standard deviation σ and mean zero, erf(a/(σ√2)) “is the probability that the error of a single measurement lies between −a and a”; table: erf 1 = 0.842700793.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.15", "url": "https://www.teclamairme.com/#r-17.15", "chapter": 17, "status": "confirmed", "claim": "Ai(0)=1/(3^(2/3)Γ(2/3))≈0.355028; Airy functions solve the Airy equation and occur in linear-potential quantum problems. For x > 0 Ai decays exponentially and Bi grows; for x < 0 both oscillate.", "sources": [ { "title": "NIST DLMF §9.2, Airy Functions", "url": "https://dlmf.nist.gov/9.2", "own_work": false, "note": "gives the Airy equation and Ai(0) special value.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §9.16, Physical Applications", "url": "https://dlmf.nist.gov/9.16", "own_work": false, "note": "documents Airy-function applications including quantum-mechanical turning-point problems; Ai “first appears as an integral in two articles by G.B. Airy on the intensity of light in the neighborhood of a caustic.”", "fetched": [ "2026-09-09" ] }, { "title": "Airy function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Airy_function", "own_work": false, "note": "read as raw wikitext: “When x is positive, Ai(x) is positive, convex, and decreasing exponentially to zero, while Bi(x) is positive, convex, and increasing exponentially. When x is negative, Ai(x) and Bi(x) oscillate around zero”; “the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential)”; “The Airy function underlies the form of the intensity near an optical directional caustic, such as that of the rainbow.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.16", "url": "https://www.teclamairme.com/#r-17.16", "chapter": 17, "status": "confirmed", "claim": "K(k)=∫_0^(π/2)dθ/√(1−k²sin²θ), so K(0)=π/2; K gives the finite-amplitude pendulum period, whereas ellipse arc length uses the second-kind integral E.", "sources": [ { "title": "NIST DLMF §19.2, Legendre’s Integrals", "url": "https://dlmf.nist.gov/19.2", "own_work": false, "note": "defines K and E.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §19.6, Special Cases", "url": "https://dlmf.nist.gov/19.6", "own_work": false, "note": "gives K(0)=π/2.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §22.19, Physical Applications", "url": "https://dlmf.nist.gov/22.19", "own_work": false, "note": "gives the pendulum period as 4K(sin(α/2)).", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §19.30, Lengths of Plane Curves", "url": "https://dlmf.nist.gov/19.30", "own_work": false, "note": "gives ellipse arc length using E, resolving the inherited K/E mix-up.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.17", "url": "https://www.teclamairme.com/#r-17.17", "chapter": 17, "status": "confirmed", "claim": "The Fresnel integral S(x)=∫_0^x sin(πt²/2)dt tends to 1/2, and S together with C parametrizes the Cornu spiral used in diffraction. Cornu showed in 1874 that diffraction intensity can be read off a graph of the spiral; the same curve is Euler’s of 1744.", "sources": [ { "title": "NIST DLMF §7.2, Fresnel Integrals", "url": "https://dlmf.nist.gov/7.2", "own_work": false, "note": "defines S and C, and 7.2.9 gives lim_{x→∞} C(x) = lim_{x→∞} S(x) = ½.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §7.12(ii), Fresnel Asymptotics", "url": "https://dlmf.nist.gov/7.12.ii", "own_work": false, "note": "gives the large-z asymptotic expansions of the auxiliary functions f and g, referring to 7.5.3–7.5.4 for C and S (the limits at infinity are on the §7.2 line).", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §7.20, Mathematical Applications", "url": "https://dlmf.nist.gov/7.20", "own_work": false, "note": "identifies the parametric Cornu spiral.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §7.21, Physical Applications", "url": "https://dlmf.nist.gov/7.21", "own_work": false, "note": "connects Fresnel integrals and the Cornu spiral to diffraction.", "fetched": [ "2026-09-09" ] }, { "title": "Euler spiral, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler_spiral", "own_work": false, "note": "read as raw wikitext: named for “elastic springs (“Euler spiral,” 1744), graphical computations in light diffraction (“Cornu spiral,” 1874)”; “In 1874, Alfred Marie Cornu showed that diffraction intensity could be read off a graph of the spiral”; the diagram of a Cornu spiral is used for plane-wave diffraction from a knife-edge.", "fetched": [ "2026-09-22" ] }, { "title": "R. D. Blandford and K. S. Thorne, Applications of Classical Physics, Ch. 8 “Diffraction,” §8.4.3 (Caltech Ph136, 2012)", "url": "http://www.pmaweb.caltech.edu/Courses/ph136/yr2012/1208.1.K.pdf", "own_work": false, "note": "for a straight edge, “Long before the occultation, we can approximate σmin = −∞ and σmax = +∞ …, so ∆Eσ is given by the arrow from (−1/2, −1/2) to (1/2, 1/2)”; and “The flux |ψ|2 ∝ |∆Eσ|2 is proportional to the squared length of the arrow on the Cornu spiral.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "" }, { "id": "r-17.18", "url": "https://www.teclamairme.com/#r-17.18", "chapter": 17, "status": "confirmed", "claim": "The Gauss hypergeometric series is Σ(a)_n(b)_nz^n/((c)_nn!); parameter choices produce logarithmic and inverse-trigonometric special cases and negative integer parameters give polynomial cases.", "sources": [ { "title": "NIST DLMF §15.2, Gauss Series", "url": "https://dlmf.nist.gov/15.2", "own_work": false, "note": "gives the defining series, analytic continuation and polynomial cases.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §15.4, Special Cases", "url": "https://dlmf.nist.gov/15.4", "own_work": false, "note": "gives logarithm, arctangent and arcsine parameter choices.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.19", "url": "https://www.teclamairme.com/#r-17.19", "chapter": 17, "status": "confirmed", "claim": "Li_s(z)=Σz^n/n^s specializes to −ln(1−z) at s=1 and ζ(s) at z=1 where convergent; Li_2(1/2)=π²/12−(ln2)²/2. Li₁(z) = −ln(1 − z).", "sources": [ { "title": "NIST DLMF §25.12, Polylogarithms", "url": "https://dlmf.nist.gov/25.12", "own_work": false, "note": "gives the definition and the special case “ζ(s) = Li_{s}(1)”; its reflection formula 25.12.6, Li₂(x) + Li₂(1 − x) = π²/6 − (ln x) ln(1 − x), gives Li₂(½) = π²/12 − (ln 2)²/2 at x = ½ (Li₁ is on the Polylogarithm line).", "fetched": [ "2026-09-09" ] }, { "title": "Polylogarithm, Wikipedia", "url": "https://en.wikipedia.org/wiki/Polylogarithm", "own_work": false, "note": "read as raw wikitext: “Li₁(z) = −ln(1−z)” and “Li₁(½) = ln 2.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.20", "url": "https://www.teclamairme.com/#r-17.20", "chapter": 17, "status": "confirmed", "claim": "For the standard Jacobi convention, θ₃(0;i)=Γ(1/4)/(√2π^(3/4))=π^(1/4)/Γ(3/4); theta functions connect to elliptic functions and the zeta functional equation. In the convention ϑ(z; τ) = Σ exp(πin²τ + 2πinz), the relation ϑ(0; −1/τ) = (−iτ)^½ ϑ(0; τ) is what Riemann used for the zeta functional equation.", "sources": [ { "title": "MathWorld, Jacobi Theta Functions", "url": "https://mathworld.wolfram.com/JacobiThetaFunctions.html", "own_work": false, "note": "explicitly gives θ₃(e^−π)=π^(1/4)/Γ(3/4); “The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions.”", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §5.5, Functional Relations", "url": "https://dlmf.nist.gov/5.5", "own_work": false, "note": "Euler’s reflection formula makes the two displayed gamma forms identical.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §20.9, Relations to Other Functions", "url": "https://dlmf.nist.gov/20.9", "own_work": false, "note": "connects theta functions to elliptic functions and the Riemann zeta function.", "fetched": [ "2026-09-09" ] }, { "title": "Theta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Theta_function", "own_work": false, "note": "read as raw wikitext: “ϑ(z; τ) = Σ_{n=−∞}^{∞} exp(πin²τ + 2πinz)”; “The relation ϑ(0; −1/τ) = (−iτ)^½ ϑ(0; τ) was used by Riemann to prove the functional equation for the Riemann zeta function, by means of the Mellin transform.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.21", "url": "https://www.teclamairme.com/#r-17.21", "chapter": 17, "status": "confirmed", "claim": "Legendre, Hermite and Laguerre are classical orthogonal-polynomial families; Hermite occurs in oscillator eigenfunctions, generalized Laguerre in hydrogenic radial eigenfunctions, and Legendre in spherical harmonics. Legendre polynomials are orthogonal under weight 1 on [−1, 1], Laguerre under e^(−x)x^α, Hermite under e^(−x²).", "sources": [ { "title": "NIST DLMF §18.3, Orthogonal Polynomials: Definitions", "url": "https://dlmf.nist.gov/18.3", "own_work": false, "note": "defines the classical families and their weights (Table 18.3.1: the orthogonality intervals and weight functions) (the link formerly pointed to the chapter’s table of contents).", "fetched": [ "2026-09-24" ] }, { "title": "NIST DLMF §18.39, Applications in the Physical Sciences", "url": "https://dlmf.nist.gov/18.39", "own_work": false, "note": "gives the oscillator and hydrogen-atom eigenfunction applications.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §14.30, Spherical and Spheroidal Harmonics", "url": "https://dlmf.nist.gov/14.30", "own_work": false, "note": "relates spherical harmonics to the Ferrers (associated Legendre) functions 𝖯_{l}^{m}(cos θ); it does not name the Legendre polynomials themselves.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §18.3, Classical orthogonal polynomials: definitions", "url": "https://dlmf.nist.gov/18.3", "own_work": false, "note": "“Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization”; its weights: Legendre w(x) = 1, Laguerre w(x) = e^(−x)x^α, Hermite w(x) = e^(−x²).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.22", "url": "https://www.teclamairme.com/#r-17.22", "chapter": 17, "status": "confirmed", "claim": "The upper and lower incomplete gamma functions sum to Γ(s); γ(1/2, x²)=√π erf(x), and regularized incomplete gamma functions supply chi-square distribution functions.", "sources": [ { "title": "NIST DLMF §8.2, Incomplete Gamma Functions", "url": "https://dlmf.nist.gov/8.2", "own_work": false, "note": "defines upper and lower functions and their sum.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §8.4, Special Values", "url": "https://dlmf.nist.gov/8.4", "own_work": false, "note": "gives the error-function identity.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §8.23, Statistical Applications", "url": "https://dlmf.nist.gov/8.23", "own_work": false, "note": "identifies chi-square distributions as particular regularized incomplete-gamma forms.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.23", "url": "https://www.teclamairme.com/#r-17.23", "chapter": 17, "status": "judgement", "claim": "The arithmetic classifications of μ, γ and erf(1) are treated conservatively: explicit recent sources leave μ and γ irrationality unresolved; the cited 2023 analysis proves ∫_0^1e^(−x²)dx transcendental but does not settle its ratio 2/√π=erf(1).", "sources": [ { "title": "Wikipedia, List of mathematical constants", "url": "https://en.wikipedia.org/wiki/List_of_mathematical_constants", "own_work": false, "note": "the Soldner constant μ’s row marks its rationality, algebraicity and period status all as unknown (“?”).", "fetched": [ "2026-09-24" ] }, { "title": "Wolf, 6+∞ New Expressions for the Euler–Mascheroni Constant (2025)", "url": "https://cmst.eu/wp-content/uploads/files/10.12921_cmst.2025.0000011_WOLF_acc.pdf", "own_work": false, "note": "explicitly states that γ irrationality is unknown.", "fetched": [ "2026-09-09" ] }, { "title": "MathOverflow, arithmetic of ∫₀¹e^(−x²)dx (2023)", "url": "https://mathoverflow.net/questions/461028/is-the-integral-of-e-x2-from-0-to-1-known-to-be-irrational", "own_work": false, "note": "supplies a detailed Siegel–Shidlovsky proof for the unnormalized integral; it does not classify erf(1), so the page makes only a documented literature-survey judgement for that ratio.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.24", "url": "https://www.teclamairme.com/#r-17.24", "chapter": 17, "status": "judgement", "claim": "Chudnovsky’s algebraic-independence work in the 1970s makes π algebraically independent of Γ(1/4), and likewise of Γ(1/3); this proves the gamma values and the displayed Ai(0) combination transcendental. The result is Chudnovsky’s 1976 Doklady paper (Waldschmidt, Philibert); the 1975 that Wikipedia’s Lemniscate constant page gives is the date of his abstract in the Notices of the AMS, which announced the result (two events, not a dispute); the 1984 date is the later AMS monograph that collected it.", "sources": [ { "title": "G. V. Chudnovsky, Algebraic independence of the values of elliptic function at algebraic points", "url": "https://doi.org/10.1007/BF01390068", "own_work": false, "note": "the primary author’s 1980 paper cites the relevant 1976 Doklady paper by title, issue and pages.", "fetched": [ "2026-09-09" ] }, { "title": "Michel Waldschmidt, Transcendental Number Theory: the State of the Art (KIAS, Seoul, February 2004)", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/KIAS-02-2004.pdf", "own_work": false, "note": "(under the heading “G.V. Chudnovskij (1976)”): “The two numbers Γ(1/4) and π are algebraically independent and the two numbers Γ(1/3) and π are algebraically independent.”", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Lemniscate constant (revision of 13 September 2026)", "url": "https://en.wikipedia.org/w/index.php?title=Lemniscate_constant&oldid=1374703971", "own_work": false, "note": "a permanent link to the revision read. Read raw on 24 September 2026: “In 1975, Gregory Chudnovsky proved that π and ϖ are algebraically independent over Q,” citing his abstract in the Notices of the AMS 22 (1975), p. A-486; the book dates the result by the 1976 Doklady paper, which that abstract announced. History: a 23 September 2026 reading reported this sentence gone; the page has had no edit since 13 September, so that reading was mistaken and the sentence was there throughout.", "fetched": [ "2026-09-24" ] }, { "title": "Georges Philibert, Une mesure d’indépendance algébrique, Annales de l’Institut Fourier 38 (1988), no. 3, 85–103", "url": "https://doi.org/10.5802/aif.1142", "own_work": false, "note": "cited for its reference list, which distinguishes Chudnovsky’s 1976 Doklady paper (entry [2]) from the 1984 AMS monograph (entry [4]).", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-17.25", "url": "https://www.teclamairme.com/#r-17.25", "chapter": 17, "status": "confirmed", "claim": "The epigraph is quoted from the 1999 Cambridge University Press preface by George Andrews, Richard Askey and Ranjan Roy, where the authors attribute the remark to Paul Turán.", "sources": [ { "title": "Cambridge University Press, Special Functions — Preface", "url": "https://www.cambridge.org/core/books/abs/special-functions/preface/7A3FFF3EE7FEB76075E47BA4C4A44152", "own_work": false, "note": "prints the sentence, author list and attribution to Turán; print publication year 1999.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-17.26", "url": "https://www.teclamairme.com/#r-17.26", "chapter": 17, "status": "confirmed", "claim": "The gamma function is the unique log-convex extension of the factorial: Bohr and Mollerup, 1922. Γ is not the only analytic extension of the factorial: any analytic function vanishing on the positive integers, such as k sin(mπx), can be added. Substituting t = u² in Euler’s integral turns Γ(½) into the Gaussian integral, 2∫₀^∞ e^(−u²) du = √π.", "sources": [ { "title": "Bohr–Mollerup theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem", "own_work": false, "note": "proved in 1922 by Harald Bohr and Johannes Mollerup; “Γ(x) is the only function that satisfies f(x + 1) = x f(x) with log(f(x)) convex and also with f(1) = 1,” for positive f on x > 0.", "fetched": [ "2026-09-10" ] }, { "title": "Gamma function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gamma_function", "own_work": false, "note": "read as raw wikitext: “it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin(mπx) for an integer m.”; “Perhaps the best-known value of the gamma function at a non-integer argument is Γ(1/2) = √π, which can be found … simply by making the substitution t = u² in the integral definition of the gamma function, resulting in a Gaussian integral”; with t = cx², “if z = 1/2, c = 1 we get Γ(1/2) = 2∫₀^∞ e^(−t²) dt = √π” (formulas as typeset on the page).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.27", "url": "https://www.teclamairme.com/#r-17.27", "chapter": 17, "status": "confirmed", "claim": "The gamma reflection formula is Γ(z)Γ(1−z) = π/sin(πz) for z ≠ 0, ±1, …; at z = 1/3 it gives Γ(1/3)Γ(2/3) = 2π/√3, the identity the chapter’s Ai(0) transcendence argument uses beside Chudnovsky’s theorem.", "sources": [ { "title": "NIST DLMF §5.5(ii), Functional Relations: Reflection", "url": "https://dlmf.nist.gov/5.5", "own_work": false, "note": "equation 5.5.3 states Γ(z)Γ(1−z) = π/sin(πz) for z ≠ 0, ±1, …; §5.5(i) carries the recurrence and §5.5(iv) the Bohr–Mollerup theorem.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Own arithmetic checked: sin(π/3) = √3/2, so the reflection formula at z = 1/3 gives Γ(1/3)Γ(2/3) = 2π/√3 ≈ 3.6275987, which matches the direct product of the two gamma values to full double precision. Added in the V14 prose pass because Chapter ᚐ(A)’s paragraph 7 invokes “the reflection formula” by name, and no reference already in the chapter states it: r-17.3 cites DLMF §5.5 but only for the recurrence Γ(z+1) = zΓ(z). Card ai0 invokes the same identity under 17.24 and should cite this entry too." }, { "id": "r-17.28", "url": "https://www.teclamairme.com/#r-17.28", "chapter": 17, "status": "confirmed", "claim": "The defining equations of the four families the chapter names and had no family card for: Bessel’s equation z²w″ + zw′ + (z² − ν²)w = 0 with Jν its solution of the first kind; the Airy equation w″ = zw with Ai and Bi its standard solutions; erf z = (2/√π)∫₀ᶻ e^(−t²) dt with erfc z = 1 − erf z; and the Fresnel integrals C(z) = ∫₀ᶻ cos(πt²/2) dt and S(z) = ∫₀ᶻ sin(πt²/2) dt.", "sources": [ { "title": "NIST DLMF §10.2, Bessel Functions: Definitions", "url": "https://dlmf.nist.gov/10.2", "own_work": false, "note": "equation 10.2.1 gives Bessel’s differential equation z²(d²w/dz²) + z(dw/dz) + (z² − ν²)w = 0, and 10.2.2 gives the ascending series defining Jν(z), the Bessel function of the first kind.", "fetched": [ "2026-09-11" ] }, { "title": "NIST DLMF §9.2, Airy Functions: Differential Equation", "url": "https://dlmf.nist.gov/9.2", "own_work": false, "note": "equation 9.2.1 gives the Airy equation d²w/dz² = zw, names Ai(z) and Bi(z) as its standard solutions, records that all solutions are entire, and gives Ai(0) = 1/(3^(2/3)Γ(2/3)) = 0.35502 80538….", "fetched": [ "2026-09-11" ] }, { "title": "NIST DLMF §7.2, Error Functions and Fresnel Integrals: Definitions", "url": "https://dlmf.nist.gov/7.2", "own_work": false, "note": "7.2.1 gives erf z = (2/√π)∫₀ᶻ e^(−t²) dt, 7.2.2 gives erfc z = (2/√π)∫ᶻ^∞ e^(−t²) dt = 1 − erf z, and 7.2.7 and 7.2.8 give the Fresnel integrals C(z) = ∫₀ᶻ cos(πt²/2) dt and S(z) = ∫₀ᶻ sin(πt²/2) dt.", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Added for Q17.4, answered by the default. The glossary carried family cards for hypergeometric, polylog, theta, orthogonal polynomials and incomplete gamma — none of which the prose mentions — and none for Bessel, Airy, error or Fresnel, which it does. The four value cards (j01, j1zero, ai0, erf1, fresnelS) each cite an entry for their own value, and no entry carried the defining equations the family cards need. This one does." }, { "id": "r-17.29", "url": "https://www.teclamairme.com/#r-17.29", "chapter": 17, "status": "confirmed", "claim": "The Cauchy principal value, named after Augustin-Louis Cauchy, assigns a value to an improper integral that is otherwise undefined by excluding a symmetric neighbourhood of the singularity and taking the limit as that neighbourhood shrinks; the two divergent halves are made to cancel by being cut away at the same rate. It is also the way the distribution p.v.(1/x) is defined. Approached asymmetrically, the integral can be made to equal any real number; li(x) for x > 1 is a Cauchy principal value.", "sources": [ { "title": "Cauchy principal value, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cauchy_principal_value", "own_work": false, "note": "read as raw HTML: “the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non-singular domain,” with the symmetric-limit formulation and the distribution p.v.(1/x).", "fetched": [ "2026-09-17" ] }, { "title": "Cauchy Principal Value, MathWorld", "url": "https://mathworld.wolfram.com/CauchyPrincipalValue.html", "own_work": false, "note": "the symmetric-exclusion limit written out.", "fetched": [ "2026-09-17" ] }, { "title": "Alan Reynolds, Introduction to Complex Analysis (PHYS30041 course notes): Cauchy principal value", "url": "https://alan-reynolds.quarto.pub/introduction-to-complex-analysis-phys30041/cpv.html", "own_work": false, "note": "“By approaching the singularity asymmetrically in other ways, we can make the value of the integral equal to any real number we choose!”", "fetched": [ "2026-09-22" ] }, { "title": "Logarithmic integral function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Logarithmic_integral_function", "own_work": false, "note": "read as raw wikitext: “The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value”; “the prime number theorem states that: π(x) ∼ li(x).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.30", "url": "https://www.teclamairme.com/#r-17.30", "chapter": 17, "status": "confirmed", "claim": "Cites the definition of the Lambert W function as the multivalued inverse of w↦we^w and its two real branches. W solves equations such as x^x = z and is used in combinatorics and in delay differential equations; Lambert studied the related equation in 1758, Euler a special case in 1783, and Corless, Gonnet, Hare, Jeffrey and Knuth gave the modern treatment in 1996.", "sources": [ { "title": "Lambert W function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lambert_W_function", "own_work": false, "note": "defines W as the inverse of w e^w, its branches W0 and W-1, and the branch point at (-1/e, -1); “It is useful in combinatorics, for instance, in the enumeration of trees” and “also occurs in the solution of delay differential equations, such as y′(t) = a y(t − 1).”", "fetched": [ "2026-09-22" ] }, { "title": "Lambert W functions, B. Ćurgus, Western Washington University", "url": "https://faculty.curgus.wwu.edu/Courses/Math_pages/Math_138/Lambert_W_function.html", "own_work": false, "note": "carries x^x solved by W: “Solve x^x = y for x > 0 and appropriate y > 0.” with solution “x = exp(W0(ln y)) = ln y / W0(ln y)”", "fetched": [ "2026-09-22" ] }, { "title": "Corless, Knuth and Lambert W, Nick Higham", "url": "https://nhigham.com/2016/08/23/corless-knuth-and-lambert-w/", "own_work": false, "note": "carries the 1996 date of the canonical reference: “Robert M. Corless, Gaston N. Gonnet, D. E. G. Hare and David J. Jeffrey and Donald Knuth, On the Lambert W Function, Adv. in Comput. Math. 5, 329-359, 1996”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.31", "url": "https://www.teclamairme.com/#r-17.31", "chapter": 17, "status": "confirmed", "claim": "Cites the beta function’s definition as an integral and its identity B(a, b)=Γ(a)Γ(b)/Γ(a+b). Euler introduced the integral; the function normalizes the beta distribution, appears in the Veneziano amplitude, and has B(½, ½) = π.", "sources": [ { "title": "Beta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Beta_function", "own_work": false, "note": "gives Euler’s integral definition and the identity relating B to the gamma function; “it was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano.”", "fetched": [ "2026-09-22" ] }, { "title": "Introduction to the Gamma Function, P. Sebah and X. Gourdon", "url": "https://scipp-legacy.pbsci.ucsc.edu/~haber/archives/physics116A10/gamma.pdf", "own_work": false, "note": "carries Euler’s integral and B(½, ½): “Euler gave (30) in 1730” and “8.1 Special values B(1/2, 1/2) = π”", "fetched": [ "2026-09-22" ] }, { "title": "Beta distribution, Wikipedia", "url": "https://en.wikipedia.org/wiki/Beta_distribution", "own_work": false, "note": "carries the normalization: “The beta function, B, is a normalization constant to ensure that the total probability is 1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.33", "url": "https://www.teclamairme.com/#r-17.33", "chapter": 17, "status": "confirmed", "claim": "Cites the digamma function’s definition as the logarithmic derivative of the gamma function and its special values. It obeys ψ(x+1) = ψ(x) + 1/x, with ψ(1) = −γ and ψ(n+1) = Hₙ − γ, and has the asymptotic expansion ψ(z) ∼ ln z − 1/(2z) − ….", "sources": [ { "title": "Digamma function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Digamma_function", "own_work": false, "note": "defines ψ = Γ’/Γ, gives ψ(1) = -γ and the harmonic-number recurrence.", "fetched": [ "2026-09-22" ] }, { "title": "§5.11 Asymptotic Expansions, NIST Digital Library of Mathematical Functions", "url": "https://dlmf.nist.gov/5.11", "own_work": false, "note": "carries the two expansions side by side: 5.11.1 “Ln Γ(z) ∼ (z−1/2) ln z − z + 1/2 ln(2π) + Σ B2k/(2k(2k−1)z^(2k−1))” and 5.11.2 “ψ(z) ∼ ln z − 1/(2z) − Σ B2k/(2k z^(2k))”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-17.35", "url": "https://www.teclamairme.com/#r-17.35", "chapter": 17, "status": "confirmed", "claim": "Describes the Cornu spiral as the curve traced by plotting the Fresnel integrals against each other, with curvature proportional to arc length, and its use in diffraction analysis. Euler studied its curvature property in 1744 and Alfred Cornu used it for diffraction; with C and S the integrals of cos(πt²/2) and sin(πt²/2) it winds toward (½, ½) and (−½, −½). Also called a clothoid, it is used for transition curves in railway and highway engineering.", "sources": [ { "title": "Euler spiral, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler_spiral", "own_work": false, "note": "describes the Cornu spiral / Euler spiral, its curvature-proportional-to-arc-length property, and Cornu’s diffraction application; “This curve is also referred to as a clothoid or Cornu spiral”; “They are also widely used in railway and highway engineering to design transition curves between straight and curved sections of railways or roads,” so that “the centripetal acceleration increases smoothly with the traveled distance” instead of causing “lateral jerk.”", "fetched": [ "2026-09-22" ] }, { "title": "The Cornu Spiral, HyperPhysics, Georgia State University", "url": "http://hyperphysics.gsu.edu/hbase/phyopt/cornu.html", "own_work": false, "note": "carries the two limit points: “The eyes of the spirals at (.5,.5) and (-.5, -.5) represent image plane points at infinity.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.1", "url": "https://www.teclamairme.com/#r-18.1", "chapter": 18, "status": "confirmed", "claim": "For Re(s)>1, the Riemann zeta function is the series sum of n^(-s); elsewhere it is defined by analytic continuation, is meromorphic, and has its only singularity at s=1, a simple pole with residue 1. An analytic continuation is unique: two analytic functions equal on a set with a limit point are equal throughout.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.2 Definition and Expansions", "url": "https://dlmf.nist.gov/25.2", "own_work": false, "note": "DLMF states the defining series for Re(s)>1 and says the continuation is meromorphic with only a simple pole at s=1 and residue 1.", "fetched": [ "2026-09-09" ] }, { "title": "NIST DLMF §1.10, Functions of a complex variable", "url": "https://dlmf.nist.gov/1.10", "own_work": false, "note": "“If f₂(z), analytic in D₂, equals f₁(z) on an arc in D = D₁ ∩ D₂, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f₂(z) is called an analytic continuation of f₁(z).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.2", "url": "https://www.teclamairme.com/#r-18.2", "chapter": 18, "status": "confirmed", "claim": "The special values are ζ(0)=-1/2, ζ(2)=π²/6, ζ(4)=π⁴/90, and ζ(6)=π⁶/945; more generally the positive even values are rational multiples of powers of π involving Bernoulli numbers. ζ(0) = −1/2 follows from the functional equation and the residue at s = 1. Euler found the values ζ(2n) in E41 (written 1735, published 1740) and gave the Bernoulli-number form in his 1755 book. For odd positive integers no such simple expression is known.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.6 Integer Arguments", "url": "https://dlmf.nist.gov/25.6", "own_work": false, "note": "equations 25.6.1 and 25.6.2 list these values and the general even-value formula.", "fetched": [ "2026-09-09" ] }, { "title": "MIT 18.785 Number Theory I (Fall 2015), Lecture 16", "url": "https://math.mit.edu/classes/18.785/2015fa/LectureNotes16.pdf", "own_work": false, "note": "read from the PDF: “We can determine the value of ζ(0) via the functional equation. We know that ζ(s) has a pole of residue 1 at s = 1 … thus ζ(0) = −1/2.”", "fetched": [ "2026-09-22" ] }, { "title": "Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856", "url": "https://arxiv.org/abs/1303.1856", "own_work": false, "note": "read from the PDF: “In his 1755 book on differential calculus Euler [E212, Part II, Chap. 6, Arts. 141–153] gave closed forms for ζ(2n) explicitly expressed in terms of the Bernoulli numbers.”", "fetched": [ "2026-09-22" ] }, { "title": "Euler Archive, E41, De summis serierum reciprocarum", "url": "https://scholarlycommons.pacific.edu/euler-works/41/", "own_work": false, "note": "“Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function”; Written Date 1735, Published Date 1740.", "fetched": [ "2026-09-22" ] }, { "title": "Bernoulli number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bernoulli_number", "own_work": false, "note": "its table gives B₆ = 1/42.", "fetched": [ "2026-09-22" ] }, { "title": "Riemann zeta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_zeta_function", "own_work": false, "note": "read as raw wikitext: “For odd positive integers, no such simple expression is known.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.3", "url": "https://www.teclamairme.com/#r-18.3", "chapter": 18, "status": "judgement", "claim": "Euler solved the Basel sum as π²/6 and later identified the zeta Euler product in 1737; the problem is named for Basel, the home city of Euler and of the Bernoullis, who had failed at it, and solving it made Euler’s reputation; the historical date of the Basel result is convention-sensitive because the work was written/presented in 1734–1735 and published in 1740.", "sources": [ { "title": "Euler Archive / Pacific University, De summis serierum reciprocarum", "url": "https://scholarlycommons.pacific.edu/euler-works/41/", "own_work": false, "note": "the source identifies Euler’s paper and gives its dates: Written Date 1735, Published Date 1740.", "fetched": [ "2026-09-09" ] }, { "title": "Nature, The Zeta-Function of Riemann", "url": "https://www.nature.com/articles/126600c0", "own_work": false, "note": "the historical note dates Euler’s prime-product observation to 1737.", "fetched": [ "2026-09-09" ] }, { "title": "Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 (Bull. AMS 50 (2013) 527–628)", "url": "https://arxiv.org/abs/1303.1856", "own_work": false, "note": "the same survey, free on arXiv: Euler solved the Basel problem by 1735, and his three proofs that ζ(2) = π²/6 (E41, read 5 December 1735) were published in 1740.", "fetched": [ "2026-09-22" ] }, { "title": "Wikipedia, Basel problem", "url": "https://en.wikipedia.org/wiki/Basel_problem", "own_work": false, "note": "the problem was posed by Pietro Mengoli in 1650; it is named for Basel as the home town of Euler and of the Bernoulli family, who had attempted it without success; and Euler’s 1734 solution brought him immediate fame at twenty-eight.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Sources distinguish discovery, presentation, and publication dates. We keep the snapshot’s “found by Euler in 1734” as a discovery date, but the review should not silently imply that the paper was published in 1734." }, { "id": "r-18.4", "url": "https://www.teclamairme.com/#r-18.4", "chapter": 18, "status": "confirmed", "claim": "Euler’s product is ζ(s)=∏p(1-p^(-s))^(-1) for Re(s)>1, because expanding the geometric factors and using unique factorization lists each positive integer once. The proof uses only the geometric series and the fundamental theorem of arithmetic; at s = 1 the divergence of the harmonic series gives infinitely many primes.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.2", "url": "https://dlmf.nist.gov/25.2", "own_work": false, "note": "the zeta-function section includes the Euler product representation and its domain.", "fetched": [ "2026-09-09" ] }, { "title": "Riemann zeta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_zeta_function", "own_work": false, "note": "read as raw wikitext: “The proof of Euler’s identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler’s formula, which becomes ∏_p p/(p−1), implies that there are infinitely many primes.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.5", "url": "https://www.teclamairme.com/#r-18.5", "chapter": 18, "status": "confirmed", "claim": "The functional equation relates ζ(s) to ζ(1-s) through the sine and gamma factors; its sine factor gives the trivial zeros ζ(-2n)=0 for n≥1, and the completed xi function is symmetric under s↔1-s. ξ is entire and its zeros are exactly the nontrivial zeros of ζ; Riemann established the functional equation in 1859.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.4 Reflection Formulas", "url": "https://dlmf.nist.gov/25.4", "own_work": false, "note": "equations 25.4.2–25.4.4 give the reflection formula and xi-function definition/symmetry.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions, §25.10 Zeros", "url": "https://dlmf.nist.gov/25.10", "own_work": false, "note": "DLMF explicitly states that the functional equation implies ζ(-2n)=0 and calls these the trivial zeros.", "fetched": [ "2026-09-09" ] }, { "title": "Riemann zeta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_zeta_function", "own_work": false, "note": "read as raw wikitext: “The xi function is an entire function whose zeros are exactly the non-trivial zeros of the Riemann zeta function”; “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude.””", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.6", "url": "https://www.teclamairme.com/#r-18.6", "chapter": 18, "status": "confirmed", "claim": "At s=1 the harmonic series diverges and ζ has a simple pole with residue 1, its only singularity; consequently the reciprocal function 1/ζ(s), after analytic continuation as a reciprocal, has a removable singularity and takes the value 0 at s=1. The divergence of the harmonic series was first proved by Nicole Oresme in 1350.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.2", "url": "https://dlmf.nist.gov/25.2", "own_work": false, "note": "DLMF states the simple pole and residue 1: “It is a meromorphic function whose only singularity in ℂ is a simple pole at s = 1, with residue 1.”", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions, §5.2 Gamma Function", "url": "https://dlmf.nist.gov/5.2", "own_work": false, "note": "the analogous reciprocal-at-a-pole convention is stated for 1/Γ; the value here follows directly from the Laurent expansion of ζ at its simple pole.", "fetched": [ "2026-09-09" ] }, { "title": "Harmonic series (mathematics), Wikipedia", "url": "https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)", "own_work": false, "note": "read as raw wikitext: “The divergence of the harmonic series was first proven in 1350 by Nicole Oresme.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Part 2 item on 1/ζ(1): we retain the existing wording. This is the value of the analytically continued reciprocal function at the removable singularity, not a claim that the divergent series has a finite value." }, { "id": "r-18.7", "url": "https://www.teclamairme.com/#r-18.7", "chapter": 18, "status": "confirmed", "claim": "The first nontrivial zero has height about 14.134725, the second about 21.022040, and the third about 25.010858; Gram published the first list in 1903. Gram’s 1903 computation showed the first 10 zeros, those of height at most 50, on the critical line. Montgomery’s 1973 pair-correlation conjecture uses the normalization 2π/log T, the average spacing near height T; Dyson pointed out that it matches random Hermitian matrices.", "sources": [ { "title": "A. M. Odlyzko, Tables of zeros of the Riemann zeta function", "url": "https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html", "own_work": false, "note": "(file zeros1, first three lines): the first three nontrivial heights are 14.134725142, 21.022039639 and 25.010857580.", "fetched": [ "2026-09-10" ] }, { "title": "Riemann hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_hypothesis", "own_work": false, "note": "read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021); its table of zero computations opens with 1859?, 3 zeros: “B. Riemann used the Riemann–Siegel formula (unpublished, but reported in Siegel 1932).”", "fetched": [ "2026-09-22" ] }, { "title": "Montgomery’s pair correlation conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture", "own_work": false, "note": "read as raw wikitext: a conjecture “made by Hugh Montgomery (1973)”; “(The factor 2π/log(T) is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about T.)”; “which, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices.”", "fetched": [ "2026-09-22" ] }, { "title": "C. L. Siegel, On Riemann’s Nachlass for Analytic Number Theory (1932), tr. Barkan and Sklar, arXiv:1810.05198", "url": "https://arxiv.org/pdf/1810.05198", "own_work": false, "note": "“For the smallest positive zero he finds the value of a1 = 14.1386; Gram has calculated the value 14.1347, different by less than 3 parts per thousand.”", "fetched": [ "2026-10-01" ] } ], "editorial_note": "" }, { "id": "r-18.8", "url": "https://www.teclamairme.com/#r-18.8", "chapter": 18, "status": "confirmed", "claim": "Rigorous computation by Platt and Trudgian verified the Riemann hypothesis up to height 3,000,175,332,800, which is the lowest 12,363,153,437,138 nontrivial zeros. This claim line used to open with “Hardy proved in 1914 that infinitely many zeta zeros lie on the critical line,” which neither source below carries: DLMF §25.10 says only that ζ(½ + it) has infinitely many zeros with t real, naming nobody, and the full text of Platt and Trudgian never mentions Hardy. 18.22 was written to supply the attribution and the year, and the claim line was narrowed to match on 17 September 2026. Their paper is Bulletin of the London Mathematical Society 53 (2021), pp. 792–797, published online 22 January 2021, doi:10.1112/blms.12460, from the preprint arXiv:2004.09765 of 21 April 2020. Gram checked the first 10 zeros in 1903; Gourdon reached 10¹³ in 2004; Platt and Trudgian’s verification (preprint 2020) appeared in 2021.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.10 Zeros", "url": "https://dlmf.nist.gov/25.10", "own_work": false, "note": "DLMF states that there are infinitely many zeros on the critical line (because Z(t) changes sign infinitely often, ζ(½ + it) has infinitely many zeros with t real) and describes their computation via the function Z(t) and the Riemann–Siegel formula; it does not name Hardy.", "fetched": [ "2026-09-09" ] }, { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765)", "url": "https://arxiv.org/abs/2004.09765", "own_work": false, "note": "Theorem 1 reads “The Riemann hypothesis is true up to height 3 000 175 332 800. That is, the lowest 12 363 153 437 138 non-trivial zeroes ρ have ℜρ = 1/2.”", "fetched": [ "2026-09-09" ] }, { "title": "Riemann hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_hypothesis", "own_work": false, "note": "read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.9", "url": "https://www.teclamairme.com/#r-18.9", "chapter": 18, "status": "confirmed", "claim": "Apéry proved ζ(3) irrational in 1978; whether ζ(3) is transcendental remains open.", "sources": [ { "title": "Waldschmidt, Zeta Values", "url": "https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ZetaValuesRIMS2009.pdf", "own_work": false, "note": "Theorem 4 identifies Apéry’s 1978 irrationality theorem and gives ζ(3)’s value.", "fetched": [ "2026-09-09" ] }, { "title": "OEIS A002117", "url": "https://oeis.org/A002117", "own_work": false, "note": "the entry records Apéry’s 1978 irrationality result and that the rational-multiple-of-π³ question is unresolved.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.10", "url": "https://www.teclamairme.com/#r-18.10", "chapter": 18, "status": "unconfirmed", "claim": "The value ζ(1/2) is approximately -1.4603545088; no source cited here settles whether it is irrational.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.10", "url": "https://dlmf.nist.gov/25.10", "own_work": false, "note": "DLMF confirms the critical-line framework but does not establish an arithmetic classification for ζ(1/2).", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Riemann Zeta Function", "url": "https://mathworld.wolfram.com/RiemannZetaFunction.html", "own_work": false, "note": "gives ζ(1/2) = −1.46035450880…, a negative number.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The numerical value and its sign are confirmed; what remains unconfirmed is the arithmetic status. No source settles whether ζ(1/2) is irrational, so the status-open wording is retained. The formula carries |ζ(½)|, and the tip, the chapter key and the card now say so explicitly, because ζ(1/2) itself is negative." }, { "id": "r-18.11", "url": "https://www.teclamairme.com/#r-18.11", "chapter": 18, "status": "confirmed", "claim": "The prime-counting function satisfies π(10)=4, π(100)=25, and π(10^6)=78498; the notation π(x) here counts primes, not the circle constant.", "sources": [ { "title": "Prime-counting function, Encyclopaedia/standard reference summary", "url": "https://en.wikipedia.org/wiki/Prime-counting_function", "own_work": false, "note": "the definition and the listed values are given explicitly; its table runs to π(10²⁹), and “The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively.” (exponents as typeset on the page).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.12", "url": "https://www.teclamairme.com/#r-18.12", "chapter": 18, "status": "confirmed", "claim": "The prime number theorem π(x)~x/log x was conjectured by Gauss and Legendre and proved independently by Hadamard and de la Vallée Poussin in 1896 using the zero-free line Re(s)=1. Wiener showed that the prime number theorem is equivalent to ζ having no zeros on Re(s) = 1.", "sources": [ { "title": "MacTutor, Jacques Hadamard biography", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Hadamard/", "own_work": false, "note": "MacTutor dates the proof to 1896 and credits Hadamard and de la Vallée Poussin using complex analysis.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions, §25.10", "url": "https://dlmf.nist.gov/25.10", "own_work": false, "note": "DLMF connects the absence of zeros on Re(s)=1 with the prime number theorem and cites both 1896 proofs.", "fetched": [ "2026-09-09" ] }, { "title": "Riemann Hypothesis, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/RiemannHypothesis.html", "own_work": false, "note": "“Wiener showed that the prime number theorem is literally equivalent to the assertion that the Riemann zeta function ζ(s) has no zeros on σ = 1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.13", "url": "https://www.teclamairme.com/#r-18.13", "chapter": 18, "status": "confirmed", "claim": "For the Mertens function M(x)=Σn≤x μ(n), the Riemann hypothesis is equivalent to M(x)=O(x^(1/2+ε)) for every ε>0, while the stronger Mertens conjecture |M(x)|<√x was disproved by Odlyzko and te Riele in 1985.", "sources": [ { "title": "Wolfram MathWorld, Mertens Conjecture", "url": "https://mathworld.wolfram.com/MertensConjecture.html", "own_work": false, "note": "the page states the RH equivalence and records the disproof of the stronger conjecture: “The Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their proof is indirect and does not produce a specific counterexample.”", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Mertens conjecture", "url": "https://en.wikipedia.org/wiki/Mertens_conjecture", "own_work": false, "note": "the history and Odlyzko–te Riele 1985 disproof are stated.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.14", "url": "https://www.teclamairme.com/#r-18.14", "chapter": 18, "status": "confirmed", "claim": "Dirichlet introduced L-functions in 1837 to prove that every arithmetic progression with coprime first term and difference contains infinitely many primes; the nonvanishing L(1, χ) step is central.", "sources": [ { "title": "Encyclopedia of Mathematics, Dirichlet L-function", "url": "https://encyclopediaofmath.org/wiki/Dirichlet-L-function", "own_work": false, "note": "the entry dates Dirichlet’s introduction of L-functions to 1837 and states their role in primes in arithmetic progressions.", "fetched": [ "2026-09-09" ] }, { "title": "MIT 18.785 Lecture Notes, Dirichlet L-functions", "url": "https://math.mit.edu/classes/18.785/2016fa/LectureNotes18.pdf", "own_work": false, "note": "the notes state Dirichlet’s 1837 theorem and the coprime progression condition.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.15", "url": "https://www.teclamairme.com/#r-18.15", "chapter": 18, "status": "confirmed", "claim": "Gram points are defined by θ(t)=nπ for the Riemann–Siegel theta function; g₀, the first above the lowest zero, is approximately 17.8456, and Gram’s law is only a rule of thumb rather than a theorem: in the long run it fails for about a quarter of all Gram intervals.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.10 Zeros", "url": "https://dlmf.nist.gov/25.10", "own_work": false, "note": "DLMF defines the real Hardy Z-function and the Riemann–Siegel theta function used to locate critical-line zeros.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Riemann–Siegel theta function", "url": "https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function", "own_work": false, "note": "the page gives the Gram-point convention and the first point near 17.8456; “Today we know that, in the long run, Gram’s law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta function.”", "fetched": [ "2026-09-09" ] }, { "title": "Gram Point, Wolfram MathWorld", "url": "https://mathworld.wolfram.com/GramPoint.html", "own_work": false, "note": "a Gram point gₙ satisfies θ(gₙ) = nπ, numbered from n = 0; its table gives g₀ = 17.8455995404.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.16", "url": "https://www.teclamairme.com/#r-18.16", "chapter": 18, "status": "confirmed", "claim": "At s=2, ζ(2) times the reciprocal Euler product is 1, and the p=2 Euler factor is (1-2^-2)^-1=4/3; multiplying all prime factors reconstructs π²/6. The full product equals ζ(2) = π²/6.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §25.2 and §25.6", "url": "https://dlmf.nist.gov/25.2", "own_work": false, "note": "the Euler product and ζ(2)=π²/6 are stated in the DLMF sections.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "At p=2, (1-2^-2)^-1=(1-1/4)^-1=4/3; multiplying ζ(2) by ∏p(1-p^-2) cancels the product exactly.", "fetched": [] }, { "title": "NIST DLMF §25.6, Zeta function: integer arguments", "url": "https://dlmf.nist.gov/25.6", "own_work": false, "note": "25.6.1: ζ(2) = π²/6.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-18.17", "url": "https://www.teclamairme.com/#r-18.17", "chapter": 18, "status": "confirmed", "claim": "Riemann’s memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse appeared in the Monatsberichte der Berliner Akademie in November 1859; in it he argues that ξ(t) can vanish only for the imaginary part of t between ½i and −½i, that is for Re(s) between 0 and 1 inclusive, and records the hypothesis as “very probable” while setting the proof aside. Its length is given as six manuscript pages or as nine printed ones.", "sources": [ { "title": "Clay Mathematics Institute, Riemann’s 1859 Manuscript", "url": "https://www.claymath.org/collections/riemanns-1859-manuscript/", "own_work": false, "note": "“Just six manuscript pages in length,” first published in the Monatsberichte der Berliner Akademie, November 1859.", "fetched": [ "2026-09-09" ] }, { "title": "On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins (Clay Mathematics Institute)", "url": "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf", "own_work": false, "note": "carries the epigraph verbatim — “it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” — and the argument confining the roots to the closed strip.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, On the Number of Primes Less Than a Given Magnitude", "url": "https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude", "own_work": false, "note": "calls it “a seminal 9-page paper” published in the November 1859 Monatsberichte.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The two live sources count different objects: Clay counts Riemann’s handwritten manuscript at six pages, Wikipedia the printed article at nine. Neither supports the “eight pages” the text used to give, so the text now says only that the paper runs to fewer than ten pages, which both sources support." }, { "id": "r-18.18", "url": "https://www.teclamairme.com/#r-18.18", "chapter": 18, "status": "confirmed", "claim": "The regularized value ζ(−1) = −1/12 is the one used to compute the Casimir force for a scalar field in one dimension; the three-dimensional parallel-plate calculation regularizes with ζ(−3) = 1/120 and gives an energy per unit area of −ħcπ²/720a³.", "sources": [ { "title": "Wikipedia, 1 + 2 + 3 + 4 + ⋯", "url": "https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF", "own_work": false, "note": "“The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension,” and the same value is what fixes the twenty-six dimensions of bosonic string theory.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Casimir effect", "url": "https://en.wikipedia.org/wiki/Casimir_effect", "own_work": false, "note": "the parallel-plate derivation runs through “ζ(−3) = 1/120” and yields ⟨E⟩/A = −ħcπ²/720a³.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The page had attributed the two-metal-plate calculation to ζ(−1) = −1/12. That is the one-dimensional scalar case; the plates need ζ(−3). Both are now named." }, { "id": "r-18.19", "url": "https://www.teclamairme.com/#r-18.19", "chapter": 18, "status": "confirmed", "claim": "The Riemann Hypothesis is one of the seven Clay Millennium Prize Problems, each carrying a prize of $1,000,000, and a proof would sharpen the known estimates for how the primes are distributed.", "sources": [ { "title": "Clay Mathematics Institute, Millennium Problems", "url": "https://www.claymath.org/millennium-problems/", "own_work": false, "note": "“the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem,” and the Riemann Hypothesis is one of the seven listed.", "fetched": [ "2026-09-09" ] }, { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.1", "url": "https://arxiv.org/abs/2004.09765", "own_work": false, "note": "“Of course, on the Riemann hypothesis much more is known: Schoenfeld proved that |ψ(x) − x| ≤ (1/8π)√x log²x, (x > 59),” a sharper error term than the unconditional bounds the same section collects.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.20", "url": "https://www.teclamairme.com/#r-18.20", "chapter": 18, "status": "confirmed", "claim": "The de Bruijn–Newman constant Λ is a real number for which the Riemann hypothesis is equivalent to Λ ≤ 0; Rodgers and Tao proved Λ ≥ 0, so the hypothesis is true if and only if Λ = 0.", "sources": [ { "title": "Rodgers and Tao, The De Bruijn–Newman constant is non-negative (Forum of Mathematics Pi 8, 2020)", "url": "https://arxiv.org/abs/1801.05914", "own_work": false, "note": "the abstract defines Λ by the heat-flow deformation, states that the Riemann hypothesis is equivalent to Λ ≤ 0, and proves Newman’s conjecture Λ ≥ 0.", "fetched": [ "2026-09-09" ] }, { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.4", "url": "https://arxiv.org/abs/2004.09765", "own_work": false, "note": "“The Riemann hypothesis is equivalent to the statement that Λ ≤ 0. Rodgers and Tao proved that Λ ≥ 0.,” with Corollary 2 giving Λ ≤ 0.2.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-18.21", "url": "https://www.teclamairme.com/#r-18.21", "chapter": 18, "status": "confirmed", "claim": "Analytic continuation is unique: if f1 is analytic in a domain D1 and f2 is analytic in D2, and the two agree on an arc in D = D1 ∩ D2, or on just an infinite number of points with a limit point in D, then they agree throughout D, and f2 is called an analytic continuation of f1.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §1.10(ii) Analytic Continuation", "url": "https://dlmf.nist.gov/1.10", "own_work": false, "note": "states the uniqueness in so many words — “Let f1(z) be analytic in a domain D1. If f2(z), analytic in D2, equals f1(z) on an arc in D = D1 ∩ D2, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f2(z) is called an analytic continuation of f1(z).”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass for the new card `analcont`. Chapter ᚑ(O)’s sentence “That is analytic continuation: a well-behaved function of a complex variable is so rigid that knowing it on a small patch fixes it everywhere” carried no footnote of its own, and 18.1 (DLMF §25.2) supplies only the fact that ζ is continued, not the uniqueness theorem that makes the continuation the function rather than a function. This entry carries the theorem." }, { "id": "r-18.22", "url": "https://www.teclamairme.com/#r-18.22", "chapter": 18, "status": "confirmed", "claim": "Hardy proved in 1914 that infinitely many nontrivial zeros of the Riemann zeta function lie on the critical line; whether all of them do is still unknown.", "sources": [ { "title": "Wolfram MathWorld, Riemann Hypothesis", "url": "https://mathworld.wolfram.com/RiemannHypothesis.html", "own_work": false, "note": "states “In 1914, Hardy proved that an infinite number of values for t can be found for which ζ(1/2 + it) = 0 and t is real (Havil 2003, p. 213),” and immediately adds “However, it is not known if all nontrivial roots satisfy Re(s) = 1/2.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass. The chapter’s sentence “Hardy proved in 1914 that infinitely many zeros are on the line” was footnoted 18.8 alone, and the V12 sweep had already recorded that neither of 18.8’s sources carries the attribution or the year — DLMF §25.10 says only that ζ(½ + it) has infinitely many zeros with t real, naming nobody, and the full text of Platt and Trudgian never mentions Hardy. This entry supplies both. 18.8 continues to carry the verification height, which is what it does support." }, { "id": "r-18.23", "url": "https://www.teclamairme.com/#r-18.23", "chapter": 18, "status": "confirmed", "claim": "Euler’s product expansion first appeared in Variae observationes circa series infinitas (Eneström E72), written in 1737 and published in 1744 in Commentarii academiae scientiarum Petropolitanae 9, 160–188.", "sources": [ { "title": "Euler Archive / University of the Pacific, E72 Variae observationes circa series infinitas", "url": "https://scholarlycommons.pacific.edu/euler-works/72/", "own_work": false, "note": "the record gives “Written Date 1737,” “Published Date 1744,” the citation “Commentarii academiae scientiarum Petropolitanae, Volume 9, pp. 160-188,” and the content summary “This is the paper in which Euler product expansions first appeared.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass to carry the 1737 date the chapter uses three times. The V12 sweep recorded that the date rested on two sources a reader cannot open: originalsources.com returns a sign-in page under HTTP 200, and the 1930 Nature note gives no detail. The Euler Archive record is open, names the paper, and dates it directly. This is the companion of 18.3’s E41 record for the 1734 Basel result." }, { "id": "r-18.24", "url": "https://www.teclamairme.com/#r-18.24", "chapter": 18, "status": "confirmed", "claim": "Ramanujan wrote the identity 1 + 2 + 3 + 4 + ... = -1/12 in his second letter to G. H. Hardy, of 27 February 1913.", "sources": [ { "title": "1 + 2 + 3 + 4 + ⋯, Wikipedia", "url": "https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF", "own_work": false, "note": "read as raw HTML, § Ramanujan summation: “Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913: ‘Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich’s Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = −1/12 under my theory.” So the date, the addressee and the identity are all on that page.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 for the B6 pass. The #zetam1 card said “Ramanujan states it in his letter to Hardy of 27 February 1913” and footnoted 18.18, which is the Casimir entry: its claim line is about the regularized value used for the Casimir force and neither of its two sources mentions Ramanujan, Hardy or 1913. A reader following that marker landed on a physics calculation. Grepping references.html for “Ramanujan” before this entry existed returned r-5.9, r-6.11, r-6.12, r-6.31 and r-17.5, none of them about the letter." }, { "id": "r-18.25", "url": "https://www.teclamairme.com/#r-18.25", "chapter": 18, "status": "confirmed", "claim": "The chapter epigraph is Riemann’s, from Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse of 1859, in David Wilkins’s English translation; it is the sentence in which the Riemann hypothesis is first stated and immediately set aside.", "sources": [ { "title": "Bernhard Riemann, On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins, at the Clay Mathematics Institute", "url": "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf", "own_work": false, "note": "fetched, read as extracted PDF text. The epigraph appears word for word: “One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” The Clay page hosting it credits the transcription and translation to David Wilkins by permission and the manuscript images to the Niedersächsische Staats- und Universitätsbibliothek Göttingen.", "fetched": [ "2026-09-17", "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is the strongest of the fourteen: the quotation was checked against the translation the footer names, on the site that publishes it, and matches exactly, including the footer’s gloss that the roots are those of Riemann’s transformed function." }, { "id": "r-18.26", "url": "https://www.teclamairme.com/#r-18.26", "chapter": 18, "status": "confirmed", "claim": "The fundamental theorem of arithmetic says every integer greater than 1 is a product of primes in exactly one way up to order; it can be derived from Euclid’s Elements, Book VII propositions 30, 31 and 32 and Book IX proposition 14. It is the reason 1 is not counted as a prime, and it fails for general rings of algebraic integers, which is why unique factorization domains are singled out by name.", "sources": [ { "title": "Fundamental theorem of arithmetic, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic", "own_work": false, "note": "read as raw HTML: “The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid’s Elements”; “1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique”; and “the theorem does not hold for algebraic integers,” which is what unique factorization domains name.", "fetched": [ "2026-09-17" ] }, { "title": "Fundamental Theorem of Arithmetic, MathWorld", "url": "https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html", "own_work": false, "note": "the statement: “every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes.”", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-18.27", "url": "https://www.teclamairme.com/#r-18.27", "chapter": 18, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Euler had guessed the real-variable form of the functional equation in 1749; Riemann proved the functional equation in his 1859 paper; Gram observed in 1903 that zeros tend to fall between consecutive Gram points (Gram’s law); the Mertens conjecture was made by Stieltjes in a letter to Hermite of 1885 and in print by Mertens in 1897; Mertens conjectured |M(x)| < sqrt(x) in 1897; Kotnik and te Riele improved the upper bound for a counterexample to the Mertens conjecture in 2006, to about e^(1.59×10^40), and a later preprint lowers it to about e^(1.017×10^29), and Kim and Nguyen lowered it in 2024 to about e^(1.96×10^19); Gauss conjectured the prime number theorem around 1792; Legendre published a version of the conjecture in 1798; Chebyshev proved in 1850 that pi(x) is within constant factors of x/ln x.", "sources": [ { "title": "E352 Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques, Euler Archive (University of the Pacific)", "url": "https://scholarlycommons.pacific.edu/euler-works/352/", "own_work": false, "note": "read raw: “Euler evaluates several values of the now-named Riemann zeta function, and finds some functional relations for it. ... Published Date 1768 ... Written Date 1749.”", "fetched": [ "2026-09-18" ] }, { "title": "Riemann zeta function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_zeta_function", "own_work": false, "note": "read raw: “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude” and used to construct the analytic continuation in the first place..”", "fetched": [ "2026-09-18" ] }, { "title": "Riemann hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_hypothesis", "own_work": false, "note": "read raw: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2”; “Trudgian (2011) showed that both Gram’s law and Rosser’s rule fail in a positive proportion of cases.”; Turing’s check “that Z has the correct sign at several consecutive Gram points” “is still the usual method used.”", "fetched": [ "2026-09-18" ] }, { "title": "Mertens conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mertens_conjecture", "own_work": false, "note": "read raw: “It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Mertens (1897)”; “In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of n for which m(n) › 1.2184, but without giving any specific value for such an n..”", "fetched": [ "2026-09-18" ] }, { "title": "Prime number theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Prime_number_theorem", "own_work": false, "note": "read raw: “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.”; “Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a / (A log a + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture”; “In two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-18.28", "url": "https://www.teclamairme.com/#r-18.28", "chapter": 18, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Erdős and Selberg each published an elementary proof of the prime number theorem in 1949; Riemann’s 1859 explicit formula writes pi(x) as a sum over the zeros of zeta; de Bruijn’s 1950 deformation of the xi-related function H; Newman defined the constant Lambda in 1976; Riemann’s only paper on number theory, 1859, made s complex and continued zeta to the plane; Euler worked formally with divergent sums in the 1740s. Riemann’s memoir shows ξ(t) can vanish only for Im t between ½i and −½i, that is 0 ≤ Re(s) ≤ 1. The explicit formula’s dominant term, li(x), comes from the pole at s = 1, the nontrivial zeros control the oscillations of the primes around it, and von Koch proved that the Riemann hypothesis implies the best possible bound for the error of the prime number theorem.", "sources": [ { "title": "Prime number theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Prime_number_theorem", "own_work": false, "note": "read raw: “Several different proofs of it were found, including the “elementary” proofs of Atle Selberg (1949) and Paul Erdős (1949).”; “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.”", "fetched": [ "2026-09-18" ] }, { "title": "Riemann hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_hypothesis", "own_work": false, "note": "read raw: “their occurrence in his explicit formula for the number of primes π(x) less than or equal to a given number x, which he published in his 1859 paper “On the Number of Primes Less Than a Given Magnitude..””; “the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros”; “This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their “expected” positions.”; “Helge von Koch proved that the Riemann hypothesis implies the “best possible” bound for the error of the prime number theorem.” (read 29 September 2026 from the cached copy of 24 September).", "fetched": [ "2026-09-18" ] }, { "title": "De Bruijn–Newman constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant", "own_work": false, "note": "read raw: “De Bruijn showed in 1950 that H has only real zeros if λ≥1/2, and moreover, that if H has only real zeros for some λ, H also has only real zeros if λ is replaced by any larger value.”; “Newman proved in 1976 the existence of a constant Λ for which the “if and only if” claim holds; and this then implies that Λ is unique.”; “Brad Rodgers and Terence Tao proved that Λ ≥ 0, so the Riemann hypothesis is equivalent to Λ = 0..”", "fetched": [ "2026-09-18" ] }, { "title": "On the Number of Primes Less Than a Given Magnitude, Wikipedia", "url": "https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude", "own_work": false, "note": "read raw: “is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin..”", "fetched": [ "2026-09-18" ] }, { "title": "1 − 2 + 3 − 4 + ⋯, Wikipedia", "url": "https://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF", "own_work": false, "note": "read raw: “In a 1749 report, Leonhard Euler acknowledges that the series diverges but prepares to sum it anyway.”", "fetched": [ "2026-09-18" ] }, { "title": "Riemann, On the Number of Prime Numbers less than a Given Quantity, tr. D. R. Wilkins (Clay Mathematics Institute)", "url": "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf", "own_work": false, "note": "read from the PDF: “it follows that the function ξ(t) can only vanish if the imaginary part of t lies between ½i and −½i.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-19.1", "url": "https://www.teclamairme.com/#r-19.1", "chapter": 19, "status": "confirmed", "claim": "A nonsingular projective Weierstrass cubic has the unique point at infinity (0 : 1 : 0), and this point is the identity for the elliptic-curve group law; its homogeneous X-coordinate is 0.", "sources": [ { "title": "RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms", "url": "https://www.rfc-editor.org/rfc/rfc6090", "own_work": false, "note": "Section 3 names the point at infinity as the identity (“a special point (@,@) that represents the identity element (which is called the “point at infinity”)”) and says “The point at infinity is its own inverse”; Appendix F.2 says it is “regarded as equivalent to the homogenous coordinates (0, 1, 0).”", "fetched": [ "2026-09-09" ] }, { "title": "Elliptic curve, projective group law", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "the projective equation has the unique point O = [0:1:0] at infinity, which is the identity of the group.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.2", "url": "https://www.teclamairme.com/#r-19.2", "chapter": 19, "status": "confirmed", "claim": "Elliptic curves in short Weierstrass form have the chord-and-tangent group law: three collinear intersection points sum to the identity, and the rational points form an abelian group.", "sources": [ { "title": "Ben Lynn, Elliptic Curves: Group of Points (Stanford crypto notes)", "url": "https://crypto.stanford.edu/pbc/notes/elliptic/group.html", "own_work": false, "note": "describes the projective point at infinity, the line-through-two-points construction, and the abelian group of points: in (projective) Weierstrass form “an elliptic curve always contains exactly one point of infinity, (0, 1, 0) (‘the point at the ends of all lines parallel to the y-axis’)”; “every line intersects the curve at exactly three points (counting multiplicities)”; and “It can be verified that P + Q = Q + P, P + O = P, (−P) + P = O,” with O taken to be the point at infinity (0, 1, 0).", "fetched": [ "2026-09-09" ] }, { "title": "RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms", "url": "https://www.rfc-editor.org/rfc/rfc6090", "own_work": false, "note": "gives the affine addition and inverse formulas and the exceptional point-at-infinity cases.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.3", "url": "https://www.teclamairme.com/#r-19.3", "chapter": 19, "status": "confirmed", "claim": "The Mordell–Weil theorem says E(Q) is a finitely generated abelian group, decomposing into a finite torsion subgroup and a free part of rank r.", "sources": [ { "title": "Elliptic curve, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Elliptic_curve", "own_work": false, "note": "(§ Elliptic curves over non-closed fields): “If k is finitely generated over its prime subfield, then X(k) is a finitely-generated group (the Mordell–Weil theorem),” and “The group X(ℚ) is isomorphic to F ⊕ X(ℚ)ₜ, where X(ℚ)ₜ is a finite Abelian group and F is a free Abelian group of a certain finite rank r.”", "fetched": [ "2026-09-10" ] }, { "title": "Elliptic curve rank", "url": "https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve", "own_work": false, "note": "writes E(Q) as Z^r plus its finite torsion group and defines r as the rank.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.4", "url": "https://www.teclamairme.com/#r-19.4", "chapter": 19, "status": "confirmed", "claim": "For E: y² = x³ − x, the rational rank is 0 and E(Q) has torsion Z/2 × Z/2, namely the identity and the three points (0, 0), (1, 0), and (−1, 0); this is equivalent to 1 not being a congruent number.", "sources": [ { "title": "Poonen, Rational points on varieties, section 8.3", "url": "https://math.mit.edu/~poonen/papers/Qpoints.pdf", "own_work": false, "note": "identifies 32A2 as y² = x³ − x, states rank 0 and torsion Z/2 × Z/2, and concludes that there are four rational points.", "fetched": [ "2026-09-09" ] }, { "title": "Keith Conrad, The Congruent Number Problem (PDF)", "url": "https://kconrad.math.uconn.edu/articles/congruentnumber.pdf", "own_work": false, "note": "shows that n is not congruent exactly when the only rational points of y² = x³ − n²x have y = 0, namely (0, 0), (n, 0) and (−n, 0); so for n = 1 the curve y² = x³ − x has only those three affine points (with the point at infinity, four).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.5", "url": "https://www.teclamairme.com/#r-19.5", "chapter": 19, "status": "confirmed", "claim": "For y² = x³ + ax + b, the discriminant is −16(4a³ + 27b²); for y² = x³ − x it is 64, and the j-invariant is 1728 = 12³, reflecting complex multiplication by i. For y² = x³ + ax + b, j = 1728·4a³/(4a³ + 27b²); j(i) = 1728.", "sources": [ { "title": "LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x", "url": "https://www.lmfdb.org/EllipticCurve/Q/32a2/", "own_work": false, "note": "gives conductor 32, discriminant 64, j-invariant 1728 = 2⁶ · 3³, rank 0 and torsion Z/2 × Z/2 for this curve.", "fetched": [ "2026-09-09" ] }, { "title": "Elliptic curve invariants", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "gives the short Weierstrass discriminant −16(4a³ + 27b²), writes j as 1728·g′₂³/(g′₂³ − 27g′₃²) in lattice invariants (the a, b form is on the j-invariant line), and says the j-invariant specifies isomorphism classes of elliptic curves; the curve “is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition 4a³ + 27b² ≠ 0, that is, being square-free in x.),” and “Since the curve is smooth, hence continuous, it can be shown that this point at infinity is the identity element of a group structure.”", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "Substitute a = −1 and b = 0: Δ = −16(4(−1)^3) = 64 and j = 1728·4(−1)^3/(4(−1)^3) = 1728 = 12^3.", "fetched": [] }, { "title": "j-invariant, Wikipedia", "url": "https://en.wikipedia.org/wiki/J-invariant", "own_work": false, "note": "read as raw wikitext: “In case of a short Weierstraß elliptic curve y² = x³ + ax + b, this formula simplifies to” j = 1728·4a³/(4a³ + 27b²).", "fetched": [ "2026-09-22" ] }, { "title": "Complex multiplication, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_multiplication", "own_work": false, "note": "read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “j(Z[i]) = j(i) = 1728.”", "fetched": [ "2026-09-22" ] }, { "title": "LMFDB, elliptic curve 37.a1 (Cremona label 37a1)", "url": "https://www.lmfdb.org/EllipticCurve/Q/37a1/", "own_work": false, "note": "minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.6", "url": "https://www.teclamairme.com/#r-19.6", "chapter": 19, "status": "confirmed", "claim": "The torsion points of y² = x³ − x are the identity and the three roots of x³ − x, each of order 2.", "sources": [ { "title": "Poonen, Rational points on varieties, section 8.3", "url": "https://math.mit.edu/~poonen/papers/Qpoints.pdf", "own_work": false, "note": "states that the curve 32A2 has torsion Z/2 × Z/2 and four rational points.", "fetched": [ "2026-09-09" ] }, { "title": "RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms", "url": "https://www.rfc-editor.org/rfc/rfc6090", "own_work": false, "note": "states that a point with y = 0 doubles to the point at infinity and that the point at infinity is the identity.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.7", "url": "https://www.teclamairme.com/#r-19.7", "chapter": 19, "status": "confirmed", "claim": "The curve y² = x³ − x has eight points over F5, and Hasse’s bound is |#E(F5) − 6| ≤ 2√5. a_p = p + 1 − #E(F_p).", "sources": [ { "title": "Hasse’s theorem for elliptic curves", "url": "https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves", "own_work": false, "note": "states |#E(Fq) − (q+1)| ≤ 2√q.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For x = 0,1,4 modulo 5, x³−x = 0 and y = 0; for x = 2 the value is 1, giving y = ±1; for x = 3 the value is 4, giving y = ±2. Thus there are 7 affine points plus O, so #E(F5)=8 and |8−6|=2≤2√5.", "fetched": [] }, { "title": "Elliptic curve, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "read as raw wikitext: a_p = p + 1 − #E(F_p); the curve’s L-function “is defined as an Euler product, with one factor for every prime number p,” built from these a_p, and the proof of the Shimura–Taniyama–Weil conjecture “implies that its L-function is the L-function of a modular form whose analytic continuation is known.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.8", "url": "https://www.teclamairme.com/#r-19.8", "chapter": 19, "status": "confirmed", "claim": "The curve y² + y = x³ − x has conductor 37 and rank 1, the smallest known conductor for a rank-one elliptic curve; its L-value at 1 vanishes. 37a1 is semistable with analytic rank 1; the Antwerp tables of curves of conductor up to 200 came from Swinnerton-Dyer’s search, later extended by Cremona.", "sources": [ { "title": "Elkies, Best known conductors for elliptic curves of given rank", "url": "https://people.math.harvard.edu/~elkies/womack2001.html", "own_work": false, "note": "lists rank 1, curve [0 0 1 −1 0], conductor 37, and marks it as the smallest conductor found by exhaustive search for that rank.", "fetched": [ "2026-09-09" ] }, { "title": "Poonen, Rational points on varieties, section 8.3", "url": "https://math.mit.edu/~poonen/papers/Qpoints.pdf", "own_work": false, "note": "Poonen’s survey of rational points on curves; read again on 23 September 2026, it does not display the 37a1 model, which the LMFDB 37a1 line carries.", "fetched": [ "2026-09-09" ] }, { "title": "Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020)", "url": "https://annals.math.princeton.edu/2020/191-2/p01", "own_work": false, "note": "for a semistable curve with suitable reduction, proves that algebraic rank 1 and finite Ш give analytic rank 1, a converse to the Gross–Zagier/Kolyvagin theorem; the page does not discuss 37a1, whose L(1) = 0 is on the LMFDB lines.", "fetched": [ "2026-09-09" ] }, { "title": "LMFDB, elliptic curve 37.a1 (Cremona label 37a1)", "url": "https://www.lmfdb.org/EllipticCurve/Q/37a1/", "own_work": false, "note": "minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0; its Mordell-Weil generators table lists the point (0, 0), of order ∞.", "fetched": [ "2026-09-22" ] }, { "title": "John Cremona, The elliptic curve database for conductors to 130000 (ANTS VII)", "url": "https://johncremona.github.io/papers/ecdb_ants.pdf", "own_work": false, "note": "read from the PDF: the Antwerp Table 1 lists “All elliptic curves of conductor N ≤ 200”; “Swinnerton-Dyer searched for curves with small coefficients and kept those with conductor N ≤ 200”; the Antwerp tables are in “Antwerp IV” … “(edited by Birch and Kuyk).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.9", "url": "https://www.teclamairme.com/#r-19.9", "chapter": 19, "status": "confirmed", "claim": "Mazur’s theorem gives exactly fifteen possible torsion subgroups for elliptic curves over Q; the largest possible order is 16.", "sources": [ { "title": "Mazur theorem lecture notes, IMPA", "url": "https://w3.impa.br/~hossein/myarticles/EllipticCurvesCourse.pdf", "own_work": false, "note": "states the fifteen groups in Mazur’s classification of E(Q)_tors.", "fetched": [ "2026-09-09" ] }, { "title": "Mazur torsion theorem overview", "url": "https://wstein.org/projects/kleinerman_senior_thesis.pdf", "own_work": false, "note": "lists the cyclic groups of orders 1–10 and 12 and the four products Z/2 × Z/2n, whose largest order is 16.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.10", "url": "https://www.teclamairme.com/#r-19.10", "chapter": 19, "status": "confirmed", "claim": "Five is the smallest congruent number, represented by the rational right triangle with sides 3/2, 20/3, and 41/6; 1 through 4 are not congruent. 6 is the area of the 3-4-5 triangle; Fibonacci found 7 congruent; Fermat proved 1 is not (1640); n is congruent exactly when y² = x³ − n²x has a point of infinite order.", "sources": [ { "title": "OEIS A268602, first congruent number triangle", "url": "https://oeis.org/A268602", "own_work": false, "note": "gives the triangle 3/2, 20/3, 41/6, verifies its area is 5, and identifies 5 as the first congruent number.", "fetched": [ "2026-09-09" ] }, { "title": "The congruent number problem", "url": "https://www.johndcook.com/blog/2021/12/17/congruent-number/", "own_work": false, "note": "defines congruent numbers and independently verifies the area-5 triangle and the fact that 5 is the smallest.", "fetched": [ "2026-09-09" ] }, { "title": "Keith Conrad, The congruent number problem (University of Connecticut)", "url": "https://kconrad.math.uconn.edu/articles/congruentnumber.pdf", "own_work": false, "note": "read from the PDF: 6 is the area of a (3, 4, 5) right triangle; “Fibonacci discovered in the 13th century that 7 is congruent”; “Theorem 2 (Fermat, 1640). The number 1 is not congruent.”", "fetched": [ "2026-09-22" ] }, { "title": "Elliptic curve, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "read as raw wikitext: “n is a congruent number if and only if the elliptic curve y² = x³ − n²x has a rational point of infinite order.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.11", "url": "https://www.teclamairme.com/#r-19.11", "chapter": 19, "status": "confirmed", "claim": "Birch and Swinnerton-Dyer conjectured that the rank equals the order of vanishing of L(E, s) at s=1; this remains a Millennium Problem.", "sources": [ { "title": "Birch–Swinnerton-Dyer conjecture, MathConjectures", "url": "https://www.mathconjectures.com/conjectures/NT-ARITH-002", "own_work": false, "note": "states ord_{s=1} L(E, s) = r and labels the conjecture open.", "fetched": [ "2026-09-09" ] }, { "title": "Clay Mathematics Institute, Millennium Prize Problems", "url": "https://www.claymath.org/millennium-problems/", "own_work": false, "note": "lists Birch and Swinnerton-Dyer among the Millennium Prize Problems.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.12", "url": "https://www.teclamairme.com/#r-19.12", "chapter": 19, "status": "confirmed", "claim": "Birch and Swinnerton-Dyer’s early computational evidence came from counting points modulo primes, and Gross–Zagier and Kolyvagin proved the relevant rank-one implication in the late twentieth century.", "sources": [ { "title": "Birch–Swinnerton-Dyer conjecture", "url": "https://en.wikipedia.org/wiki/Birch%E2%80%93Swinnerton-Dyer_conjecture", "own_work": false, "note": "“In the early 1960s, Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number N​p of points modulo p for a large number of primes p on elliptic curves whose rank was known”; and Gross and Zagier in 1986 and Kolyvagin in 1989 proved the rank-zero and rank-one implications, not the vanishing of any particular L-value.", "fetched": [ "2026-09-09" ] }, { "title": "Christopher Skinner, A converse to a theorem of Gross, Zagier, and Kolyvagin (Annals of Mathematics, 2020)", "url": "https://annals.math.princeton.edu/2020/191-2/p01", "own_work": false, "note": "proves a converse to the theorem of Gross, Zagier and Kolyvagin: for certain semistable curves, algebraic rank 1 and finite Ш give analytic rank 1.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-19.13", "url": "https://www.teclamairme.com/#r-19.13", "chapter": 19, "status": "confirmed", "claim": "Every elliptic curve over Q is modular: semistable curves were handled by Wiles and Taylor–Wiles, and the full theorem was proved by Breuil, Conrad, Diamond, and Taylor in 2001; this was the bridge to Fermat’s Last Theorem via Ribet’s theorem, which Ken Ribet proved in the summer of 1986 and published in 1990. The conductor of the curve is the level of the corresponding modular form. Hasse’s conjecture that L(E, s) continues to the whole plane was shown in 1999 to follow from the modularity theorem.", "sources": [ { "title": "Breuil, Conrad, Diamond and Taylor, On the modularity of elliptic curves over Q", "url": "https://www.math.u-psud.fr/~breuil/PUBLICATIONS/STW.pdf", "own_work": false, "note": "the four authors’ preprint: “Theorem A. If E/Q is an elliptic curve, then E is modular.”; it is dated 2000, “and in revised form” January 2001; the 2001 journal publication is on the Edixhoven line.", "fetched": [ "2026-09-09" ] }, { "title": "Edixhoven, Rational elliptic curves are modular", "url": "https://www.numdam.org/article/SB_1999-2000__42__161_0.pdf", "own_work": false, "note": "records the semistable Wiles/Taylor–Wiles result, the full Breuil–Conrad–Diamond–Taylor theorem, and the consequence for Fermat’s Last Theorem; its bibliography gives “On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), no. 4, 843-939.”", "fetched": [ "2026-09-09" ] }, { "title": "Ribet’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ribet%27s_theorem", "own_work": false, "note": "read as raw HTML: “In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Modularity theorem implied FLT”, and its bibliography gives the publication as Ribet, Ken (1990), “On modular representations of Gal(Q/Q) arising from modular forms”, Inventiones Mathematicae 100 (2), pp. 431–476.", "fetched": [ "2026-09-17" ] }, { "title": "Modularity theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Modularity_theorem", "own_work": false, "note": "read as raw wikitext: “Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form.”; the parametrization “may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N.”", "fetched": [ "2026-09-22" ] }, { "title": "Elliptic curve, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "read as raw wikitext: “Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane … In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.14", "url": "https://www.teclamairme.com/#r-19.14", "chapter": 19, "status": "confirmed", "claim": "The curve y² = x³ − x has complex multiplication by i, with the extra map (x, y) ↦ (−x, iy) and endomorphism ring Z[i].", "sources": [ { "title": "LMFDB 32.a3 (Cremona 32a2), the curve y² = x³ − x", "url": "https://www.lmfdb.org/EllipticCurve/Q/32a2/", "own_work": false, "note": "records complex multiplication with discriminant −4 and geometric endomorphism ring Z[√−1].", "fetched": [ "2026-09-09" ] }, { "title": "Complex multiplication, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_multiplication", "own_work": false, "note": "read as raw wikitext: CM “is the theory of elliptic curves E that have an endomorphism ring larger than the integers”; curves Y² = 4X³ − aX have “two conjugate order-4 automorphisms” sending Y → ±iY, X → −X; “Kronecker conjectured – in what became known as the Kronecker Jugendtraum – that every abelian extension of K could be obtained by the (roots of the) equation of a suitable elliptic curve with complex multiplication”; “To this day this remains one of the few cases of Hilbert’s twelfth problem which has actually been solved.”; “j(Z[i]) = j(i) = 1728.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.15", "url": "https://www.teclamairme.com/#r-19.15", "chapter": 19, "status": "confirmed", "claim": "Elliptic-curve cryptography uses fast scalar multiplication kP while relying on the assumed hardness of recovering k from P and kP; secp256k1 is the curve used by Bitcoin, and Shor’s algorithm threatens this classical security model. Double-and-add computes kP quickly; the fastest known classical attacks need O(√n) steps; secp256k1 is y² = x³ + 7 over a 256-bit prime field and is Bitcoin’s ECDSA curve; Shor’s algorithm would break ECC on a large fault-tolerant quantum computer.", "sources": [ { "title": "RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms", "url": "https://www.rfc-editor.org/rfc/rfc6090", "own_work": false, "note": "defines elliptic-curve point multiplication and the discrete-logarithm setting used by ECC.", "fetched": [ "2026-09-09" ] }, { "title": "NIST, Post-Quantum Cryptography FAQ", "url": "https://csrc.nist.gov/projects/post-quantum-cryptography/faqs", "own_work": false, "note": "explains that Shor’s algorithm is the kind of “dramatic quantum attack” for which classical security measures stop being relevant; the FAQ does not itself name discrete logarithms, which the Elliptic-curve cryptography line carries.", "fetched": [ "2026-09-09" ] }, { "title": "Elliptic curve point multiplication, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication", "own_work": false, "note": "read as raw wikitext: “The simplest method is the double-and-add method.”", "fetched": [ "2026-09-22" ] }, { "title": "Elliptic-curve cryptography, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic-curve_cryptography", "own_work": false, "note": "read as raw wikitext: “all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard’s rho, etc.), need O(√n) steps”; “Shor’s algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a sufficiently large fault-tolerant quantum computer.”", "fetched": [ "2026-09-22" ] }, { "title": "SECG, SEC 2: Recommended Elliptic Curve Domain Parameters, version 2.0", "url": "https://www.secg.org/sec2-v2.pdf", "own_work": false, "note": "§2.4.1 secp256k1: p = 2²⁵⁶ − 2³² − 2⁹ − 2⁸ − 2⁷ − 2⁶ − 2⁴ − 1; “The curve E: y² = x³ + ax + b over Fp is defined by” a = 0, b = 7.", "fetched": [ "2026-09-22" ] }, { "title": "Bitcoin Developer Guide, Transactions", "url": "https://developer.bitcoin.org/devguide/transactions.html", "own_work": false, "note": "“Bitcoin uses the Elliptic Curve Digital Signature Algorithm (ECDSA) with the secp256k1 curve.”", "fetched": [ "2026-09-22" ] }, { "title": "Cloudflare blog, Automatic key exchange for origins", "url": "https://blog.cloudflare.com/automatic-key-exchange-for-origins/", "own_work": false, "note": "“over 95% of origins support X25519”; the post describes the move to post-quantum hybrid key agreement.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.16", "url": "https://www.teclamairme.com/#r-19.16", "chapter": 19, "status": "judgement", "claim": "The historical attributions in this chapter compress a long development: Diophantus, Newton, Poincaré, Fermat, Birch, Swinnerton-Dyer, Frey, Ribet, and Wiles are associated with the milestones stated in the prose. The tangent step was used by Diophantus and the chord-tangent construction by Newton in the 1670s; associativity is the only non-trivial part of the group law.", "sources": [ { "title": "Elliptic curve, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "points to “elliptic integral for the origin of the term,” gives the group law “by the tangent and secant method,” and says the curves “were used in Andrew Wiles’s proof of Fermat’s Last Theorem”; the historical attributions are on the other lines.", "fetched": [ "2026-09-09" ] }, { "title": "Breuil, Conrad, Diamond and Taylor, On the modularity of elliptic curves over Q", "url": "https://www.math.u-psud.fr/~breuil/PUBLICATIONS/STW.pdf", "own_work": false, "note": "confirms the modern modularity attribution and dates.", "fetched": [ "2026-09-09" ] }, { "title": "Fermat’s right triangle theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem", "own_work": false, "note": "“It is the only complete proof given by Fermat”; Fermat wrote it in the margin of his copy of Diophantus and his son Clement-Samuel published it in 1670.", "fetched": [ "2026-09-09" ] }, { "title": "Fermat’s Last Theorem", "url": "https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem", "own_work": false, "note": "“In 1984, Gerhard Frey noticed an apparent link between these two previously unrelated and unsolved problems”; Ribet completed the link in 1986; Wiles lectured 21–23 June 1993 and the corrected proof filled the May 1995 Annals of Mathematics.", "fetched": [ "2026-09-09" ] }, { "title": "Araceli Bonifant and John Milnor, On real and complex cubic curves, arXiv:1603.09018", "url": "https://arxiv.org/abs/1603.09018", "own_work": false, "note": "read from the PDF: “The “tangent” part of the chord-tangent map was used by Diophantus of Alexandria in the third century”; “in the 1670’s Newton used the “chord-tangent construction” to interpret the solutions of Diophantine equations given by Diophantus and Fermat.”", "fetched": [ "2026-09-22" ] }, { "title": "Sander Zwegers, On the group law on elliptic curves, arXiv:2401.02346", "url": "https://arxiv.org/abs/2401.02346", "own_work": false, "note": "read from the PDF: “The only non-trivial part is the associativity. One approach to this is to concentrate on elliptic curves over C and use the theory of elliptic functions, in particular the Weierstraß ℘-function”; the paper gives “a simple and elementary proof of the associativity.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "We found authoritative support for the major milestones, but the chapter compresses several independent historical threads into a single narrative and the exact priority wording for Diophantus, Newton, and Poincaré is not uniquely documented in the fetched sources. We retain the chapter’s broad chronology while treating the compressed priority language as historical judgement. The Frey, Ribet and Wiles dates given in the prose are the ones the Fermat’s Last Theorem record uses; other accounts date Frey’s observation to his 1986 paper and Ribet’s publication to 1990, which are different milestones rather than a different chronology. Settled on 17 September 2026 and not to be reopened: they are the proof and the printing of one result. Ribet proved the epsilon conjecture in the summer of 1986 and published it as “On modular representations of Gal(Q/Q) arising from modular forms,” Inventiones Mathematicae 100 (1990), pp. 431–476; both dates are now carried by 19.13 and stated on the #modularity card. No source we found fixes a year for Fermat’s own proof, so the prose names no decade for it." }, { "id": "r-19.17", "url": "https://www.teclamairme.com/#r-19.17", "chapter": 19, "status": "confirmed", "claim": "The largest known rank of an elliptic curve over Q is at least 31 (Alpöge and Howell, 2026), after rank 29 (Elkies and Klagsbrun, 2024) and rank 28 (Elkies, 2006); the largest rank known exactly and unconditionally is 20 (Elkies and Klagsbrun, 2020), and whether ranks are bounded is open. Mordell’s theorem gives no effective method for computing the rank, and it is unknown whether the numerical methods are effective for all curves.", "sources": [ { "title": "Dujella, History of elliptic curve rank records", "url": "https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html", "own_work": false, "note": "re-read raw on 17 September 2026 and unchanged: the table lists rank 28 in 2006 (Elkies), rank 29 in 2024 (Elkies–Klagsbrun) and ranks 30 and 31 in 2026 (Alpöge–Howell), the running text says “The current record is an example of elliptic curve with rank ≥ 31, found by Alpöge and Howell in 2026,” and the highest rank known unconditionally and exactly is still 20, found by Elkies–Klagsbrun in 2020.", "fetched": [ "2026-09-09" ] }, { "title": "Rank of an elliptic curve", "url": "https://en.wikipedia.org/wiki/Rank_of_an_elliptic_curve", "own_work": false, "note": "as of August 2026 the largest known rank is at least 31, exactly 31 under the generalized Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture, and there is no consensus among experts on whether ranks are bounded.", "fetched": [ "2026-09-09" ] }, { "title": "Birch and Swinnerton-Dyer conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture", "own_work": false, "note": "read as raw wikitext: “Although Mordell’s theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of a given curve. The rank can sometimes be calculated using numerical methods, but it is currently unknown if these methods are effective for all curves.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.18", "url": "https://www.teclamairme.com/#r-19.18", "chapter": 19, "status": "confirmed", "claim": "Fermat’s one surviving complete proof is his right triangle theorem, proved by infinite descent: from any Pythagorean triangle whose area is a square one can derive a smaller one. That theorem is equivalent to the statement that 1 is not a congruent number, and Fermat’s Last Theorem for exponent 4 is an immediate consequence of it.", "sources": [ { "title": "Fermat’s right triangle theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_right_triangle_theorem", "own_work": false, "note": "carries all four halves of the claim in so many words — “It is the only complete proof given by Fermat”; “Fermat’s proof is a proof by infinite descent. It shows that, from any example of a Pythagorean triangle with square area, one can derive a smaller example”; “the existence of a square congruum is equivalent to the statement that the number 1 is not a congruent number”; and “An immediate consequence of the last of these formulations is that Fermat’s Last Theorem is true in the special case that its exponent is 4.”", "fetched": [ "2026-09-10" ] }, { "title": "Proof by infinite descent, Wikipedia", "url": "https://en.wikipedia.org/wiki/Proof_by_infinite_descent", "own_work": false, "note": "defines the method — “a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction” — and records that Fermat coined the term and often used it for Diophantine equations.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass for the new card `descent` and to settle cross-chapter conflict C1. Chapter ᚆ(H) calls the descent “the one proof of that kind he left behind … covers fourth powers only” and Chapter ᚒ(U) called it “the only complete proof he left”; both sentences are about this single argument, and this entry is the source that says so. The “only complete proof” wording was already quoted under 19.16; it is repeated here so the card can stand on one entry." }, { "id": "r-19.19", "url": "https://www.teclamairme.com/#r-19.19", "chapter": 19, "status": "confirmed", "claim": "Mordell’s theorem, that the rational points of an elliptic curve over Q form a finitely generated abelian group, was proved by Louis Mordell in 1922, answering a question posed by Henri Poincare around 1901 in the memoir Sur les proprietes arithmetiques des courbes algebriques, Journal de Mathematiques Pures et Appliquees, serie 5, tome 7 (1901), pp. 161-233. Andre Weil generalised it to Jacobians of higher-genus curves over arbitrary number fields in his doctoral dissertation, published in 1928 and dated 1929 by some accounts.", "sources": [ { "title": "Mordell–Weil theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil_theorem", "own_work": false, "note": "states “The case with A an elliptic curve E and K the field of rational numbers is Mordell’s theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proven by Louis Mordell in 1922,” and gives the general statement that A(K) is a finitely-generated abelian group. Re-read raw on 17 September 2026 for Weil: “Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928,” while the same page’s infobox gives “First proof by André Weil, First proof in 1929” — the dissertation and its publication, not two claims.", "fetched": [ "2026-09-10" ] }, { "title": "H. Poincaré, Sur les propriétés arithmétiques des courbes algébriques, Journal de Mathématiques Pures et Appliquées, série 5, tome 7 (1901)", "url": "https://www.numdam.org/item/JMPA_1901_5_7__161_0/", "own_work": false, "note": "the memoir itself, free in full text, pp. 161–233, publisher Gauthier-Villars. This is the primary document behind the 1901 date.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added in the V14 prose pass. Chapter ᚒ(U)’s “In 1922 Louis Mordell proved” and the `mordellweil` card’s 1922 and 1901 dates were footnoted only at 19.3, whose two sources state the theorem but name no year for it and no questioner. This entry carries both dates in one sentence, so it also stands behind the chapter’s “stated clearly by Poincaré in 1901” in the first paragraph. Extended 17 September 2026. The Wikipedia source hedges — “a question apparently posed by Henri Poincaré around 1901” — and names no paper, so the identification of the memoir is the book’s own, made from its NUMDAM record; the memoir is now cited directly so a reader can check it. The entry was also attached to #grouplaw’s “Poincaré stated in 1901” clause, which until now carried no citation for the date at all: its only reference, 19.2, describes the chord-and-tangent construction and names neither a person nor a year." }, { "id": "r-19.20", "url": "https://www.teclamairme.com/#r-19.20", "chapter": 19, "status": "confirmed", "claim": "Barry Mazur proved the full torsion conjecture for elliptic curves over the rationals in two papers, of 1977 and 1978; the fifteen groups are the cyclic groups of order 1 to 10 and 12 together with C2n x C2 for n = 1 to 4; Loïc Merel proved in 1996 that torsion is uniformly bounded over every number field.", "sources": [ { "title": "Torsion conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Torsion_conjecture", "own_work": false, "note": "states “Barry Mazur (1977, 1978) proved the full torsion conjecture for elliptic curves over the rationals,” and lists the groups as C_n with 1 ≤ n ≤ 10, C_12, and C_2n × C_2 with 1 ≤ n ≤ 4. Re-read raw on 17 September 2026 for Merel: “Finally, Loïc Merel (1996) proved the conjecture for elliptic curves over any number field,” after Kamienny (1992) and Kamienny and Mazur (1995) had reached quadratic fields and fields of degree at most 8; the page adds that Merel proved “that there is a bound on the order of the torsion group depending only on the degree.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "Added in the V14 prose pass to carry the year the chapter uses. “Mazur proved in 1977” was footnoted 19.9 alone, and neither of 19.9’s two sources gives a date for the theorem; they state the classification only. The source here dates it to two papers rather than one, which is what the `mazur16` card already says." }, { "id": "r-19.21", "url": "https://www.teclamairme.com/#r-19.21", "chapter": 19, "status": "confirmed", "claim": "The L-function of an elliptic curve over Q is the Euler product of (1 - a_p p^-s + p^(1-2s))^-1 over the primes of good reduction, times (1 - a_p p^-s)^-1 at the primes dividing the conductor, with a_p = p + 1 - #E(F_p); the product converges only for Re(s) > 3/2, and modularity gives it an analytic continuation to the whole plane and a functional equation relating L(E, s) to L(E, 2 - s). The Nagell-Lutz theorem is named for two independent discoverers, Trygve Nagell, who published it in 1935, and Elisabeth Lutz, in 1937. Weierstrass introduced the symbol for his elliptic function in his lectures of 1862-1863.", "sources": [ { "title": "Elliptic curve, Wikipedia (§ L-functions)", "url": "https://en.wikipedia.org/wiki/Elliptic_curve", "own_work": false, "note": "read as raw HTML: the L-function is “defined as an Euler product, with one factor for every prime number p,” with “a_p = p + 1 − #E(F_p),” and given as L(E(Q), s) = ∏ over p not dividing N of (1 − a_p p^−s + p^(1−2s))^−1 times ∏ over p dividing N of (1 − a_p p^−s)^−1. “This product converges for Re(s) > 3/2 only. Hasse’s conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s). In 1999 this was shown to be a consequence of the proof of the Shimura–Taniyama–Weil conjecture.”", "fetched": [ "2026-09-17" ] }, { "title": "Nagell–Lutz theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Nagell%E2%80%93Lutz_theorem", "own_work": false, "note": "read as raw HTML: “The result is named for its two independent discoverers, the Norwegian Trygve Nagell (1895–1988) who published it in 1935, and Élisabeth Lutz (1937)”; for y² = x³ + ax² + bx + c “with integer coefficients a, b, c,” and D “the discriminant of the cubic polynomial on the right side”: “If P = (x, y) is a rational point of finite order on E, for the elliptic curve group law, then: x and y are integers; either y = 0, in which case P has order two, or else y divides D, which immediately implies that y² divides D” (formulae as typeset on the page).", "fetched": [ "2026-09-17" ] }, { "title": "Weierstrass elliptic function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Weierstrass_elliptic_function", "own_work": false, "note": "read as raw HTML: the function “is usually written with a rather special, lower case script letter ℘, which was Weierstrass’s own notation introduced in his lectures of 1862–1863,” which is the 1860s the #weierstrass card gives.", "fetched": [ "2026-09-17" ] }, { "title": "LMFDB, elliptic curve 37.a1 (Cremona label 37a1)", "url": "https://www.lmfdb.org/EllipticCurve/Q/37a1/", "own_work": false, "note": "minimal Weierstrass equation y² + y = x³ − x; “This elliptic curve is semistable. There is only one prime p of bad reduction”; analytic rank 1, so L(E, 1) = 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 17 September 2026 for the B6 pass. Three separate gaps in one chapter, all of the same shape: a card asserting something its cited entry does not carry. #LfuncE’s whole quick line is the Euler product and the only entry behind it was r-19.11, which states the Birch and Swinnerton-Dyer conjecture and its Millennium-Problem status; no entry anywhere in references.html carried an elliptic-curve Euler product, the six “Euler product” hits all being the Riemann zeta function in the prime-mileposts chapter. #tors4’s Nagell 1935 and Lutz 1937 stood on r-19.6, which is the torsion of one curve. #weierstrass’s “℘-function of the 1860s” stood on r-19.5, which is Δ = 64 and j = 1728. Not sourced and therefore not asserted: the #j1728 card said the 1728 normalisation “goes back to Klein’s work on the modular function in the 1870s,” and no source found in this pass dates Klein’s work, so the decade was removed from the card rather than left standing." }, { "id": "r-19.22", "url": "https://www.teclamairme.com/#r-19.22", "chapter": 19, "status": "confirmed", "claim": "The chapter epigraph is the opening line of the preface to Serge Lang’s Elliptic Curves: Diophantine Analysis, Grundlehren der mathematischen Wissenschaften 231, Springer-Verlag, 1978.", "sources": [ { "title": "Serge Lang, Wikipedia", "url": "https://en.wikipedia.org/wiki/Serge_Lang", "own_work": false, "note": "read as raw HTML: its bibliography gives “Lang, Serge, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften, volume 231, Springer-Verlag, Berlin–New York, 1978, ISBN 3-540-08489-4, doi 10.1007/978-3-662-07010-9,” with Alan Baker’s review in the Bulletin of the AMS (N.S.) 2 (1980), 352–354.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. The book, the series, the publisher and the year are sourced here; that the sentence is the preface’s opening line is the author’s own reading of the printed book and is offered as such." }, { "id": "r-19.23", "url": "https://www.teclamairme.com/#r-19.23", "chapter": 19, "status": "confirmed", "claim": "Homogeneous or projective coordinates were introduced by August Ferdinand Möbius in Der barycentrische Calcul of 1827. A point is a triple (X : Y : Z), not all zero, with triples counted as equal when they differ by a nonzero common scale; the ordinary affine plane is the part with Z ≠ 0, recovered as (X/Z, Y/Z), and the points with Z = 0 are the points at infinity. The advantage is that points at infinity get finite coordinates. The point at infinity, the identity of the curve’s group, has projective coordinates (0 : 1 : 0).", "sources": [ { "title": "Homogeneous coordinates, Wikipedia", "url": "https://en.wikipedia.org/wiki/Homogeneous_coordinates", "own_work": false, "note": "read as raw HTML: “homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul… They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates,” with the scaling equivalence and the Z = 0 line at infinity.", "fetched": [ "2026-09-17" ] }, { "title": "RFC 6090, Fundamental Elliptic Curve Cryptography Algorithms", "url": "https://www.rfc-editor.org/rfc/rfc6090", "own_work": false, "note": "a special point “represents the identity element (which is called the “point at infinity”)”; “The point at infinity (@,@) is regarded as equivalent to the homogenous coordinates (0, 1, 0).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-19.24", "url": "https://www.teclamairme.com/#r-19.24", "chapter": 19, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Coates and Wiles (1977): first general result towards BSD, for CM curves; Tunnell’s criterion dates from 1983; Koblitz and Miller independently proposed elliptic-curve cryptography in 1985; Taniyama stated the first version of the conjecture in 1955; Taniyama and Shimura worked on the conjecture until 1957, and Weil gave it its first serious evidence in 1967; Weil showed in 1967 that the modularity conjecture would follow from conjectured functional equations; Wiles, with Taylor, proved the semistable case in 1995; Hasse proved the bound |#E(F_p) - (p+1)| <= 2 sqrt(p) in 1933. Coates and Wiles proved their theorem in 1976 and published it in 1977; it was the first general result toward the conjecture.", "sources": [ { "title": "Birch and Swinnerton-Dyer conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture", "own_work": false, "note": "read raw: “Coates, J.; Wiles, A. (1977). “On the conjecture of Birch and Swinnerton-Dyer.” Inventiones Mathematicae 39 (3): 223–251..”", "fetched": [ "2026-09-18" ] }, { "title": "Tunnell’s theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Tunnell%27s_theorem", "own_work": false, "note": "read raw: “The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in Tunnell (1983).”; “Tunnell’s theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.”", "fetched": [ "2026-09-18" ] }, { "title": "Elliptic-curve cryptography, Wikipedia", "url": "https://en.wikipedia.org/wiki/Elliptic-curve_cryptography", "own_work": false, "note": "read raw: “The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985..”", "fetched": [ "2026-09-18" ] }, { "title": "Modularity theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Modularity_theorem", "own_work": false, "note": "read raw: “Yutaka Taniyama stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems.”; “Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve”; “In 1995, Andrew Wiles, with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves..”", "fetched": [ "2026-09-18" ] }, { "title": "Hasse’s theorem on elliptic curves, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves", "own_work": false, "note": "read raw: “It was proven by Hasse in 1933, with the proof published in a series of papers in 1936..”", "fetched": [ "2026-09-18" ] }, { "title": "Andrew Wiles, The Birch and Swinnerton-Dyer Conjecture (Clay Mathematics Institute problem description)", "url": "https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf", "own_work": false, "note": "read from the PDF: “The first general result proved was for elliptic curves with complex multiplication … This theorem was proved in 1976 and is due to Coates and Wiles.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-20.1", "url": "https://www.teclamairme.com/#r-20.1", "chapter": 20, "status": "confirmed", "claim": "The middle-thirds Cantor set is obtained by repeatedly deleting open middle thirds from [0, 1]; it consists of points with a ternary expansion using only 0 and 2, is uncountable, and has the cardinality of the interval.", "sources": [ { "title": "Wikipedia, Cantor set", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "the construction, ternary 0/2 description, and equal cardinality with [0, 1] are stated in the construction and cardinality sections.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.2", "url": "https://www.teclamairme.com/#r-20.2", "chapter": 20, "status": "confirmed", "claim": "The lengths removed in the Cantor construction form the geometric series 1/3+2/9+4/27+…=1, so the Cantor set has Lebesgue measure zero although it is nonempty and uncountable.", "sources": [ { "title": "Wikipedia, Cantor set", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "the composition section evaluates the removed geometric progression to 1 and explains why endpoints and uncountably many points remain.", "fetched": [ "2026-09-09" ] }, { "title": "Wolfram MathWorld, Measure Zero", "url": "https://mathworld.wolfram.com/MeasureZero.html", "own_work": false, "note": "defines measure zero by covers with intervals of arbitrarily small total length.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.3", "url": "https://www.teclamairme.com/#r-20.3", "chapter": 20, "status": "confirmed", "claim": "Henry John Stephen Smith described the construction in 1874 and Georg Cantor mentioned it in 1883.", "sources": [ { "title": "Wikipedia, Cantor set", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "the historical remarks identify Smith’s 1874 discovery and Cantor’s 1883 mention.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.4", "url": "https://www.teclamairme.com/#r-20.4", "chapter": 20, "status": "confirmed", "claim": "Cantor’s 1874 work established that the reals are uncountable while the real algebraic numbers are countable — the countability of the rationals is older and is not the content of that paper; see 20.22 — and the ternary-to-binary digit map explains why the Cantor set has continuum cardinality.", "sources": [ { "title": "Wikipedia, Cantor set", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "the article gives the 0/2 ternary representation, the binary-tree/cardinality argument, and the uncountability conclusion, but no date for Cantor’s 1874 paper (it dates the set to Smith, 1874, and Cantor, 1883) and nothing about algebraic numbers. This claim line said until 17 September 2026 that Cantor’s 1874 work showed “the rationals are countable”; the first theorem of that article is that the real algebraic numbers are countable, which is what 20.22 states and what the chapter’s argument needs.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.5", "url": "https://www.teclamairme.com/#r-20.5", "chapter": 20, "status": "confirmed", "claim": "Lebesgue measure extends length, is countably additive, and gives the Cantor set and the rationals in an interval measure zero; countable sets are null, while the Cantor set is uncountable. Kolmogorov’s 1933 axioms of probability take σ-additivity from Lebesgue’s measure theory.", "sources": [ { "title": "Wikipedia, Lebesgue measure", "url": "https://en.wikipedia.org/wiki/Lebesgue_measure", "own_work": false, "note": "defines Lebesgue measure and countable additivity, and states that countable sets such as the rationals have measure zero.", "fetched": [ "2026-09-09" ] }, { "title": "Null set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Null_set", "own_work": false, "note": "read as raw wikitext: “all countable sets are null. In particular, the set ℚ of rational numbers is a null set, despite being dense in ℝ.”", "fetched": [ "2026-09-22" ] }, { "title": "Probability axioms, Wikipedia", "url": "https://en.wikipedia.org/wiki/Probability_axioms", "own_work": false, "note": "read as raw wikitext: the axioms were “introduced by Russian mathematician Andrey Kolmogorov in 1933”; “The event space, F, must be a σ-algebra on Ω”; “The third axiom, σ-additivity, is relatively modern, and originates with Lebesgue’s measure theory.”", "fetched": [ "2026-09-22" ] }, { "title": "Cantor set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "“As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0.”", "fetched": [ "2026-09-29" ] } ], "editorial_note": "" }, { "id": "r-20.6", "url": "https://www.teclamairme.com/#r-20.6", "chapter": 20, "status": "confirmed", "claim": "The Cantor function is continuous and nondecreasing, constant on removed middle-third intervals, has derivative zero almost everywhere, and takes value 1/2 at 1/2. Absolute continuity is exactly the condition under which f(x) = f(a) + ∫ f′, and the Cantor function is not absolutely continuous.", "sources": [ { "title": "Wikipedia, Cantor function", "url": "https://en.wikipedia.org/wiki/Cantor_function", "own_work": false, "note": "the article describes the devil’s staircase, its constancy on removed intervals, derivative zero almost everywhere, and midpoint value.", "fetched": [ "2026-09-09" ] }, { "title": "Absolute continuity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Absolute_continuity", "own_work": false, "note": "read as raw wikitext: for f on [a, b], being absolutely continuous is equivalent to f having a derivative f′ almost everywhere that is Lebesgue integrable with f(x) = f(a) + ∫ₐˣ f′(t) dt, the “fundamental theorem of Lebesgue integral calculus”; the Cantor function is listed as not absolutely continuous.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-20.7", "url": "https://www.teclamairme.com/#r-20.7", "chapter": 20, "status": "confirmed", "claim": "Hausdorff dimension for a self-similar set with m copies scaled by 1/r is ln(m)/ln(r); the Cantor dimension is ln 2/ln 3, about 0.6309, and it is transcendental by Gelfond–Schneider. Felix Hausdorff introduced the dimension in 1918, and Gelfond and Schneider proved the theorem independently in 1934.", "sources": [ { "title": "Wikipedia, Hausdorff dimension", "url": "https://en.wikipedia.org/wiki/Hausdorff_dimension", "own_work": false, "note": "gives the self-similar dimension formula and the Cantor-set example, and dates the introduction of the dimension to Felix Hausdorff in 1918.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Gelfond–Schneider theorem", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem", "own_work": false, "note": "states that if a is algebraic other than 0 or 1 and b is algebraic irrational, every value of a^b is transcendental; applying 3^x=2 proves ln 2/ln 3 transcendental; it dates the two independent proofs, Gelfond’s and Schneider’s, to 1934.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.8", "url": "https://www.teclamairme.com/#r-20.8", "chapter": 20, "status": "confirmed", "claim": "The Smith–Volterra–Cantor set is nowhere dense and has Lebesgue measure one half when middle quarters, then middle sixteenths and so on, are removed. Sources differ on Henry Smith’s year: 1874 (the publisher’s record of the paper, November 1874 issue, and the article on the Cantor set) or 1875 (the article on this set); the book follows the publisher’s record; corrections welcome.", "sources": [ { "title": "Wikipedia, Smith–Volterra–Cantor set", "url": "https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set", "own_work": false, "note": "gives the construction, nowhere-dense property, and measure 1/2; it removes open middle subintervals of width 1/4 to the n at step n, sums the removed lengths to 1/2, and dates Smith’s discussion to 1875.", "fetched": [ "2026-09-09" ] }, { "title": "Zenodo, H. J. S. Smith, On the Integration of Discontinuous Functions", "url": "https://zenodo.org/records/1932560", "own_work": false, "note": "a free record and scan of the paper, Proc. London Math. Soc. s1-6 (1874) 140–153 (doi:10.1112/plms/s1-6.1.140), with publication date 1 November 1874.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-20.9", "url": "https://www.teclamairme.com/#r-20.9", "chapter": 20, "status": "confirmed", "claim": "The Koch curve uses four copies scaled by one third, has Hausdorff dimension ln 4/ln 3, infinite length, and its snowflake has area 8/5 of the initial triangle.", "sources": [ { "title": "Wikipedia, Koch snowflake", "url": "https://en.wikipedia.org/wiki/Koch_snowflake", "own_work": false, "note": "states the four-at-one-third construction, dimension, infinite perimeter, and limiting area 8/5.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.10", "url": "https://www.teclamairme.com/#r-20.10", "chapter": 20, "status": "confirmed", "claim": "The Sierpiński triangle consists of three copies at half scale and has Hausdorff dimension ln 3/ln 2; Wacław Sierpiński described it in 1915.", "sources": [ { "title": "Wikipedia, Sierpiński triangle", "url": "https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle", "own_work": false, "note": "gives the three-copy construction, dimension, and historical attribution to Sierpiński; “If one takes Pascal’s triangle with 2^n rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpiński triangle,” and “The states of an n-disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the nth step in the construction of the Sierpiński triangle” (exponent as typeset on the page).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.11", "url": "https://www.teclamairme.com/#r-20.11", "chapter": 20, "status": "confirmed", "claim": "The Menger sponge keeps twenty of twenty-seven subcubes at each step, has Hausdorff dimension ln 20/ln 3, zero volume, infinite surface area, and is a universal curve. The limit is neither a solid nor a surface: it has topological dimension 1, so it is a curve.", "sources": [ { "title": "Wikipedia, Menger sponge", "url": "https://en.wikipedia.org/wiki/Menger_sponge", "own_work": false, "note": "states the 20-of-27 construction, dimension, zero volume, infinite surface area, and universal-curve property.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.12", "url": "https://www.teclamairme.com/#r-20.12", "chapter": 20, "status": "confirmed", "claim": "Banach and Tarski’s 1924 theorem, using the axiom of choice, decomposes a three-dimensional ball into finitely many pieces, some of them non-measurable, that can be reassembled into two congruent balls; Robinson proved in 1947 that five pieces suffice and fewer than five do not, one of the five being a single point.", "sources": [ { "title": "Wikipedia, Banach–Tarski paradox", "url": "https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox", "own_work": false, "note": "states the 1924 publication, axiom-of-choice dependence, non-measurable pieces, and Robinson’s sharp five-piece result: doubling the ball can be done with five pieces, and fewer than five will not suffice.", "fetched": [ "2026-09-09" ] }, { "title": "R. M. Robinson, On the decomposition of spheres, Fundamenta Mathematicae 34 (1947), 246–260", "url": "https://eudml.org/doc/213130", "own_work": false, "note": "the bibliographic record dates the five-piece minimum to 1947.", "fetched": [ "2026-09-09" ] }, { "title": "Encyclopedia of Mathematics, Banach–Tarski paradox", "url": "https://encyclopediaofmath.org/wiki/Banach%E2%80%93Tarski_paradox", "own_work": false, "note": "identifies Banach and Tarski’s 1924 paper and the equidecomposability theorem in three dimensions.", "fetched": [ "2026-09-09" ] }, { "title": "MathWorld, Banach–Tarski Paradox", "url": "https://mathworld.wolfram.com/Banach-TarskiParadox.html", "own_work": false, "note": "states that five pieces are minimal and four suffice when the single point at the center is omitted, carrying the claim that one of the five pieces is a singleton.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-20.13", "url": "https://www.teclamairme.com/#r-20.13", "chapter": 20, "status": "confirmed", "claim": "Robert May’s 1976 Nature paper made the logistic map f(x)=rx(1−x) a standard example of simple rules with complicated dynamics.", "sources": [ { "title": "Wikipedia, Logistic map", "url": "https://en.wikipedia.org/wiki/Logistic_map", "own_work": false, "note": "identifies May’s 1976 Nature paper and discusses the logistic map as a simple nonlinear recurrence with complex behavior; “In the general case with r between 1 and 2, the population will quickly approach the value (r − 1)/r, independent of the initial population,” and “With r between 2 and 3, the population will also eventually approach the same value” (fraction as typeset on the page); in the demographic example, “xₙ is a number between zero and one, which represents the ratio of existing population to the maximum possible population.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-20.14", "url": "https://www.teclamairme.com/#r-20.14", "chapter": 20, "status": "confirmed", "claim": "For the logistic map, r=2 has attracting fixed point 1/2; for 1 0 there is a δ > 0 such that every finite collection of pairwise disjoint subintervals of total length under δ has Σ|f(bᵢ) − f(aᵢ)| < ε. A function on [a, b] is absolutely continuous exactly when it is the indefinite Lebesgue integral of an L¹ function (a condition stronger than uniform continuity), and that equivalence is the fundamental theorem of Lebesgue integral calculus, due to Lebesgue. The Cantor function is continuous and of bounded variation but not absolutely continuous, which is why it is differentiable almost everywhere with derivative zero and still climbs from 0 to 1. The Banach–Zaretsky theorem characterizes absolute continuity as continuity together with bounded variation and the Luzin N property.", "sources": [ { "title": "Absolute continuity, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Absolute_continuity", "own_work": false, "note": "read as raw HTML: gives the ε–δ definition over finite collections of pairwise disjoint intervals, states that the fundamental theorem of calculus holds for absolutely continuous functions and is yet another characterization of them, and gives the Cantor ternary function as the notable example that is differentiable almost everywhere and still fails it.", "fetched": [ "2026-09-17" ] }, { "title": "Absolute continuity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Absolute_continuity", "own_work": false, "note": "read as raw HTML: names the equivalence of the ε–δ condition with f(x) = f(a) + ∫ₐˣ g the “fundamental theorem of Lebesgue integral calculus, due to Lebesgue,” lists the Cantor function as uniformly continuous, of bounded variation and not absolutely continuous, and states the Banach–Zareckiĭ characterization; its opening calls absolute continuity “a regularity property of functions that is stronger than uniform continuity.”", "fetched": [ "2026-09-17" ] }, { "title": "Absolutely Continuous, MathWorld", "url": "https://mathworld.wolfram.com/AbsolutelyContinuous.html", "own_work": false, "note": "the companion notion for measures, and the Radon–Nikodym theorem that turns it into an integral.", "fetched": [ "2026-09-17" ] }, { "title": "Cantor function, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cantor_function", "own_work": false, "note": "read as raw wikitext: the function is “non-decreasing,” and though it has zero derivative almost everywhere it “goes from 0 to 1 as x goes from 0 to 1, and takes on every value in between.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-20.26", "url": "https://www.teclamairme.com/#r-20.26", "chapter": 20, "status": "confirmed", "claim": "A subset is dense when its closure is the whole space, equivalently when the interior of its complement is empty; the rationals are dense in the reals. A subset is nowhere dense when the closure has empty interior, equivalently when the complement of its closure is dense. The Baire category theorem says that in a complete metric space the complement of a countable union of nowhere dense sets is never empty. The rationals are dense and null; the Cantor set is nowhere dense and null.", "sources": [ { "title": "Nowhere-dense set, Encyclopedia of Mathematics", "url": "https://encyclopediaofmath.org/wiki/Nowhere-dense_set", "own_work": false, "note": "read as raw HTML: gives the equivalent definitions — “The closure of A has empty interior” and “The complement of the closure of A is dense” — and states the Baire category theorem in the form used here.", "fetched": [ "2026-09-17" ] }, { "title": "Dense set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dense_set", "own_work": false, "note": "read as raw HTML: cl_X A = X and int_X(X ∖ A) = ∅ as equivalent definitions, the rationals as the standard dense subset of the reals, and nowhere dense as “the interior of its closure is empty.”", "fetched": [ "2026-09-17" ] }, { "title": "Null set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Null_set", "own_work": false, "note": "read as raw wikitext: “all countable sets are null. In particular, the set ℚ of rational numbers is a null set, despite being dense in ℝ.”", "fetched": [ "2026-09-22" ] }, { "title": "Cantor set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cantor_set", "own_work": false, "note": "read as raw wikitext: “Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense”; “the Cantor set is uncountable but has Lebesgue measure 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-20.27", "url": "https://www.teclamairme.com/#r-20.27", "chapter": 20, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Volterra’s nowhere-dense set of positive measure, dated 1881.", "sources": [ { "title": "Smith–Volterra–Cantor set, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Smith%E2%80%93Volterra%E2%80%93Cantor_set", "own_work": false, "note": "read raw: “In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-21.1", "url": "https://www.teclamairme.com/#r-21.1", "chapter": 21, "status": "confirmed", "claim": "Euler’s formula is e^(iθ) = cos θ + i sin θ, and θ = π gives Euler’s identity e^(iπ) + 1 = 0; it was published in Euler’s Introductio in analysin infinitorum (1748); Roger Cotes had given the equivalent logarithmic form in 1714. Multiplying by a number of modulus 1 adds its argument and leaves the modulus unchanged.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §4.21", "url": "https://dlmf.nist.gov/4.21", "own_work": false, "note": "gives the complex trigonometric identities and de Moivre theorem; the θ = π substitution is direct arithmetic.", "fetched": [ "2026-09-09" ] }, { "title": "The Euler Archive, E101: Introductio in analysin infinitorum, volume 1", "url": "https://scholarlycommons.pacific.edu/euler-works/101/", "own_work": false, "note": "identifies Euler’s Introductio in analysin infinitorum, in which he “lays the foundations of modern mathematical analysis,” with publication date 1748, and links a free scan of the text.", "fetched": [ "2026-09-22" ] }, { "title": "Euler’s formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_formula", "own_work": false, "note": "dates the publication of e^(ix) = cos x + i sin x to the Introductio of 1748 and records Roger Cotes’s 1714 Logometria, whose geometrical argument gives ix = ln(cos x + i sin x).", "fetched": [ "2026-09-09" ] }, { "title": "Complex number, Wikipedia (multiplication in polar form)", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "read as raw wikitext: in polar form “the absolute values are multiplied and the arguments are added to yield the polar form of the product.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-21.2", "url": "https://www.teclamairme.com/#r-21.2", "chapter": 21, "status": "judgement", "claim": "Euler’s identity links five fundamental constants and has twice headed a published poll — the Mathematical Intelligencer’s of 1990 and Physics World’s of 2004 — and the Benjamin Peirce quotation comes from notes taken by his student W. E. Byerly.", "sources": [ { "title": "Euler’s identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_identity", "own_work": false, "note": "records the Peirce quotation and attributes it to a Harvard lecture; it also reports descriptions of the identity as famous/beautiful; its bibliography identifies Kasner and Newman (1940) as a source for the quotation, and it records the 1990 Mathematical Intelligencer poll naming the identity the most beautiful theorem in mathematics and the 2004 Physics World poll in which it tied with Maxwell’s equations as the greatest equation ever.", "fetched": [ "2026-09-09" ] }, { "title": "Benjamin Peirce, Wikiquote", "url": "https://en.wikiquote.org/wiki/Benjamin_Peirce", "own_work": false, "note": "sources the quotation to notes by W. E. Byerly published in R. C. Archibald, Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (1925), and again in Kasner and Newman (1940), and attaches it to Euler’s identity.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The card used to say the identity is voted the most beautiful equation whenever anyone holds a vote. No source supports a claim that broad, so we replaced it with the two polls that are actually on the record, 1990 and 2004. The quotation’s chain of transmission is now traced to its earliest printing — Byerly’s notes in Archibald (1925) — but Byerly was writing down a lecture from memory and no verbatim transcript survives, so the exact wording stays a judgement rather than a quotation from the lecturer’s own hand." }, { "id": "r-21.3", "url": "https://www.teclamairme.com/#r-21.3", "chapter": 21, "status": "confirmed", "claim": "Complex conjugation sends a + bi to a − bi, and z times its conjugate equals a² + b² = |z|²; hence i times its conjugate is 1 and the modulus of e^(iθ) is 1.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §1.9(i), Complex Numbers", "url": "https://dlmf.nist.gov/1.9#i", "own_work": false, "note": "the conjugate, “1.9.11 z̄ = x − iy,” and “1.9.12 |z̄| = |z|”; the identities follow by substituting z = a + bi and i (the line formerly pointed to §4.21(iv), which is about trigonometric functions of a complex argument).", "fetched": [ "2026-09-24" ] }, { "title": "Complex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "confirms conjugation, modulus, and the geometric interpretation of multiplication.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-21.4", "url": "https://www.teclamairme.com/#r-21.4", "chapter": 21, "status": "confirmed", "claim": "The Gaussian integers Z[i] have norm N(a+bi)=a²+b²; N(1+i)=2, and 2 factors as −i(1+i)², so 2 is not prime in Z[i]. Its units, the elements whose inverse is again a Gaussian integer, are 1, −1, i and −i. Primes congruent to 1 modulo 4 split while primes congruent to 3 modulo 4 remain Gaussian prime up to associates. Gauss introduced the ring in his second monograph on quartic (biquadratic) reciprocity, 1832.", "sources": [ { "title": "Gaussian integer, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gaussian_integer", "own_work": false, "note": "confirms the Gaussian-integer norm, its multiplicativity, the factorization of 2, and the split/inert behavior by congruence modulo 4; its opening sentence attributes the ring to Gauss’s second monograph on quartic reciprocity (1832), while its own reference list dates that monograph 1831; “The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, −1, i and −i.”", "fetched": [ "2026-09-09" ] }, { "title": "Carl Friedrich Gauss, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/", "own_work": false, "note": "links “An extract from Theoria residuorum biquadraticorum (1828 - 32),” a span consistent with a second memoir printed in 1832; the page does not date the two memoirs separately.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-21.5", "url": "https://www.teclamairme.com/#r-21.5", "chapter": 21, "status": "confirmed", "claim": "For real θ, e^(iθ) lies on the unit circle and multiplication by it rotates without changing modulus. In polar form the moduli multiply and the arguments add.", "sources": [ { "title": "Euler’s formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_formula", "own_work": false, "note": "gives e^(iθ)=cos θ+i sin θ, says e^(iφ) “traces out the unit circle in the complex plane as φ ranges through the real numbers,” and says “The polar form simplifies the mathematics when used in multiplication or powers of complex numbers”; the polar product rule on the Complex number line below is what makes multiplication by a unit factor a rotation in the complex plane.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions, §4.21(iv)", "url": "https://dlmf.nist.gov/4.21#iv", "own_work": false, "note": "supplies the modulus identities; cos²θ+sin²θ=1 gives the stated value.", "fetched": [ "2026-09-09" ] }, { "title": "Complex number, Wikipedia (multiplication in polar form)", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "read as raw wikitext: in polar form “the absolute values are multiplied and the arguments are added to yield the polar form of the product.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-21.6", "url": "https://www.teclamairme.com/#r-21.6", "chapter": 21, "status": "confirmed", "claim": "Split-complex numbers use a unit j with j²=+1 and j≠±1; they are associated with Cockle’s tessarines and have a hyperbola/Lorentz-boost interpretation. They are one of exactly three two-dimensional unital algebras over the reals, the others being the complex numbers (i²=−1) and the dual numbers (ε²=0); Clifford used them as the coefficients of his split-biquaternions in 1873. Within Cockle’s tessarines they are the subalgebra of real tessarines w + yj. Their squared modulus is x² − y² and is multiplicative, and its level sets are hyperbolas, so multiplying by a number of squared modulus 1 keeps each point on its hyperbola.", "sources": [ { "title": "Split-complex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Split-complex_number", "own_work": false, "note": "confirms the defining relation, the historical association with tessarines (Cockle, 1848), the unit hyperbola, and the sentence ‘Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane’; “the product of a number z with its conjugate is N(z) := zz∗ = x² − y²,” “It has the composition algebra property: ‖zw‖ = ‖z‖‖w‖,” and “The set of points {z : ‖z‖² = a²} is a hyperbola for every nonzero a.”", "fetched": [ "2026-09-09" ] }, { "title": "Hypercomplex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hypercomplex_number", "own_work": false, "note": "‘Up to isomorphism, there are exactly three 2-dimensional unital algebras over the reals: the ordinary complex numbers, the split-complex numbers, and the dual numbers’, and notes that every such algebra is associative and commutative.", "fetched": [ "2026-09-09" ] }, { "title": "Split-biquaternion, Wikipedia", "url": "https://en.wikipedia.org/wiki/Split-biquaternion", "own_work": false, "note": "‘This algebra was introduced by William Kingdon Clifford in an 1873 article for the London Mathematical Society’, with split-complex coefficients.", "fetched": [ "2026-09-09" ] }, { "title": "Bicomplex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Bicomplex_number", "own_work": false, "note": "read as raw wikitext: “The tessarines are now best known for their subalgebra of real tessarines t = w + y j, also called split-complex numbers, which express the parametrization of the unit hyperbola.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-21.7", "url": "https://www.teclamairme.com/#r-21.7", "chapter": 21, "status": "confirmed", "claim": "Dual numbers have ε²=0 with ε≠0; their first-order algebra gives f(x+ε)=f(x)+f′(x)ε, which underlies forward-mode automatic differentiation. William Clifford introduced them in 1873.", "sources": [ { "title": "Dual number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Dual_number", "own_work": false, "note": "confirms the dual-number definition, nilpotent ε, the derivative expansion, and use in automatic differentiation, and opens ‘Dual numbers were introduced in 1873 by William Clifford’.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-21.8", "url": "https://www.teclamairme.com/#r-21.8", "chapter": 21, "status": "judgement", "claim": "The complex plane identifies a+bi with (a, b); Caspar Wessel first described a complex number as a point of the plane, in a memoir presented to the Copenhagen Academy in 1797 and published in 1799; Jean-Robert Argand independently issued a pamphlet in 1806, with a further paper in 1814; Gauss is dated at 21.4, which rules on 1831 for the announcement and 1832 for the printing; and complex arithmetic cannot be given an order compatible with field arithmetic, though it keeps the associative, commutative and distributive laws and division by nonzero numbers. Wessel’s paper was presented to the Royal Danish Academy in 1797; Gauss’s announcement of the second memoir appeared in the Göttingische gelehrte Anzeigen of 23 April 1831. Gauss’s 1831 work largely established modern notation and terminology, the term complex number included.", "sources": [ { "title": "Complex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "confirms the coordinate-plane model, the Wessel/Argand/Gauss history, and the non-orderability of the complex field. Re-read raw on 17 September 2026 for the years: “The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799 … Wessel’s memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers,” and the page quotes the Encyclopedia of Mathematics: “The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the ‘Argand diagram’, came into use after the publication in 1806 and 1814 of papers by J. R. Argand”; of Gauss: “It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology,” and he “introduced the term complex number for a + bi”; the complex numbers support “the same basic arithmetic operations of addition, subtraction, multiplication, and division, satisfying the same associative, commutative, and distributive laws,” and “Unlike the reals, C is not an ordered field … in any ordered field, the square of any element is necessarily positive, so i² = −1 precludes the existence of an ordering on C.”", "fetched": [ "2026-09-09" ] }, { "title": "Euler’s formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/Euler%27s_formula", "own_work": false, "note": "confirms the historical description of Wessel’s complex-plane representation.", "fetched": [ "2026-09-09" ] }, { "title": "Caspar Wessel, Wikipedia", "url": "https://en.wikipedia.org/wiki/Caspar_Wessel", "own_work": false, "note": "read as raw wikitext: “His fundamental paper, Om directionens analytiske betegning (On the Analytical Representation of Direction), was presented in 1797 to the Royal Danish Academy of Sciences and Letters.”", "fetched": [ "2026-09-22" ] }, { "title": "Deutsches Textarchiv, TEI header of Gauß, Anzeige von Theoria residuorum biquadraticorum, commentatio secunda", "url": "https://www.deutschestextarchiv.de/api/tei_header/gauss_theoria_1831", "own_work": false, "note": "the archive’s record, served as plain XML: “Göttingische gelehrte Anzeigen, 23. April 1831.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "The broad historical sequence is supported, but exact priority language such as ‘made respectable’ and ‘fixed the modern view’ is interpretive, so we record it as a judgement call." }, { "id": "r-21.9", "url": "https://www.teclamairme.com/#r-21.9", "chapter": 21, "status": "confirmed", "claim": "De Moivre’s formula (cos θ+i sin θ)^n=cos(nθ)+i sin(nθ) gives (cos(π/3)+i sin(π/3))^6=1 and describes the sixth roots of unity on a regular hexagon. De Moivre worked with the relation from 1707 onward, including in the Miscellanea Analytica of 1730. Sources differ on whether he wrote it in this form: MacTutor says it appears so in his paper of 1722, the De Moivre’s formula article says he never stated it; corrections welcome. It follows from Euler’s formula and the exponential law.", "sources": [ { "title": "NIST Digital Library of Mathematical Functions, §4.21, De Moivre’s Theorem", "url": "https://dlmf.nist.gov/4.21", "own_work": false, "note": "equation 4.21.34 states de Moivre’s theorem for integer n; substituting θ=π/3 and n=6 gives cos 2π+i sin 2π=1.", "fetched": [ "2026-09-09" ] }, { "title": "Roots of unity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Root_of_unity", "own_work": false, "note": "confirms the geometric regular-polygon description of roots of unity.", "fetched": [ "2026-09-09" ] }, { "title": "De Moivre’s formula, Wikipedia", "url": "https://en.wikipedia.org/wiki/De_Moivre%27s_formula", "own_work": false, "note": "names the formula for de Moivre ‘although he never stated it in his works’, cites his 1707 paper and the Miscellanea Analytica of 1730, where he considers cos θ and cos nθ explicitly, and derives the formula from Euler’s.", "fetched": [ "2026-09-09" ] }, { "title": "Abraham de Moivre, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/De_Moivre/", "own_work": false, "note": "on the formula for (cos x + i sin x)ⁿ: “It appears in this form in a paper which de Moivre published in 1722, but a closely related formula had appeared in an earlier paper which de Moivre published in 1707.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-21.10", "url": "https://www.teclamairme.com/#r-21.10", "chapter": 21, "status": "confirmed", "claim": "The fundamental theorem of algebra gives n complex roots counted with multiplicity for a degree-n polynomial; z²+1 has the two roots i and −i. d’Alembert attempted a proof in 1746; Gauss’s 1799 proof had a topological gap that Ostrowski filled only in 1920; Argand’s proof of 1806 was the first rigorous one and the first statement of the theorem for complex rather than real coefficients. Gauss’s 1799 proof was his doctoral dissertation (Helmstedt).", "sources": [ { "title": "Fundamental theorem of algebra, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra", "own_work": false, "note": "states existence of a complex root and the degree-counting formulation with multiplicity, and its history section gives ‘A first attempt at proving the theorem was made by d’Alembert in 1746’, Gauss’s 1799 proof with ‘a topological gap, only filled by Alexander Ostrowski in 1920’, and ‘The first rigorous proof was published by Argand … in 1806 … it was also here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients’.", "fetched": [ "2026-09-09" ] }, { "title": "NIST Digital Library of Mathematical Functions, §1.11", "url": "https://dlmf.nist.gov/1.11", "own_work": false, "note": "provides standard polynomial/root background; z²+1=(z−i)(z+i) is direct algebra.", "fetched": [ "2026-09-09" ] }, { "title": "Carl Friedrich Gauss, MacTutor History of Mathematics", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/", "own_work": false, "note": "the Duke of Brunswick “requested that Gauss submit a doctoral dissertation to the University of Helmstedt”; “Gauss’s dissertation was a discussion of the fundamental theorem of algebra.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-21.11", "url": "https://www.teclamairme.com/#r-21.11", "chapter": 21, "status": "confirmed", "claim": "arg(i)=π/2 under the principal argument in (−π, π], and π/2 is transcendental because π is transcendental and nonzero rational scaling preserves transcendence.", "sources": [ { "title": "Complex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "confirms the argument convention and that i lies at angle π/2.", "fetched": [ "2026-09-09" ] }, { "title": "Lindemann–Weierstrass theorem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "records the transcendence of π; if π/2 were algebraic, multiplying by 2 would make π algebraic.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-21.12", "url": "https://www.teclamairme.com/#r-21.12", "chapter": 21, "status": "judgement", "claim": "The imaginary unit satisfies i²=−1; the historical card’s dates for Cardano, Descartes, Euler, Wessel, Argand, and Gauss describe the development of imaginary numbers and their geometric representation. Gauss in 1831 introduced the term complex number and suggested that direct, inverse and lateral units would have been better names than positive, negative and imaginary.", "sources": [ { "title": "Complex number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Complex_number", "own_work": false, "note": "confirms the definition and gives the standard historical account of imaginary/complex numbers and their representations; “The term “imaginary” for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The elementary definition is confirmed. The many exact historical attributions in the card are compressed chronology, and the fetched overview does not establish every individual date and priority claim at the wording’s precision; we leave them as a judgement call rather than silently upgrading them." }, { "id": "r-21.13", "url": "https://www.teclamairme.com/#r-21.13", "chapter": 21, "status": "confirmed", "claim": "The extended complex plane adjoins one point at infinity, is conformally represented by the Riemann sphere, and Möbius transformations act on it; 1/0 is represented as infinity in the extended setting. The idea is Riemann’s, of 1857.", "sources": [ { "title": "Riemann sphere, Wikipedia", "url": "https://en.wikipedia.org/wiki/Riemann_sphere", "own_work": false, "note": "confirms the one-point compactification, stereographic sphere model, Möbius transformations, and the extended-value convention 1/0 = ∞, and cites Riemann’s Theorie der Abel’schen Functionen of 1857 as the source of the construction.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-21.14", "url": "https://www.teclamairme.com/#r-21.14", "chapter": 21, "status": "judgement", "claim": "The two-square identity is a norm-composition identity associated with Diophantus, Brahmagupta, and Fibonacci; it is the two-dimensional member of the composition-of-squares story. Brahmagupta’s identity is in the Brāhmasphuṭasiddhānta, written c. 628; Degen found the eight-square identity around 1818; Hurwitz proved in 1898 that such formulas exist only in 1, 2, 4 and 8 dimensions.", "sources": [ { "title": "Brahmagupta’s identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Brahmagupta%27s_identity", "own_work": false, "note": "confirms the identity and its connection with sums of two squares and Brahmagupta.", "fetched": [ "2026-09-09" ] }, { "title": "Four-square identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Four-square_identity", "own_work": false, "note": "redirects to Euler’s four-square identity, which dates Euler’s letter to Goldbach 4 May 1748 and states that such an identity is possible only for n = 1, 2, 4 or 8 by Hurwitz’s theorem.", "fetched": [ "2026-09-09" ] }, { "title": "Brahmagupta–Fibonacci identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity", "own_work": false, "note": "gives the identity, its first appearance in Diophantus’s Arithmetica (III, 19), Brahmagupta’s rediscovery and generalization to Pell’s equation, its introduction to western Europe by Fibonacci in the Book of Squares of 1225, the multiplicativity of the complex modulus it expresses, and the continuation through Euler’s four squares, Degen’s eight squares and Hurwitz’s classification, without their years; read again on 23 September 2026, the page carries neither 1818 nor 1898, which the Degen and Hurwitz lines carry; it dates Diophantus: “The identity first appeared in Diophantus’ Arithmetica (III, 19), of the third century A.D.”", "fetched": [ "2026-09-09" ] }, { "title": "Brāhmasphuṭasiddhānta, Wikipedia", "url": "https://en.wikipedia.org/wiki/Br%C4%81hmasphu%E1%B9%ADasiddh%C4%81nta", "own_work": false, "note": "read as raw wikitext: the work “is a main work of Brahmagupta, written c. 628,” and its contents include “Brahmagupta’s identity.”", "fetched": [ "2026-09-22" ] }, { "title": "Degen’s eight-square identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity", "own_work": false, "note": "read as raw wikitext: “First discovered by Carl Ferdinand Degen around 1818.”", "fetched": [ "2026-09-22" ] }, { "title": "Hurwitz’s theorem (composition algebras), Wikipedia", "url": "https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)", "own_work": false, "note": "read as raw wikitext: “multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The algebraic identity, its norm interpretation and the whole chain Diophantus → Brahmagupta → Fibonacci → Euler → Degen → Hurwitz are confirmed by the Brahmagupta–Fibonacci identity article. What that article does not carry is the year 628 for Brahmagupta, which the Brāhmasphuṭasiddhānta line now carries (the work was “written c. 628” and contains Brahmagupta’s identity)." }, { "id": "r-21.15", "url": "https://www.teclamairme.com/#r-21.15", "chapter": 21, "status": "confirmed", "claim": "The chapter’s mixed-type calculation is internally consistent: the numerator is three zeros, from ℂ, from the dual numbers and from a real-coordinate reading, each read as the scalar 0, and the denominator is the positive real 8+π/2, and the factor therefore returns the ordinary real integer 0.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Using Euler’s identity, ε²=0, Re(i)=0, i·conjugate(i)=1, (1+i)(1−i)=2, |e^(iθ)|=1, j²=1, de Moivre=1, and #{z:z²+1=0}=2 gives numerator 0 and denominator 1+2+1+1+1+2+π/2=8+π/2>0; the quotient is 0.", "fetched": [] } ], "editorial_note": "This explicitly checks the Part 2 mixed-type note: the algebra is performed in the stated systems, then the result comes home as an ordinary real number." }, { "id": "r-21.16", "url": "https://www.teclamairme.com/#r-21.16", "chapter": 21, "status": "confirmed", "claim": "Bombelli’s L’algebra (1572) supplied multiplication rules for square roots of negative numbers and used them to solve particular cases of the irreducible cubic, obtaining its three real roots. Cardano called such a solution sophistic, ‘as subtle as it is useless’.", "sources": [ { "title": "Veronica Gavagna, Rafael Bombelli, Treccani, Il Contributo italiano alla storia del Pensiero: Scienze", "url": "https://www.treccani.it/enciclopedia/rafael-bombelli_(Il-Contributo-italiano-alla-storia-del-Pensiero:-Scienze)/", "own_work": false, "note": "dates L’algebra to 1572 and explains that Cardano had encountered the irreducible cubic earlier, while Bombelli introduced new signs and multiplication rules for the negative-root quantities and thereby obtained the three real roots in particular cases.", "fetched": [ "2026-09-11" ] }, { "title": "Ars Magna (Cardano book), Wikipedia", "url": "https://en.wikipedia.org/wiki/Ars_Magna_(Cardano_book)", "own_work": false, "note": "read as raw wikitext: Cardano called the solution “sophistic,” and “Cardano then says that this answer is “as subtle as it is useless,”” the answer being 5 ± √−15; and “The Ars Magna (The Great Art, 1545) is an important Latin-language book on algebra written by Gerolamo Cardano.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added in the V14 prose pass. The prose had called Bombelli the first person to need the trick, while the chapter’s own imaginary-unit card says Cardano had already computed with such roots in 1545. Treccani distinguishes the two episodes: Cardano encountered the obstacle; Bombelli made an arithmetic of the quantities and solved particular irreducible cubics. The prose now says Bombelli made the trick work, not that he encountered it first." }, { "id": "r-21.17", "url": "https://www.teclamairme.com/#r-21.17", "chapter": 21, "status": "confirmed", "claim": "The chapter epigraph is Benjamin Peirce’s remark to his Harvard class on Euler’s identity, from notes by his student W. E. Byerly published in R. C. Archibald’s Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography of 1925, and also in Kasner and Newman’s Mathematics and the Imagination of 1940.", "sources": [ { "title": "Benjamin Peirce, Wikiquote", "url": "https://en.wikiquote.org/wiki/Benjamin_Peirce", "own_work": false, "note": "read as raw HTML: “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth. On Euler’s identity, e^(iπ) + 1 = 0. as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman,” which is the footer line word for word.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "Added 17 September 2026 to source the chapter epigraph, which carried no marker. The last of the four secondhand ones: what survives is a student’s notes printed forty-five years later, and the claim line says so." }, { "id": "r-21.18", "url": "https://www.teclamairme.com/#r-21.18", "chapter": 21, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Lindemann proved pi transcendental in 1882; Descartes coined the word ‘imaginary’ in La Géométrie (1637); Newton was using the word ‘imaginary’ by 1670; Euler introduced the letter i in a paper of 1777, printed 1794.", "sources": [ { "title": "Lindemann–Weierstrass theorem, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Lindemann%E2%80%93Weierstrass_theorem", "own_work": false, "note": "read raw: “Lindemann proved in 1882 that e^α is transcendental for every non-zero algebraic number α, thereby establishing that π is transcendental (see below)..”", "fetched": [ "2026-09-18" ] }, { "title": "Earliest Known Uses of Some of the Words of Mathematics (I), Jeff Miller, hosted by MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathword/i/", "own_work": false, "note": "read raw: “The terms IMAGINARY and REAL were introduced in French by Rene Descartes (1596-1650) in “La Geometrie” (1637):.”", "fetched": [ "2026-09-18" ] }, { "title": "Imaginary unit, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Imaginary_unit", "own_work": false, "note": "read raw: “The name imaginary is generally credited to René Descartes, and Isaac Newton used the term as early as 1670..”", "fetched": [ "2026-09-18" ] }, { "title": "Earliest Uses of Symbols for Constants, Jeff Miller, hosted by MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/constants/", "own_work": false, "note": "read raw: “This was first used by Leonhard Euler (1707-1783) in a memoir presented in 1777 but not published until 1794 in his “Institutionum calculi integralis..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-21.20", "url": "https://www.teclamairme.com/#r-21.20", "chapter": 21, "status": "confirmed", "claim": "States Fermat’s theorem on sums of two squares, that an odd prime is a sum of two squares exactly when it is congruent to 1 mod 4, stated by Fermat around 1640 and first proved by Euler, who announced his proof in 1747 and 1749 and published it between 1752 and 1755. Its smallest examples are 5 and 13 = 2² + 3², and the representation is unique up to order and sign.", "sources": [ { "title": "Fermat’s theorem on sums of two squares, Wikipedia", "url": "https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares", "own_work": false, "note": "states the theorem and its p ≡ 1 (mod 4) condition, dates Fermat’s claim to 1640, and says Euler announced a proof in letters to Goldbach of 1747 and 1749, succeeded in 1749 and published it in 1752–55; “Fermat’s theorem on sums of two squares is strongly related with the theory of Gaussian primes,” and “a prime number p is either a Gaussian prime or the norm of a Gaussian prime. Fermat’s theorem asserts that the first case occurs when p = 4k + 3, and that the second case occurs when p = 4k + 1 and p = 2.”", "fetched": [ "2026-09-22" ] }, { "title": "Pick’s theorem and sums of two squares, Keith Conrad (UConn)", "url": "https://kconrad.math.uconn.edu/blurbs/ugradnumthy/Picksumofsq.pdf", "own_work": false, "note": "states uniqueness: “Furthermore, a representation of a prime as a2 +b2 in Z is unique up to the order and signs of a and b.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-22.1", "url": "https://www.teclamairme.com/#r-22.1", "chapter": 22, "status": "confirmed", "claim": "Hamilton’s quaternion discovery occurred on 16 October 1843 while walking with his wife along Dublin’s Royal Canal; he cut i² = j² = k² = ijk = −1 into Brougham/Broom Bridge, and later described the event in his 5 August 1865 letter to his son Archibald, in which he records that the knife-cut inscription ‘has long since mouldered away’. A stone plaque on the bridge carries the formula in its place, and mathematicians walk each year from Dunsink Observatory to the bridge to mark the discovery.", "sources": [ { "title": "Trinity College Dublin, Letters describing the Discovery of Quaternions", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html", "own_work": false, "note": "the letter is dated 5 August 1865; Hamilton describes the breakfast ‘multiply triplets’ question, the 16 October walk with his wife, the formula, the bridge carving, and the later fading of the inscription.", "fetched": [ "2026-09-09" ] }, { "title": "Trinity College Dublin, Hamilton’s Research on Quaternions", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quaternions.html", "own_work": false, "note": "confirms the Royal Canal walk, 16 October 1843, the formula, and the Broom Bridge inscription.", "fetched": [ "2026-09-09" ] }, { "title": "Broom Bridge", "url": "https://en.wikipedia.org/wiki/Broom_Bridge", "own_work": false, "note": "gives the bridge’s three names (Broom, Broome and Brougham), quotes the plaque — ‘Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = ijk = −1 & cut it on a stone of this bridge’ — notes that the plaque was moved higher under the railing after vandalism, and records the annual commemorative walk from Dunsink Observatory to the site.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.2", "url": "https://www.teclamairme.com/#r-22.2", "chapter": 22, "status": "confirmed", "claim": "The quaternion algebra is four-dimensional over the reals, generated by 1, i, j, k with i² = j² = k² = ijk = −1; it is associative and division is noncommutative, with every nonzero quaternion invertible.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "gives the four-dimensional basis, multiplication rules, noncommutativity, associativity, and the conjugate-over-norm inverse for every nonzero quaternion.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.3", "url": "https://www.teclamairme.com/#r-22.3", "chapter": 22, "status": "confirmed", "claim": "For q = a + bi + cj + dk, quaternion conjugation is q̄ = a − bi − cj − dk and q q̄ = q̄ q = a² + b² + c² + d² = ‖q‖²; the norm is multiplicative.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "defines conjugation, the Euclidean norm, multiplicativity, and the inverse formula.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.4", "url": "https://www.teclamairme.com/#r-22.4", "chapter": 22, "status": "confirmed", "claim": "A unit quaternion q = cos(θ/2) + u sin(θ/2) rotates a vector v through v ↦ qvq⁻¹; q = (1 + k)/√2 is the 90-degree rotation about the k-axis and sends i to j.", "sources": [ { "title": "Quaternions and spatial rotation", "url": "https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation", "own_work": false, "note": "gives the axis-angle unit quaternion cos(θ/2)+u sin(θ/2), the conjugation map qvq⁻¹, and the use of unit quaternions for 3D rotations.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "With q=(1+k)/√2 and q⁻¹=(1−k)/√2, q i q⁻¹ = ((1+k)i(1−k))/2 = j under ij=k, ji=−k, k i=j, i k=−j.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-22.5", "url": "https://www.teclamairme.com/#r-22.5", "chapter": 22, "status": "confirmed", "claim": "The eight elements ±1, ±i, ±j, ±k form the quaternion group Q₈ of order 8; it is non-abelian. Of the five groups of order eight only Q₈ and the dihedral symmetry group of the square are non-abelian, and Q₈ is the smallest Hamiltonian group — a non-abelian group every subgroup of which is normal — a class Richard Dedekind investigated in 1897 and named after Hamilton.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "identifies the positive and negative basis vectors as the eight-element quaternion group and displays ij=k while ji=−k.", "fetched": [ "2026-09-09" ] }, { "title": "Quaternion group", "url": "https://en.wikipedia.org/wiki/Quaternion_group", "own_work": false, "note": "defines Q₈ as the non-abelian group of order eight with elements ±1, ±i, ±j, ±k.", "fetched": [ "2026-09-09" ] }, { "title": "List of small groups", "url": "https://en.wikipedia.org/wiki/List_of_small_groups", "own_work": false, "note": "lists the five groups of order eight — Z₈, Z₄ × Z₂, Z₂³, the dihedral group D₄ and Q₈ — of which only D₄ and Q₈ are non-abelian.", "fetched": [ "2026-09-09" ] }, { "title": "Dedekind group", "url": "https://en.wikipedia.org/wiki/Dedekind_group", "own_work": false, "note": "‘A non-abelian Dedekind group is called a Hamiltonian group’; Dedekind investigated these groups in 1897 and named the non-abelian ones after William Rowan Hamilton, and Q₈ is given as the smallest example.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.6", "url": "https://www.teclamairme.com/#r-22.6", "chapter": 22, "status": "confirmed", "claim": "Unit quaternions are used for spatial orientation and rotations in applications including computer graphics, robotics, navigation, flight dynamics and spacecraft attitude control, and avoid gimbal lock associated with Euler-angle systems. Quaternions have had a revival since the late twentieth century, chiefly for describing spatial rotations.", "sources": [ { "title": "Quaternions and spatial rotation", "url": "https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation", "own_work": false, "note": "lists the applications — computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites and crystallographic texture analysis — and gives the axis-angle form and the composition rule; read again on 23 September 2026 it also says “Quaternions also avoid a phenomenon called gimbal lock.” Read on 24 September 2026, it says of Euler angles that “any such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock. We can avoid this by using four Euclidean coordinates”; “In video games and other applications, one is often interested in “smooth rotations””; and, of the composition rule, “This is Rodrigues’ rotation formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840.”", "fetched": [ "2026-09-09" ] }, { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "‘quaternions have had a revival since the late 20th century, primarily due to their utility in describing spatial rotations’, lists the applications in graphics, robotics and attitude control, and states that ‘unlike Euler angles, they are not susceptible to gimbal lock’. This is the source for the gimbal-lock claim; it goes on “For this reason, quaternions are used in computer graphics,” and “it is common for the attitude control systems of spacecraft to be commanded in terms of quaternions.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.7", "url": "https://www.teclamairme.com/#r-22.7", "chapter": 22, "status": "confirmed", "claim": "Euler’s four-square identity predates Hamilton — he wrote it out in a letter to Goldbach of 4 May 1748 — and Lagrange used it in 1770 to prove that every positive integer is a sum of four squares. The quaternion proof of that theorem runs through the Hurwitz quaternions, those whose four coordinates are all integers or all half-integers, whose norm satisfies a Euclidean property and so admits division with a small remainder. Hurwitz published this number theory in the Göttingen Nachrichten of 1896 and expounded it in his Vorlesungen of 1919.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "identifies Euler’s four-square identity as a precursor: “Important precursors to this work included Euler’s four-square identity (1748),” and states “the norm is multiplicative”; the reading of the identity as that multiplicativity is on the Euler’s four-square identity line.", "fetched": [ "2026-09-09" ] }, { "title": "Four-square theorem", "url": "https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem", "own_work": false, "note": "states Lagrange’s theorem that every nonnegative integer is a sum of four squares, dates the proof to 1770, and carries a ‘Proof using the Hurwitz integers’ section resting on the Euclidean property of the Hurwitz norm.", "fetched": [ "2026-09-09" ] }, { "title": "Euler’s four-square identity", "url": "https://en.wikipedia.org/wiki/Euler%27s_four-square_identity", "own_work": false, "note": "‘Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach’, reads the identity as the multiplicativity of the quaternion absolute value, and records that Lagrange used it to prove the four-square theorem.", "fetched": [ "2026-09-09" ] }, { "title": "Hurwitz quaternion", "url": "https://en.wikipedia.org/wiki/Hurwitz_quaternion", "own_work": false, "note": "defines a Hurwitz quaternion as one whose components are either all integers or all half-integers, credits Adolf Hurwitz with introducing them and cites his Vorlesungen über die Zahlentheorie der Quaternionen of 1919, and states that Euclidean division is possible on them.", "fetched": [ "2026-09-09" ] }, { "title": "Adolf Hurwitz, Vorlesungen über die Zahlentheorie der Quaternionen (1919), Vorwort", "url": "https://archive.org/stream/vorlesungenber00hurwuoft/vorlesungenber00hurwuoft_djvu.txt", "own_work": false, "note": "Hurwitz opens the preface “Das vorliegende kleine Werk hat den Zweck, die Zahlentheorie der Quaternionen, wie ich sie in den Nachrichten von der Königl. Gesellschaft der Wissenschaften zu Göttingen aus dem Jahre 1896 veröffentlicht habe, einem weiteren Kreise zugänglich zu machen,” and signs it ‘Zürich, Mai 1919’.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Two dates are in play for the Hurwitz integers. Wikipedia writes ‘introduced by Adolf Hurwitz (1919)’ and cites the Vorlesungen; Hurwitz’s own preface to that book says he had published the theory in the Göttingen Nachrichten of 1896. We give both, since 1896 is where it was published and 1919 is the book everyone reads." }, { "id": "r-22.8", "url": "https://www.teclamairme.com/#r-22.8", "chapter": 22, "status": "confirmed", "claim": "Gauss had found the same algebra in an unpublished 1819 note later printed in 1900, while Rodrigues described a four-parameter rotation formula in 1840; Hamilton supplied the quaternion number-system formulation. The conjugation form q v q⁻¹ reached print first in Arthur Cayley’s paper in the Philosophical Magazine of February 1845; Hamilton had given the same formula in a lecture of November 1844 and communicated it to Charles Graves in October 1844, and in print he conceded Cayley’s priority ‘as respects the printing of the formula’. Cayley published the form; he is not credited with finding it.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "identifies Gauss’s 1819 unpublished discovery and 1900 publication, and Rodrigues’s 1840 rotation parametrization, while distinguishing Hamilton’s algebraic treatment.", "fetched": [ "2026-09-09" ] }, { "title": "History of quaternions", "url": "https://en.wikipedia.org/wiki/History_of_quaternions", "own_work": false, "note": "summarizes Rodrigues’s 1840 rotation work and Hamilton’s independent 1843 discovery. It does not discuss the conjugation formula, and its 1845 Cayley reference is to octonions, not to rotations.", "fetched": [ "2026-09-09" ] }, { "title": "History of Topics in Special Relativity · Lorentz transformation (Quaternions)", "url": "https://en.wikiversity.org/wiki/History_of_Topics_in_Special_Relativity/Lorentz_transformation_(Quaternions)", "own_work": false, "note": "prints Hamilton’s (cos a/2 + α sin a/2)β(cos a/2 − α sin a/2) from the abstract of his November 1844 lecture, published 1845/47, beside Cayley’s equivalent formula in the Philosophical Magazine of February 1845, notes that Hamilton had sent his to Charles Graves in October 1844, and says that in a footnote added before printing Hamilton showed his formula equivalent to Cayley’s; read again on 23 September 2026, it does not print the words “as respects the printing of the formula” quoted in the claim.", "fetched": [ "2026-09-09" ] }, { "title": "W. R. Hamilton, On Quaternions, Proceedings of the Royal Irish Academy 3 (1847), 1–16, read 11 November 1844 (ed. D. R. Wilkins, Trinity College Dublin)", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.pdf", "own_work": false, "note": "read as extracted PDF text: the rotation formula is “as that which Mr. Cayley has deduced … and has published … in the Philosophical Magazine for February, 1845,” and Hamilton “gladly recognizes his priority, as respects the printing of the formula just now referred to.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-22.9", "url": "https://www.teclamairme.com/#r-22.9", "chapter": 22, "status": "confirmed", "claim": "The Pauli matrices were introduced in 1927 in connection with electron spin, and the matrices −iσ₁, −iσ₂, −iσ₃ satisfy the quaternion multiplication relations. Pauli formalized spin in 1927 and used the Pauli matrices to represent the spin operators.", "sources": [ { "title": "Pauli matrices", "url": "https://en.wikipedia.org/wiki/Pauli_matrices", "own_work": false, "note": "gives the three matrices, their multiplication relation, and the quaternion embedding by −iσ₁, −iσ₂, −iσ₃; it gives no year (1927 is on the Spin line below).", "fetched": [ "2026-09-09" ] }, { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "explains the embedding of quaternion units into 2×2 complex matrices, and says “in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions (in the form of the famous Pauli spin matrices).”", "fetched": [ "2026-09-09" ] }, { "title": "Spin (physics), Wikipedia", "url": "https://en.wikipedia.org/wiki/Spin_(physics)", "own_work": false, "note": "read as raw wikitext: “In 1927, Pauli formalized the theory of spin using the theory of quantum mechanics invented by Erwin Schrödinger and Werner Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-22.10", "url": "https://www.teclamairme.com/#r-22.10", "chapter": 22, "status": "confirmed", "claim": "The unit quaternions form S³ and are isomorphic to SU(2); conjugation gives a two-to-one covering of SO(3), with q and −q representing the same rotation. A spinor returns to its original state only after a 720° rotation, because of the two-to-one correspondence between SU(2) and SO(3); the Lie algebra su(2) is three-dimensional.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "states that unit quaternions form S³, are isomorphic to SU(2), and universally cover SO(3).", "fetched": [ "2026-09-09" ] }, { "title": "Quaternions and spatial rotation", "url": "https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation", "own_work": false, "note": "gives the conjugation rotation map and the axis-angle interpretation.", "fetched": [ "2026-09-09" ] }, { "title": "Spinor, Wikipedia", "url": "https://en.wikipedia.org/wiki/Spinor", "own_work": false, "note": "read as raw wikitext: “It takes a rotation of 720° for a spinor to go back to its original state.”", "fetched": [ "2026-09-22" ] }, { "title": "Pauli matrices, Wikipedia (SU(2) and su(2))", "url": "https://en.wikipedia.org/wiki/Pauli_matrices", "own_work": false, "note": "read as raw wikitext: spin-½ particles “must be rotated by an angle of 4π in order to return to their original configuration. This is due to the two-to-one correspondence between SU(2) and SO(3)”; “the Lie algebra su₂ is the three-dimensional real algebra spanned by the set {iσₖ}.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-22.11", "url": "https://www.teclamairme.com/#r-22.11", "chapter": 22, "status": "confirmed", "claim": "Hamilton introduced the term versor for a unit quaternion in the 1840s and set it out with its companion term tensor in his Lectures on Quaternions of 1853; (1+i+j+k)/2 is a unit quaternion and conjugation by it cycles i, j, k, which is a 120-degree rotation about the axis (1, 1, 1).", "sources": [ { "title": "Versor", "url": "https://en.wikipedia.org/wiki/Versor", "own_work": false, "note": "defines a versor as a quaternion of norm one and says it ‘was introduced by William Rowan Hamilton in the 1840s in the context of his quaternion theory’.", "fetched": [ "2026-09-09" ] }, { "title": "Classical Hamiltonian quaternions", "url": "https://en.wikipedia.org/wiki/Classical_Hamiltonian_quaternions", "own_work": false, "note": "gives Hamilton’s tensor-and-versor vocabulary, defines a versor as a quaternion of tensor 1 and as the quotient of two equal-length vectors, and quotes Lectures on Quaternions (1853) for the definition of tensor.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "The norm of (1+i+j+k)/2 is √4/2=1. Direct multiplication by q and q⁻¹ gives q i q⁻¹=j, q j q⁻¹=k, and q k q⁻¹=i.", "fetched": [] }, { "title": "W. R. Hamilton, Lectures on Quaternions (Dublin, 1853), Internet Archive", "url": "https://archive.org/details/lecturesonquater00hami", "own_work": false, "note": "read from the full text: the book, whose preface is signed “Observatory of T. C. D., June, 1853,” decomposes the product or quotient of two inclined lines “into two factors, namely, a tensor and a versor,” and uses versor throughout.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-22.12", "url": "https://www.teclamairme.com/#r-22.12", "chapter": 22, "status": "confirmed", "claim": "The numerical values in the chapter’s factor: in the numerator ijk = −1, so ijk + 1 = 0 exactly; in the denominator ‖1+i+j+k‖=2, ij−ji=2k, (1+i+j+k)(1−i−j−k)=4, and |Q₈|=8, so the denominator is 128k and is nonzero.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Using i²=j²=k²=−1, ij=k, ji=−k and the conjugate-norm identity: ijk = k·k = −1, so ijk + 1 = 0; the norm is √(1+1+1+1)=2; ij−ji=k−(−k)=2k; the conjugate product is norm squared 4; multiplying 2·2k·4·8 gives 128k. The chapter’s pair of quarter-turns as well: (1+i)(1+j)/2 = (1 + j + i + ij)/2 = (1 + i + j + k)/2 and (1+j)(1+i)/2 = (1 + i + j + ji)/2 = (1 + i + j − k)/2, which differ because ij = k and ji = −k.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-22.13", "url": "https://www.teclamairme.com/#r-22.13", "chapter": 22, "status": "confirmed", "claim": "The quaternion fraction is legal because every nonzero quaternion has an inverse, and 0 divided by a nonzero quaternion is 0; left and right division agree in this special case.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "gives the inverse formula and identifies the quaternions as a division algebra.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For any nonzero d, 0·d⁻¹=d⁻¹·0=0 by distributivity and the zero laws; hence both left and right quotients of 0 by 128k are 0.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-22.14", "url": "https://www.teclamairme.com/#r-22.14", "chapter": 22, "status": "confirmed", "claim": "The formula qvq⁻¹ composes spatial rotations, and quaternion multiplication is noncommutative, so the order of rotations matters.", "sources": [ { "title": "Quaternions and spatial rotation", "url": "https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation", "own_work": false, "note": "explains composition by quaternion multiplication and noncommutativity of spatial rotations.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.15", "url": "https://www.teclamairme.com/#r-22.15", "chapter": 22, "status": "confirmed", "claim": "Hamilton’s quaternions were influential in mathematical physics before vector analysis displaced them in many applications; Gibbs and Heaviside developed vector analysis, including the dot and cross products used today.", "sources": [ { "title": "Quaternion", "url": "https://en.wikipedia.org/wiki/Quaternion", "own_work": false, "note": "describes quaternion use in physics and the later displacement by vector analysis associated with Gibbs, Heaviside and Helmholtz.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-22.16", "url": "https://www.teclamairme.com/#r-22.16", "chapter": 22, "status": "judgement", "claim": "The chapter calls Q₈ the ‘smallest group in which Hamilton’s minus sign lives’. Read against Hamilton’s relations that is exact: a group holding elements i, j, k with i² = j² = k² = ijk = −1 and −1 ≠ 1 must already hold all eight of ±1, ±i, ±j, ±k. It is not a claim that Q₈ is the smallest non-abelian group, which it is not, since non-abelian groups of order six exist.", "sources": [ { "title": "Quaternion group", "url": "https://en.wikipedia.org/wiki/Quaternion_group", "own_work": false, "note": "confirms the precise definition and order eight of Q₈.", "fetched": [ "2026-09-09" ] }, { "title": "Own mathematical reading", "url": null, "own_work": true, "note": "Q₈ is the group presented by ⟨i, j, k | i² = j² = k² = ijk⟩, in which the common value is central of order two and is what Hamilton wrote as −1. Its only quotient in which −1 survives is Q₈ itself, because every proper nontrivial normal subgroup of Q₈ — {±1}, ⟨i⟩, ⟨j⟩ and ⟨k⟩ — contains −1. So ‘smallest’ is exact for these relations, and only for these relations: non-abelian groups of order 6 exist.", "fetched": [] } ], "editorial_note": "We kept the wording. It is exact for the relations Hamilton cut into the bridge, and the only reading it will not bear — that Q₈ is the smallest non-abelian group of all — is set out above so nobody has to guess which one is meant." }, { "id": "r-22.17", "url": "https://www.teclamairme.com/#r-22.17", "chapter": 22, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Hamilton’s quaternions (every nonzero quaternion has an inverse), 1843; Hamilton found the four-square identity again (Graves later pointed out to him that it was not new) as a statement about quaternions in 1843.", "sources": [ { "title": "Quaternion, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Quaternion", "own_work": false, "note": "read raw: “Quaternions were first described by the Irish mathematician and physicist William Rowan Hamilton in 1843”; “Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler’s four-square identity (1748) and Olinde Rodrigues’ parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra..”", "fetched": [ "2026-09-18" ] }, { "title": "W. R. Hamilton, letter to John T. Graves, 17 October 1843 (Phil. Mag. 1844), TCD transcription", "url": "https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/QLetter/QLetter.html", "own_work": false, "note": "“We have, then, this first law for the multiplication of two quaternions together, that the modulus of the product is equal to the product of the moduli of the factors”; his footnote: “Mr. Graves has since pointed out to the writer that this theorem of ordinary algebra is not new.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-23.1", "url": "https://www.teclamairme.com/#r-23.1", "chapter": 23, "status": "confirmed", "claim": "The octonions are an eight-dimensional real algebra and are nonassociative; their multiplication can be encoded by the seven-point Fano plane.", "sources": [ { "title": "John C. Baez, The Octonions, Bulletin of the AMS 39 (2002)", "url": "https://math.ucr.edu/home/baez/octonions/", "own_work": false, "note": "Baez identifies the octonions as “the largest of the four normed division algebras,” “somewhat neglected due to their nonassociativity”; the Fano-plane multiplication mnemonic is not on this page but in §2.1, “The Fano Plane,” linked from its table of contents under “Constructing the Octonions.”", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, arXiv:math/0105155, introduction", "url": "https://arxiv.org/html/math/0105155v1", "own_work": false, "note": "the chapter epigraph is verbatim from the introduction, ‘The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.’ The book quotes the second sentence and trims its opening ‘But’. The survey appeared as Bulletin of the American Mathematical Society 39 (2002), 145–205.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-23.2", "url": "https://www.teclamairme.com/#r-23.2", "chapter": 23, "status": "confirmed", "claim": "Graves discovered the eight-dimensional algebra in December 1843 and described it to Hamilton in a letter of 26 December 1843; Cayley found it independently and got it into print first, in the Philosophical Magazine of March 1845; Hamilton vouched for Graves’s priority in a note of 14 June 1847. The eight-square identity itself had been found earlier still, by C. F. Degen in 1818. The doubling is the Cayley–Dickson construction.", "sources": [ { "title": "John C. Baez, The Octonions, §2 Constructing the Octonions", "url": "https://math.ucr.edu/home/baez/octonions/node3.html", "own_work": false, "note": "gives the multiplication table and the index-cycling and doubling identities. The Graves/Cayley history is on node1, cited below in this same reference.", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Octonion", "url": "https://en.wikipedia.org/wiki/Octonion", "own_work": false, "note": "gives the Cayley–Dickson construction and the independent Graves/Cayley history.", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, history", "url": "https://math.ucr.edu/home/baez/octonions/node1.html", "own_work": false, "note": "Graves wrote to Hamilton on 26 December 1843 describing the octaves (the page gives the letter’s date, not a day of discovery), sending three further letters in January 1844; Hamilton offered to publicize the work and kept postponing it; ‘In March of 1845, he [Cayley] published a paper in the Philosophical Magazine entitled “On Jacobi’s Elliptic Functions, in Reply to the Rev. B. Bronwin; and on Quaternions”’, with the octonions tacked on at the end; Graves added a priority postscript to a paper in the next issue of the same journal; and ‘On June 14th, 1847, Hamilton contributed a short note to the Transactions of the Royal Irish Academy, vouching for Graves’ priority’, by which time the name Cayley numbers had stuck. Re-read raw on 17 September 2026 for the Degen sentence, which the #onorm card and the p11 brace label both assert and which no entry carried: ‘Still worse, Graves later found that his eight squares theorem had already been discovered by C. F. Degen in 1818’.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The chronology is no longer a judgement call: Baez’s history gives every step of it, and the page now says what the sources say. Graves found them first, in December 1843; Cayley found them independently and printed them first, in March 1845; Hamilton vouched for Graves’s priority on 14 June 1847, too late for the name. Cayley’s March 1845 octonion paper is a different paper from the February 1845 Philosophical Magazine paper cited at 22.8 for quaternion rotation, and the two must not be collapsed into one 1845 Cayley item." }, { "id": "r-23.3", "url": "https://www.teclamairme.com/#r-23.3", "chapter": 23, "status": "confirmed", "claim": "The octonions are alternative: an associator vanishes when two arguments coincide, and Artin’s theorem says any two octonions generate an associative subalgebra.", "sources": [ { "title": "John C. Baez, The Octonions, §1.1 Preliminaries (alternative laws and Artin’s theorem)", "url": "https://math.ucr.edu/home/baez/octonions/node2.html", "own_work": false, "note": "states the octonions are alternative and records Artin’s theorem that any two elements generate an associative subalgebra.", "fetched": [ "2026-09-09" ] }, { "title": "Alternative algebra, Wikipedia", "url": "https://en.wikipedia.org/wiki/Alternative_algebra", "own_work": false, "note": "gives the left alternative law x(xy) = (xx)y, the right alternative law (yx)x = y(xx) and the flexible law (xy)x = x(yx), and states ‘Artin’s theorem states that in an alternative algebra the subalgebra generated by any two elements is associative’, crediting it to Artin and citing Schafer, with no publication year.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The card used to add that Artin’s theorem was ‘published by Zorn in 1930 and credited by him to Artin’. That is true of the Artin–Zorn theorem — a finite alternative division ring is a field — and not of the two-generator theorem the card is about; Wikipedia and the nLab both credit the latter to Artin through Schafer and give it no year. The attribution to Artin stands; the year and the Zorn publication credit are gone." }, { "id": "r-23.4", "url": "https://www.teclamairme.com/#r-23.4", "chapter": 23, "status": "confirmed", "claim": "Under the chapter’s Cayley–Dickson convention, (e₁e₂)e₄ − e₁(e₂e₄) = 2e₇, while the Fano-plane labeling in Baez’s survey assigns a different nonzero triple.", "sources": [ { "title": "Own computation using the chapter’s Cayley–Dickson product", "url": null, "own_work": true, "note": "Computed from (a, b)(c, d) = (ac − d̄b, da + bc̄) and the quaternion basis: (e₁e₂)e₄ = e₇ and e₁(e₂e₄) = −e₇, hence the difference is 2e₇. The same product gives the seven lines {1,2,3}, {1,4,5}, {1,6,7}, {2,4,6}, {2,5,7}, {3,4,7}, {3,5,6}; under Baez’s table (e_i e_{i+1} = e_{i+3}, so e₁e₂ = e₄), e₁, e₂, e₄ lie on a line and (e₁e₂)e₃ − e₁(e₂e₃) = −2e₆ (recomputed 23 September 2026).", "fetched": [] }, { "title": "John C. Baez, The Octonions, §2.1 The Fano plane", "url": "https://math.ucr.edu/home/baez/octonions/node4.html", "own_work": false, "note": "gives the Fano-plane labeling of the octonion multiplication table. The labeling is a convention, so relabeling changes which basis triple displays a chosen associator — that observation is this book’s, not the survey’s.", "fetched": [ "2026-09-10" ] }, { "title": "John C. Baez, The Octonions, §2 Constructing the Octonions (multiplication table)", "url": "https://math.ucr.edu/home/baez/octonions/node3.html", "own_work": false, "note": "the index-cycling and doubling rules, “Together with a single nontrivial product like e₁e₂ = e₄, these facts are enough to recover the whole multiplication table.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-23.5", "url": "https://www.teclamairme.com/#r-23.5", "chapter": 23, "status": "confirmed", "claim": "Moufang identities include (zx)(yz) = z((xy)z), and the unit octonions form the standard Moufang loop despite octonion multiplication being nonassociative.", "sources": [ { "title": "John C. Baez, The Octonions, §3 Octonionic Projective Geometry", "url": "https://math.ucr.edu/home/baez/octonions/node8.html", "own_work": false, "note": "cited for Ruth Moufang’s 1933 non-Desarguesian projective plane: “in 1933 Ruth Moufang constructed a remarkable example of a non-Desarguesian projective plane using the octonions”; the same page says a projective plane comes from a skew field if and only if it satisfies “the axiom of Desargues.” The survey never states the Moufang identities and never uses the word loop; those are carried by the Wikipedia entry below.", "fetched": [ "2026-09-10" ] }, { "title": "EUDML record for Moufang, Zur Struktur von Alternativkörpern (1935)", "url": "https://eudml.org/doc/159732", "own_work": false, "note": "bibliographic record confirms Moufang’s 1935 paper on the structure of alternative fields.", "fetched": [ "2026-09-09" ] }, { "title": "Moufang loop, Wikipedia", "url": "https://en.wikipedia.org/wiki/Moufang_loop", "own_work": false, "note": "opens ‘Moufang loops were introduced by Ruth Moufang (1935)’, lists the identities z(x(zy)) = ((zx)z)y, x(z(yz)) = ((xz)y)z, (zx)(yz) = (z(xy))z and (zx)(yz) = z((xy)z), and states ‘The nonzero octonions form a nonassociative Moufang loop under octonion multiplication’; and, of alternative algebras, “Multiplication in an alternative ring or alternative algebra R obey the Moufang identities.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-23.6", "url": "https://www.teclamairme.com/#r-23.6", "chapter": 23, "status": "confirmed", "claim": "The octonion norm is multiplicative, giving the eight-square composition identity; the all-ones octonion has squared norm 8. Graves showed in his letter of 26 December 1843 that the octaves form a normed division algebra. The eight-square identity was first found by Carl Ferdinand Degen around 1818 and rediscovered by Graves (1843) and Cayley (1845).", "sources": [ { "title": "John C. Baez, The Octonions, normed division algebras", "url": "https://math.ucr.edu/home/baez/octonions/node2.html", "own_work": false, "note": "states the normed-division-algebra norm is multiplicative and identifies the octonions as the eight-dimensional case.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For 1 + e₁ + ··· + e₇, the coordinate norm square is 1² + seven·1² = 8.", "fetched": [] }, { "title": "John C. Baez, The Octonions, §1.1 Preliminaries (history)", "url": "https://math.ucr.edu/home/baez/octonions/node1.html", "own_work": false, "note": "“On December 26th, he wrote to Hamilton describing a new 8-dimensional algebra, which he called the ‘octaves’. He showed that they were a normed division algebra.”", "fetched": [ "2026-09-22" ] }, { "title": "Degen’s eight-square identity, Wikipedia", "url": "https://en.wikipedia.org/wiki/Degen%27s_eight-square_identity", "own_work": false, "note": "read as raw wikitext: “First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845).” It opens: “Degen’s eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-23.7", "url": "https://www.teclamairme.com/#r-23.7", "chapter": 23, "status": "confirmed", "claim": "Hurwitz’s theorem restricts real normed division algebras to dimensions 1, 2, 4 and 8, namely the reals, complexes, quaternions and octonions. Two events: the 1, 2, 4, 8 sums-of-squares theorem is Hurwitz’s Göttingen paper of 1898; the classification of the normed composition algebras is his posthumous paper in Mathematische Annalen 88 (1923).", "sources": [ { "title": "Geometry of the Octonions, The Hurwitz Theorem", "url": "https://octonions.geometryof.org/GO/book/go/hurwitz.html", "own_work": false, "note": "states that the only real composition/normed division algebras with positive-definite norm are R, C, H and O, of dimensions 1, 2, 4 and 8.", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, normed division algebras", "url": "https://math.ucr.edu/home/baez/octonions/node2.html", "own_work": false, "note": "dates the underlying Hurwitz result to 1898 and states the four-algebra classification.", "fetched": [ "2026-09-09" ] }, { "title": "Hurwitz’s theorem (composition algebras), Wikipedia", "url": "https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)", "own_work": false, "note": "‘Hurwitz’s theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898’, citing A. Hurwitz, ‘Über die Composition der quadratischen Formen von beliebig vielen Variabeln’, Goett. Nachr. (1898), 309–316; the general composition-algebra theorem is the posthumous paper, Math. Ann. 88 (1923), 1–25.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Hurwitz has two papers here and, in this book, two theorems. The 1, 2, 4, 8 result the chapter uses is the Göttingen paper of 1898, published in 1898. The card previously read ‘proved in 1898 (published 1923)’, which is wrong: 1923 is the posthumous Mathematische Annalen paper, a different and more general piece of work. The card now says 1898, which is what the chapter prose says and what Chapter ᚔ(I) says at 21.14. Hurwitz’s quaternion integers of 1896/1919, cited in Chapter ᚕ(J) at 22.7, are a third thing again." }, { "id": "r-23.8", "url": "https://www.teclamairme.com/#r-23.8", "chapter": 23, "status": "confirmed", "claim": "The compact exceptional group G₂ is the automorphism group of the octonions, has dimension 14, and is the smallest of the exceptional Lie groups.", "sources": [ { "title": "Geometry of the Octonions, G2", "url": "https://sites.science.oregonstate.edu/coursewikis/GO/dokue6fd.html?id=book%3Acontent%3Ag2&rev=1411236840", "own_work": false, "note": "identifies Aut(O) with G₂ and counts 14 independent automorphisms.", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, exceptional Lie groups", "url": "https://math.ucr.edu/home/baez/octonions/node14.html", "own_work": false, "note": "‘the smallest of the exceptional Lie groups, G₂, is the automorphism group of the octonions’, and derives dim G₂ = 14 from Spin(7)/G₂ = S⁷.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-23.9", "url": "https://www.teclamairme.com/#r-23.9", "chapter": 23, "status": "confirmed", "claim": "The Fano plane is the projective plane over the two-element field, with seven points and seven lines, three points on each line; oriented lines encode octonion products.", "sources": [ { "title": "John C. Baez, The Octonions, §2.1 (author’s own HTML)", "url": "https://math.ucr.edu/home/baez/octonions/node4.html", "own_work": false, "note": "the Fano plane has 7 points and 7 lines, three points to a line, is the projective plane over Z₂, and gives the rule e_i e_j = e_k for cyclically ordered triples (the paper: Bull. AMS 39 (2002) 145–205, doi:10.1090/S0273-0979-01-00934-X; also arXiv:math/0105155).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-23.10", "url": "https://www.teclamairme.com/#r-23.10", "chapter": 23, "status": "confirmed", "claim": "The Fano plane was described by Gino Fano in 1892, and its automorphism group has order 168. Fano’s 1892 paper on the foundations of projective geometry constructs the model of a projective plane now named after him.", "sources": [ { "title": "Wikipedia, Fano plane", "url": "https://en.wikipedia.org/wiki/Fano_plane", "own_work": false, "note": "gives the seven-point/seven-line plane, its order-168 automorphism group, and the historical attribution to Gino Fano; the Fano plane “is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point.”", "fetched": [ "2026-09-09" ] }, { "title": "Finite geometry, Wikipedia", "url": "https://en.wikipedia.org/wiki/Finite_geometry", "own_work": false, "note": "‘In 1892, Gino Fano was the first to consider such a finite geometry’, citing Fano, ‘Sui postulati fondamentali della geometria proiettiva’, Giornale di Matematiche 30 (1892), 106–132.", "fetched": [ "2026-09-09" ] }, { "title": "A. Collino, A. Conte, A. Verra, On the life and scientific work of Gino Fano, arXiv:1311.7177", "url": "https://arxiv.org/abs/1311.7177", "own_work": false, "note": "read from the PDF: Fano’s ideas “are neatly expressed in his paper of 1892 on foundations of projective geometry, where a famous model of projective plane, named today the Fano plane, is in particular constructed.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The Fano plane article carries the seven points and seven lines and the collineation group of order 168 = 2³·3·7, but it does not give a year for Fano. The year 1892, which the card asserts, comes from the finite-geometry article and the Fano paper it cites." }, { "id": "r-23.11", "url": "https://www.teclamairme.com/#r-23.11", "chapter": 23, "status": "confirmed", "claim": "The Coxeter integral octonions have 240 norm-one elements, corresponding to the 240 roots of E₈ and forming a Moufang loop; those 240 roots also realize the eight-dimensional kissing number. E₈ is a simple Lie algebra and Lie group of dimension 248.", "sources": [ { "title": "John C. Baez, Integer Octonions", "url": "https://math.ucr.edu/home/baez/octonions/integers/", "own_work": false, "note": "links the 240 smallest integral octonions with the E₈ root polytope.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Octonion", "url": "https://en.wikipedia.org/wiki/Octonion", "own_work": false, "note": "gives Kirmse (1924) and Coxeter (1946), records that Kirmse wrongly counted eight maximal orders where there are seven because his integers are not closed under multiplication, and states that each of the seven has 240 elements of minimum nonzero norm 1 forming a Moufang loop of order 240: “The Kirmse integers and the seven maximal orders are all isometric to the E8 lattice rescaled by a factor of 1/√2.”", "fetched": [ "2026-09-09" ] }, { "title": "Kissing number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Kissing_number", "own_work": false, "note": "the kissing number in eight dimensions is 240, realized by the E₈ lattice, and was settled by Levenshtein (1979) and independently by Odlyzko and Sloane (1979).", "fetched": [ "2026-09-09" ] }, { "title": "Holger P. Petersson, Integral octonions (2019)", "url": "https://www.fernuni-hagen.de/mi/fakultaet/emeriti/docs/petersson/ass.-rem.-int.-oct.pdf", "own_work": false, "note": "Dickson’s ‘A new simple theory of hypercomplex integers’ of 1923 already constructs an algebra of integral octonions isomorphic to Coxeter’s of 1946, so the Coxeter order is more than twenty years older than is usually assumed.", "fetched": [ "2026-09-09" ] }, { "title": "E8 (mathematics), Wikipedia", "url": "https://en.wikipedia.org/wiki/E8_(mathematics)", "own_work": false, "note": "read as raw wikitext: the smallest nontrivial representation of E₈ is “the adjoint representation (of dimension 248) acting on the Lie algebra E₈ itself.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The card’s dated names all sit here now: Dickson 1923 from Petersson, Kirmse 1924 and Coxeter 1946 and the seven-not-eight maximal orders from the Octonion article, and the kissing number from its own article. The kissing number used to ride inside a parenthesis whose only marker was 23.12, which is Viazovska on packing density — a different theorem, proved by different people in a different decade." }, { "id": "r-23.12", "url": "https://www.teclamairme.com/#r-23.12", "chapter": 23, "status": "confirmed", "claim": "Viazovska proved in 2016 that the E₈ lattice gives the optimal sphere packing in eight dimensions.", "sources": [ { "title": "Maryna Viazovska, The sphere packing problem in dimension 8", "url": "https://arxiv.org/abs/1603.04246", "own_work": false, "note": "the 2016 paper’s abstract states that no packing of unit balls in R⁸ has density greater than the E₈-lattice packing.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-23.13", "url": "https://www.teclamairme.com/#r-23.13", "chapter": 23, "status": "confirmed", "claim": "A norm-compatible vector cross product exists only in dimensions 3 and 7; the seven-dimensional one is obtained from the imaginary octonions and is preserved by G₂ rather than all of SO(7).", "sources": [ { "title": "Brown and Gray, Vector Cross Products, Commentarii mathematici Helvetici 42 (1967)", "url": "https://eudml.org/doc/139340", "own_work": false, "note": "bibliographic record for the primary classification paper on vector cross products.", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, G2 and the seven-dimensional cross product", "url": "https://math.ucr.edu/home/baez/octonions/node14.html", "own_work": false, "note": "relates the imaginary octonions to the seven-dimensional cross product and G₂ symmetry.", "fetched": [ "2026-09-09" ] }, { "title": "Seven-dimensional cross product, Wikipedia", "url": "https://en.wikipedia.org/wiki/Seven-dimensional_cross_product", "own_work": false, "note": "‘Given the properties of bilinearity, orthogonality and magnitude, a nonzero cross product exists only in three and seven dimensions’; cites Robert B. Brown and Alfred Gray, ‘Vector cross products’, Commentarii Mathematici Helvetici 42 (1967), 222–236, and W. S. Massey, ‘Cross products of vectors in higher dimensional Euclidean spaces’, The American Mathematical Monthly 90 (1983), 697–701; states that the seven-dimensional product does not satisfy the Jacobi identity and is invariant under G₂ rather than SO(7); “The restriction to 0, 1, 3 and 7 dimensions is related to Hurwitz’s theorem, that normed division algebras are only possible in 1, 2, 4 and 8 dimensions.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-23.14", "url": "https://www.teclamairme.com/#r-23.14", "chapter": 23, "status": "confirmed", "claim": "The Cayley–Dickson process continues past the octonions to sedenions, where norm multiplicativity fails and zero divisors occur. The sedenions are not alternative and hence not a composition algebra: the norm is no longer multiplicative.", "sources": [ { "title": "Wikipedia, Sedenion", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "states that sedenions have zero divisors and are no longer a division algebra.", "fetched": [ "2026-09-09" ] }, { "title": "John C. Baez, The Octonions, §2.2 The Cayley–Dickson construction", "url": "https://math.ucr.edu/home/baez/octonions/node5.html", "own_work": false, "note": "the construction “provides a nice way to obtain the sequence” of the four normed division algebras (printed as an image); the next algebra “is called the sedenions,” and “the sedenions, and thus all the rest, have zero divisors.”", "fetched": [ "2026-09-22" ] }, { "title": "Cayley–Dickson construction, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction", "own_work": false, "note": "read as raw wikitext: the sedenions retain power associativity “but loses the property of being an alternative algebra and hence cannot be a composition algebra.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-23.15", "url": "https://www.teclamairme.com/#r-23.15", "chapter": 23, "status": "confirmed", "claim": "The displayed Moufang and associator values under the chosen table are 0 and 2e₇ respectively, and the denominator evaluates to 224e₇ after the associator is parenthesized as a factor.", "sources": [ { "title": "Own computation under the chapter’s Cayley–Dickson convention", "url": null, "own_work": true, "note": "Computed the two numerator differences and denominator: 2e₇ · 8 · 14 = 224e₇; the repeated-entry alternative-law term and the Moufang difference both evaluate to 0.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-23.16", "url": "https://www.teclamairme.com/#r-23.16", "chapter": 23, "status": "judgement", "claim": "Moufang’s identities were developed in connection with projective-plane work in the 1930s.", "sources": [ { "title": "John C. Baez, The Octonions, projective planes", "url": "https://math.ucr.edu/home/baez/octonions/node8.html", "own_work": false, "note": "dates Moufang’s remarkable non-Desarguesian projective-plane example to 1933.", "fetched": [ "2026-09-09" ] }, { "title": "EUDML record for Moufang (1935)", "url": "https://eudml.org/doc/159732", "own_work": false, "note": "records the 1935 paper on alternative fields associated with the Moufang identities.", "fetched": [ "2026-09-09" ] }, { "title": "Moufang loop, Wikipedia", "url": "https://en.wikipedia.org/wiki/Moufang_loop", "own_work": false, "note": "opens ‘Moufang loops were introduced by Ruth Moufang (1935)’, lists the identities z(x(zy)) = ((zx)z)y, x(z(yz)) = ((xz)y)z, (zx)(yz) = (z(xy))z and (zx)(yz) = z((xy)z), and states ‘The nonzero octonions form a nonassociative Moufang loop under octonion multiplication’.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "Two papers, one year on the page. Baez dates the non-Desarguesian projective plane to 1933; the identities themselves are the 1935 paper, and the Moufang loop article dates their introduction to 1935 as well. So the page’s ‘found the identities in 1935 while studying projective planes’ has both halves behind it, and what stays a judgement is only the compression of a two-paper story into one year." }, { "id": "r-23.17", "url": "https://www.teclamairme.com/#r-23.17", "chapter": 23, "status": "judgement", "claim": "The octonions occur in connections with mathematical physics, including string-theoretic work.", "sources": [ { "title": "John C. Baez, The Octonions", "url": "https://math.ucr.edu/home/baez/octonions/", "own_work": false, "note": "abstract lists connections to spinors, Bott periodicity, exceptional Lie groups, and supersymmetry.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The mathematical connections are established, but the prose’s phrase ‘in a way nobody has fully explained’ is interpretive. We cite the connections and leave the explanatory-status wording as authorial voice." }, { "id": "r-23.18", "url": "https://www.teclamairme.com/#r-23.18", "chapter": 23, "status": "confirmed", "claim": "Frobenius classified the finite-dimensional associative division algebras over the reals as ℝ, ℂ and ℍ; the proof is dated 1877 and the paper appeared in Crelle’s Journal in 1878.", "sources": [ { "title": "Frobenius theorem (real division algebras), Wikipedia", "url": "https://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)", "own_work": false, "note": "opens ‘the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877’, characterizes the finite-dimensional associative division algebras over the reals as the reals, the complex numbers and the quaternions, and cites F. G. Frobenius (1878), ‘Über lineare Substitutionen und bilineare Formen’, Journal für die reine und angewandte Mathematik 84, 1–63.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "The #hurwitz card’s Frobenius sentence stood on 23.7, which is Hurwitz’s theorem and does not mention him. They are different classifications and easy to swap: Frobenius assumes associativity and no norm, Hurwitz assumes a multiplicative norm and no associativity, which is why the octonions fall out of one and not the other. Hence its own entry rather than a stretch of 23.7. The source dates the proof to 1877 and the printed paper to 1878; the card says 1877, and both are recorded here so a later pass does not read one as a correction of the other." }, { "id": "r-23.19", "url": "https://www.teclamairme.com/#r-23.19", "chapter": 23, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Graves’s octonions (‘octaves’), a normed division algebra, 1843; Baez’s survey ‘The Octonions’ is dated 2002.", "sources": [ { "title": "Octonion, Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Octonion", "own_work": false, "note": "read raw: “The octonions were discovered in December 1843 by John T. Graves, inspired by his friend William Rowan Hamilton’s discovery of quaternions..”", "fetched": [ "2026-09-18" ] }, { "title": "The Octonions, John C. Baez (author’s page)", "url": "https://math.ucr.edu/home/baez/octonions/", "own_work": false, "note": "read raw: “Published in Bull. Amer. Math. Soc. 39 (2002), 145-205..”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-24.1", "url": "https://www.teclamairme.com/#r-24.1", "chapter": 24, "status": "confirmed", "claim": "The Cayley–Dickson construction doubles dimension at each step: from octonions it produces the 16-dimensional sedenions and can continue to dimensions 32, 64, and every power of two.", "sources": [ { "title": "Wikipedia, Cayley–Dickson construction", "url": "https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction", "own_work": false, "note": "describes repeated doubling, dimension 2^n, and the sequence through sedenions and beyond.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.2", "url": "https://www.teclamairme.com/#r-24.2", "chapter": 24, "status": "confirmed", "claim": "The real, complex, quaternion and octonion composition/division algebras have no zero divisors; sedenions do, so nonzero elements can multiply to zero and cancellation and division fail. The counts of two-unit-sum zero divisors (42 supports, 84 splittings, 168 signed pairs, 336 ordered) are the book’s own enumeration, set out at 24.8.", "sources": [ { "title": "Baez, The Octonions", "url": "https://math.ucr.edu/home/baez/octonions/node2.html", "own_work": false, "note": "states that the only real normed division algebras have dimensions 1, 2, 4 and 8 and explains the multiplicative norm/division property.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Sedenion", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "identifies sedenions as 16-dimensional and states that they have zero divisors, unlike the lower Cayley–Dickson algebras.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.3", "url": "https://www.teclamairme.com/#r-24.3", "chapter": 24, "status": "confirmed", "claim": "Under the chapter’s Cayley–Dickson sign convention, (e₃ + e₁₀)(e₆ − e₁₅) is zero, while both factors are nonzero; the product has zero in all sixteen coordinates. Its four unit products are e₃e₆ = e₅, e₃e₁₅ = e₁₂, e₁₀e₆ = e₁₂ and e₁₀e₁₅ = e₅, and for a = e₃ + e₁₀, b = e₆ − e₁₅, a² = −2, so (aa)b = −2b while a(ab) = 0.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Implemented the displayed Cayley–Dickson pair-product rule and the chapter’s basis-index convention in an exact integer-coordinate calculation: the two 16-coordinate factors are nonzero and their product is the all-zero vector. The same calculation gives e₃e₆ = e₅, e₃e₁₅ = e₁₂, e₁₀e₆ = e₁₂ and e₁₀e₁₅ = e₅, so ab = e₅ − e₁₂ + e₁₂ − e₅ = 0; a² = −2, so a has the two-sided inverse −a/2; and (aa)b = −2b while a(ab) = 0, so the regrouping a⁻¹(ab) = (a⁻¹a)b, an instance of the alternative law, fails for this pair (re-run 29 September 2026).", "fetched": [] }, { "title": "Wikipedia, Sedenion", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "gives explicit nontrivial zero-divisor examples of the two-unit-sum form in the standard sedenion basis.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.4", "url": "https://www.teclamairme.com/#r-24.4", "chapter": 24, "status": "confirmed", "claim": "The displayed sedenion zero-divisor factors each have norm √2, so their norm product is 2, while the norm of their zero product is 0.", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Each factor has exactly two coefficients ±1 and fourteen zero coefficients, so the Euclidean sedenion norm is sqrt(1²+1²)=√2; their product is 0 by citation 24.3, whose norm is 0, and √2·√2=2.", "fetched": [] }, { "title": "Wikipedia, Sedenion", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "identifies the sedenions as the 16-dimensional Cayley–Dickson algebra and records that they have zero divisors and are not a normed division algebra. It does not define the norm or discuss multiplicativity; the norm arithmetic above is this book’s own.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.5", "url": "https://www.teclamairme.com/#r-24.5", "chapter": 24, "status": "confirmed", "claim": "Hurwitz’s theorem restricts real normed division/composition algebras with multiplicative positive norm to dimensions 1, 2, 4 and 8; the result is associated with Hurwitz’s 1898 work.", "sources": [ { "title": "Baez, The Octonions", "url": "https://math.ucr.edu/home/baez/octonions/node2.html", "own_work": false, "note": "states the four possible dimensions and identifies the first theorem with Hurwitz’s 1898 paper.", "fetched": [ "2026-09-09" ] }, { "title": "EUDML record, Über die Composition der quadratischen Formen von beliebig vielen Variablen", "url": "https://eudml.org/doc/58420", "own_work": false, "note": "bibliographic record for Hurwitz’s 1898 composition paper.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.6", "url": "https://www.teclamairme.com/#r-24.6", "chapter": 24, "status": "confirmed", "claim": "Sedenions are noncommutative, nonassociative, nonalternative, but power-associative; all Cayley–Dickson algebras are flexible and power-associative. Schafer’s 1954 result is the flexibility; the power associativity of every Cayley–Dickson algebra is stated by the Cayley–Dickson construction page without attribution.", "sources": [ { "title": "Wikipedia, Sedenion", "url": "https://en.wikipedia.org/wiki/Sedenion", "own_work": false, "note": "“the sedenions form a 16-dimensional noncommutative and nonassociative algebra”; “Unlike the octonions, the sedenions are not an alternative algebra”; its infobox lists power associativity among the main properties.", "fetched": [ "2026-09-24" ] }, { "title": "Wikipedia, Cayley–Dickson construction", "url": "https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction", "own_work": false, "note": "states that the construction produces power-associative algebras and cites Schafer for flexibility.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.7", "url": "https://www.teclamairme.com/#r-24.7", "chapter": 24, "status": "confirmed", "claim": "The computation (aa)a − a(aa) = 0 for a = e₃ + e₁₀ is a valid instance of power associativity; the flexible identity is (ab)a = a(ba).", "sources": [ { "title": "Own computation", "url": null, "own_work": true, "note": "Using the same exact sedenion multiplication table as citation 24.3, both bracketings of a·a·a for a=e₃+e₁₀ give the same 16-coordinate vector, so their difference is zero.", "fetched": [] }, { "title": "Wikipedia, Flexible algebra", "url": "https://en.wikipedia.org/wiki/Flexible_algebra", "own_work": false, "note": "defines the flexible identity and attributes the Cayley–Dickson result to Schafer (1954).", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-24.8", "url": "https://www.teclamairme.com/#r-24.8", "chapter": 24, "status": "confirmed", "claim": "The simple two-unit-sum zero-divisor enumeration has 42 four-unit supports, 84 unordered pair splittings, 168 signed unordered factor pairs, and 336 when factor order is counted; no support contains the real unit, and each support splits into a vanishing product in exactly two of its three possible ways.", "sources": [ { "title": "Own enumeration", "url": null, "own_work": true, "note": "Enumerated all pairs (e_i ± e_j)(e_k ± e_l) with distinct imaginary indices 1≤i ω: “The sum of ordinals 1 + ω equals ω”; and “The surreal number ω − 1 is not an ordinal; the ordinal ω is not the successor of any ordinal.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.11", "url": "https://www.teclamairme.com/#r-27.11", "chapter": 27, "status": "judgement", "claim": "The empty-set symbol ∅ was introduced by Bourbaki, specifically André Weil, in 1939, inspired by the Norwegian/Danish letter Ø.", "sources": [ { "title": "Wikipedia, Empty set", "url": "https://en.wikipedia.org/wiki/Empty_set", "own_work": false, "note": "reports the Bourbaki/André Weil attribution and the 1939 date, with Ø as the inspiration.", "fetched": [ "2026-09-09" ] }, { "title": "Jeff Miller, Earliest Uses of Symbols of Set Theory and Logic (MacTutor)", "url": "https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/", "own_work": false, "note": "quotes Weil’s own account: “I had been personally responsible for its adoption … The symbol came from the Norwegian alphabet, with which I alone among the Bourbaki group was familiar” (André Weil, The Apprenticeship of a Mathematician, Birkhäuser 1992, p. 114).", "fetched": [ "2026-09-22" ] } ], "editorial_note": "We found the 1939 Bourbaki/Weil attribution in the reference summary and a first-person Weil account of the Norwegian origin, but not a primary publication scan establishing the exact first printed use. We retain the chapter’s compact historical wording as a judgement call." }, { "id": "r-27.12", "url": "https://www.teclamairme.com/#r-27.12", "chapter": 27, "status": "confirmed", "claim": "The empty set has one subset, so its power set has cardinality 1 and |P(∅)| − 1 = 0; Cantor’s theorem says every power set is strictly larger than its base set.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Basic Set Theory", "url": "https://plato.stanford.edu/entries/set-theory/basic-set-theory.html", "own_work": false, "note": "proves that P(A) is not bijectable with A; the empty-set case has the single subset ∅.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Von Neumann universe", "url": "https://en.wikipedia.org/wiki/Von_Neumann_universe", "own_work": false, "note": "shows the cumulative hierarchy beginning with the empty set and its power set.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.13", "url": "https://www.teclamairme.com/#r-27.13", "chapter": 27, "status": "confirmed", "claim": "Eilenberg and Mac Lane introduced the general notions of category, functor, and natural transformation in their 1945 paper General Theory of Natural Equivalences.", "sources": [ { "title": "Wikipedia, Category theory", "url": "https://en.wikipedia.org/wiki/Category_theory", "own_work": false, "note": "dates the general introduction of categories, functors, and natural transformations to the authors’ 1945 paper.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.14", "url": "https://www.teclamairme.com/#r-27.14", "chapter": 27, "status": "confirmed", "claim": "In Set, the empty set is an initial object and there is exactly one function from ∅ to every set X.", "sources": [ { "title": "Wikipedia, Initial and terminal objects", "url": "https://en.wikipedia.org/wiki/Initial_and_terminal_objects", "own_work": false, "note": "identifies ∅ as the unique initial object in Set.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Function (mathematics)", "url": "https://en.wikipedia.org/wiki/Function_(mathematics)", "own_work": false, "note": "the empty function is the unique function with empty domain to any codomain.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.15", "url": "https://www.teclamairme.com/#r-27.15", "chapter": 27, "status": "confirmed", "claim": "ZFC is the usual Zermelo–Fraenkel framework with choice; the axiom of choice is associated with Zermelo’s 1904 well-ordering proof and is independent of ZF. Skolem and Fraenkel formalized Separation and introduced Replacement, and von Neumann added Foundation.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Set Theory", "url": "https://plato.stanford.edu/entries/set-theory/", "own_work": false, "note": "describes ZFC, the role of AC, and the independence results for AC and CH; “Further work by Skolem and Fraenkel led to the formalization of the Separation axiom in terms of formulas of first-order, instead of the informal notion of property, as well as to the introduction of the axiom of Replacement,” and “A further addition, by von Neumann, of the axiom of Foundation, led to the standard axiom system of set theory, known as the Zermelo-Fraenkel axioms plus the Axiom of Choice, or ZFC.”", "fetched": [ "2026-09-09" ] }, { "title": "Stanford Encyclopedia of Philosophy, Zermelo’s Axiomatization of Set Theory", "url": "https://plato.stanford.edu/entries/zermelo-set-theory/", "own_work": false, "note": "gives the historical setting for Zermelo’s well-ordering work and the axiomatization of set theory.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.16", "url": "https://www.teclamairme.com/#r-27.16", "chapter": 27, "status": "confirmed", "claim": "Russell’s paradox arises from the collection of sets that are not members of themselves and helped force axiomatization of set theory.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Set Theory", "url": "https://plato.stanford.edu/entries/set-theory/", "own_work": false, "note": "discusses the paradoxes and the axiomatic development of set theory.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Russell’s paradox", "url": "https://en.wikipedia.org/wiki/Russell%27s_paradox", "own_work": false, "note": "states the self-membership contradiction and its historical consequences.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.17", "url": "https://www.teclamairme.com/#r-27.17", "chapter": 27, "status": "confirmed", "claim": "A strongly inaccessible cardinal is uncountable, regular, and a strong limit; its existence is not provable in ZFC under the usual consistency assumptions.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Set Theory", "url": "https://plato.stanford.edu/entries/set-theory/", "own_work": false, "note": "defines strongly inaccessible cardinals by regularity and strong-limit behavior and explains why their existence is an additional axiom beyond ZFC: “One cannot prove in ZFC that there exists a regular limit cardinal κ, for if κ is such a cardinal, then L_κ is a model of ZFC, and so ZFC would prove its own consistency, contradicting Gödel’s second incompleteness theorem.” It defines the term: “An infinite cardinal κ is called regular if it is not the union of less than κ smaller cardinals.” (Symbols as rendered on the page.)", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.18", "url": "https://www.teclamairme.com/#r-27.18", "chapter": 27, "status": "confirmed", "claim": "The ordinal ε₀ is the least positive fixed point of α ↦ ω^α, and Gentzen’s 1936 consistency proof for Peano arithmetic uses transfinite induction up to ε₀.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Proof Theory", "url": "https://plato.stanford.edu/entries/proof-theory/", "own_work": false, "note": "defines ε₀ as the least α with ω^α = α and explains the descending-ordinal argument in Gentzen’s consistency proof for PA.", "fetched": [ "2026-09-10" ] }, { "title": "Wikipedia, Gentzen’s consistency proof", "url": "https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof", "own_work": false, "note": "dates the published proof to 1936 and identifies transfinite induction up to ε₀ as its ordinal-theoretic ingredient.", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-27.19", "url": "https://www.teclamairme.com/#r-27.19", "chapter": 27, "status": "judgement", "claim": "Gregory H. Moore’s scholarly history traces the generalized continuum hypothesis through the work of Charles Sanders Peirce, who introduced the beth notation. Peirce used the beth notation in a letter to Cantor of December 1900.", "sources": [ { "title": "Gregory H. Moore, Early History of the Generalized Continuum Hypothesis: 1878–1938", "url": "https://doi.org/10.2178/bsl/1318855631", "own_work": false, "note": "the 2011 Bulletin of Symbolic Logic history; its abstract (the full text is not free): “This paper explores how the Generalized Continuum Hypothesis (GCH) arose from Cantor’s Continuum Hypothesis in the work of Peirce, Jourdain, Hausdorff, Tarski, and how GCH was used up to Gödel’s relative consistency result.” The beth notation and the December 1900 letter are on the Open Logic line.", "fetched": [ "2026-09-10" ] }, { "title": "Open Logic Project, Set Theory, Cardinal arithmetic (chapter PDF)", "url": "https://builds.openlogicproject.org/content/set-theory/card-arithmetic/ch.pdf", "own_work": false, "note": "read from the PDF: “Peirce introduced the “ℶ” notation; this is beth, which is the second letter in the Hebrew alphabet,” with the footnote “Peirce used this notation in a letter to Cantor of December 1900.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "Judgement retained: Moore is a specialist scholarly source and directly supports the attribution, but the primary letter scan was not available for inspection. The card therefore attributes the report to Moore rather than presenting an independently verified transcription." }, { "id": "r-27.20", "url": "https://www.teclamairme.com/#r-27.20", "chapter": 27, "status": "confirmed", "claim": "Cantor’s transfinite set theory developed from his 1874 work through the 1883 introduction of transfinite ordinals and the 1895/1897 papers; he introduced aleph notation in 1895, while the diagonal paper appeared in 1891/92. Cardinality was in use in Cantor’s work from 1874 and 1878; the 1895 and 1897 papers were a well-organized presentation of his results on cardinals and ordinals.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, The Early Development of Set Theory", "url": "https://plato.stanford.edu/entries/settheory-early/", "own_work": false, "note": "dates Cantor’s foundational 1874 paper, the 1883 introduction of transfinite ordinals, the 1895 introduction of aleph notation, the 1895/1897 transfinite-number papers and the diagonal paper’s date, 1892: “In the same short paper (1892), Cantor presented his famous proof” that the reals are “non-denumerable by the method of diagonalisation”; the 1891 is on the next line. The “second number class” “was formed by ω and all numbers following it (including ω^ω, and many more) that have only a denumerable set of predecessors,” and “his theorems justified calling the cardinality of the “second number class” ℵ₁” (symbols as rendered on the page).", "fetched": [ "2026-09-10" ] }, { "title": "Cantor’s diagonal argument, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument", "own_work": false, "note": "“Georg Cantor published this proof in 1891,” its first reference giving the paper as Cantor (1891), “Ueber eine elementare Frage der Mannigfaltigkeitslehre,” Jahresbericht der Deutschen Mathematiker-Vereinigung 1: 75–78.", "fetched": [ "2026-09-24" ] }, { "title": "Cantorsche Normalform, Wikipedia (German)", "url": "https://de.wikipedia.org/wiki/Cantorsche_Normalform", "own_work": false, "note": "read as raw wikitext: “Er wurde erstmals 1897 von Cantor für gewisse Ordinalzahlen bewiesen,” citing Cantor’s Beiträge zur Begründung der transfiniten Mengenlehre.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-27.21", "url": "https://www.teclamairme.com/#r-27.21", "chapter": 27, "status": "confirmed", "claim": "Hilbert’s 1926 paper Über das Unendliche contains the quoted defense of Cantor’s paradise, on p. 170.", "sources": [ { "title": "Wikipedia (German), Cantors Paradies", "url": "https://de.wikipedia.org/wiki/Cantors_Paradies", "own_work": false, "note": "quotes Hilbert’s sentence and cites it to Über das Unendliche, Mathematische Annalen 95 (1926), p. 170: “Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.”", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-27.22", "url": "https://www.teclamairme.com/#r-27.22", "chapter": 27, "status": "confirmed", "claim": "Gödel’s completely detailed proof that the axiom of choice and the generalized continuum hypothesis, and so the continuum hypothesis itself, are consistent with the axioms of set theory is the 1940 monograph, which followed his briefer earlier announcements.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Kurt Gödel", "url": "https://plato.stanford.edu/entries/goedel/", "own_work": false, "note": "names “The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory” as the 1940 monograph in which Gödel “gave a completely detailed proof of the two theorems,” the earlier announcement having been “lacking in a great many details”; its bibliography lists [1940] “The Consistency of the Continuum Hypothesis,” “Annals of Mathematics Studies, Volume 3, Princeton: Princeton University Press.”", "fetched": [ "2026-09-11" ] } ], "editorial_note": "Neither source at 27.4 carries the year 1940: both give Gödel 1938, and the Continuum Hypothesis entry prints no Gödel bibliography entry at all. The chapter’s prose and the ch and cont cards all name the 1940 monograph, so the monograph is dated here." }, { "id": "r-27.23", "url": "https://www.teclamairme.com/#r-27.23", "chapter": 27, "status": "confirmed", "claim": "ℵω is the first singular cardinal, because it is the union of countably many smaller cardinals.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Set Theory", "url": "https://plato.stanford.edu/entries/set-theory/", "own_work": false, "note": "states “The first singular cardinal is ℵω, as it is the union of countably-many smaller cardinals, namely ℵω = ⋃{ℵn : n<ω},” and separately that one cannot prove in ZFC that a regular limit cardinal exists.", "fetched": [ "2026-09-11" ] } ], "editorial_note": "27.6’s two sources carry ℵω’s singularity and its cofinality ω but neither calls it the first singular cardinal; that superlative is carried here. It is a statement made inside ZFC, where every successor aleph is regular — in ZF alone ℵ1 can be singular." }, { "id": "r-27.24", "url": "https://www.teclamairme.com/#r-27.24", "chapter": 27, "status": "confirmed", "claim": "A category consists of objects, arrows between them, an associative composition and an identity arrow on each object; objects are atomic in the definition and are characterized by their relations to other objects.", "sources": [ { "title": "Wikipedia, Category theory", "url": "https://en.wikipedia.org/wiki/Category_theory", "own_work": false, "note": "gives the definition as a class ob(C) of objects, a class hom(C) of morphisms or arrows, a composition ∘: hom(b, c) × hom(a, b) → hom(a, c) and an identity morphism 1x on every object x, and states that objects are considered atomic so that “one can characterize these objects in terms of their relations to other objects, as given by the morphisms”; “A prototypical example of a category is Set, the category of sets and functions between them.”", "fetched": [ "2026-09-11" ] } ], "editorial_note": "" }, { "id": "r-27.25", "url": "https://www.teclamairme.com/#r-27.25", "chapter": 27, "status": "confirmed", "claim": "An initial object of a category C is an object I such that for every object X there is precisely one morphism I → X, and any two initial objects are related by a unique isomorphism.", "sources": [ { "title": "Wikipedia, Initial and terminal objects", "url": "https://en.wikipedia.org/wiki/Initial_and_terminal_objects", "own_work": false, "note": "states “An initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X,” that a terminal object has exactly one morphism the other way, that two different initial objects have a unique isomorphism between them, and that the empty set is the unique initial object in Set.", "fetched": [ "2026-09-11" ] } ], "editorial_note": "" }, { "id": "r-27.26", "url": "https://www.teclamairme.com/#r-27.26", "chapter": 27, "status": "confirmed", "claim": "Two sets have the same cardinality exactly when there is a bijection between them, and Dedekind defined an infinite set as one that can be put in one-to-one correspondence with a proper subset of itself; the even numbers are as many as the whole numbers. Doubling, f(n) = 2n, is a bijection from the natural numbers to the even numbers.", "sources": [ { "title": "Stanford Encyclopedia of Philosophy, Infinity", "url": "https://plato.sydney.edu.au/entries/infinity/index.html", "own_work": false, "note": "states “When there is a bijection between two sets, we say that they have the same cardinality,” gives Galileo’s paradox that “the squares of natural numbers are a proper subset of the natural numbers, but they have the same cardinality since they can be put in one-to-one correspondence,” and records Dedekind’s 1884 definition of an infinite set as one admitting a bijection with a proper subset of itself.", "fetched": [ "2026-09-11" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "n ↦ 2n pairs every whole number with exactly one even number and leaves nothing over on either side, so the even numbers are equinumerous with the whole numbers; the source above carries the same fact for the squares.", "fetched": [ "2026-09-11" ] }, { "title": "Cardinality, Wikipedia", "url": "https://en.wikipedia.org/wiki/Cardinality", "own_work": false, "note": "read as raw wikitext: the even numbers have the cardinality of the natural numbers “since the function f(n) = 2n is a bijection from ℕ to E”; Hilbert’s hotel moves “room n to room 2n.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "27.1’s source characterizes cardinality by bijection but contains no equinumerous-with-a-proper-subset statement and no even-numbers example, so the chapter’s opening paragraph is carried here instead." }, { "id": "r-27.27", "url": "https://www.teclamairme.com/#r-27.27", "chapter": 27, "status": "confirmed", "claim": "A well-order is a total order in which every non-empty subset has a least element; every element except a possible greatest one has a unique successor, and some elements other than the least may have no predecessor. Two ordered sets have the same order type when there is an order isomorphism between them, and the order type of a well-ordered set is by definition an ordinal. ω is the smallest limit ordinal; 1 + ω = ω, which is not equal to ω + 1.", "sources": [ { "title": "Well-order, Wikipedia", "url": "https://en.wikipedia.org/wiki/Well-order", "own_work": false, "note": "read as raw HTML: “a well-order… on a set S is a total ordering on S with the property that every non-empty subset of S has a least element… Every element s of a well-ordered set, except a possible greatest element, has a unique successor… There may be elements, besides the least element, that have no predecessor.”", "fetched": [ "2026-09-17" ] }, { "title": "Order type, Wikipedia", "url": "https://en.wikipedia.org/wiki/Order_type", "own_work": false, "note": "two ordered sets share an order type exactly when an order isomorphism exists between them, and the order types of well-ordered sets are the ordinals.", "fetched": [ "2026-09-17" ] }, { "title": "Well Ordered Set, MathWorld", "url": "https://mathworld.wolfram.com/WellOrderedSet.html", "own_work": false, "note": "the same least-element condition.", "fetched": [ "2026-09-17" ] }, { "title": "Limit ordinal, Wikipedia", "url": "https://en.wikipedia.org/wiki/Limit_ordinal", "own_work": false, "note": "read as raw wikitext: “Limit ordinals are those that are non-zero and have no predecessor, such as ω or ω²”; “the smallest limit ordinal is ω, the smallest ordinal greater than every natural number.”", "fetched": [ "2026-09-22" ] }, { "title": "Infinity, Stanford Encyclopedia of Philosophy", "url": "https://plato.stanford.edu/entries/infinity/", "own_work": false, "note": "“1 + ω is the order type of a single element, followed by a copy of the natural numbers, which is in fact the same as the order type of the natural numbers! Thus, 1 + ω = ω, which is not equal to ω + 1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-27.28", "url": "https://www.teclamairme.com/#r-27.28", "chapter": 27, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: The continuum hypothesis is Cantor’s conjecture of 1878; ZFC grew from Zermelo’s 1908 axiomatization.", "sources": [ { "title": "Continuum hypothesis, Wikipedia", "url": "https://en.wikipedia.org/wiki/Continuum_hypothesis", "own_work": false, "note": "the page’s words: “Georg Cantor advanced the continuum hypothesis in 1878.”", "fetched": [ "2026-09-18" ] }, { "title": "The Early Development of Set Theory, Stanford Encyclopedia of Philosophy", "url": "https://plato.stanford.edu/entries/settheory-early/", "own_work": false, "note": "read raw: “As a result of his incisive analysis, in 1908 he published his axiom system, showing how it blocked the known paradoxes.”", "fetched": [ "2026-09-18" ] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-27.34", "url": "https://www.teclamairme.com/#r-27.34", "chapter": 27, "status": "confirmed", "claim": "States that Cantor’s second number class is the set of the infinite countable ordinals, itself uncountable, with cardinality ℵ₁. Georg Cantor introduced it in 1883; together with the first number class it makes up ω₁, the set of all countable ordinals. With countable choice, the supremum of every countable set of countable ordinals is countable, so no countable sequence is cofinal in ω₁.", "sources": [ { "title": "Ordinal number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ordinal_number", "own_work": false, "note": "describes Cantor’s number classes: the second number class is the countably infinite ordinals, and together with the finite ordinals it makes up the countable ordinals, of cardinality ℵ₁.", "fetched": [ "2026-09-22" ] }, { "title": "Aleph number, Wikipedia", "url": "https://en.wikipedia.org/wiki/Aleph_number", "own_work": false, "note": "carries ω₁ as the set of all countable ordinals: “ℵ1 is the cardinality of the set of all countable ordinal numbers. This set is denoted by ω1 (or sometimes Ω).”", "fetched": [ "2026-09-22" ] }, { "title": "First uncountable ordinal, Wikipedia", "url": "https://en.wikipedia.org/wiki/First_uncountable_ordinal", "own_work": false, "note": "“the elements of ω1 are the countable ordinals (including finite ordinals), of which there are uncountably many”; and, if the axiom of countable choice holds, “the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal,” so no countable sequence of countable ordinals reaches ω₁.", "fetched": [ "2026-09-24" ] } ], "editorial_note": "" }, { "id": "r-27.35", "url": "https://www.teclamairme.com/#r-27.35", "chapter": 27, "status": "confirmed", "claim": "States that for every set X there is exactly one function from the empty set to X, the empty function, making ∅ the initial object of Set.", "sources": [ { "title": "Function (mathematics), Wikipedia", "url": "https://en.wikipedia.org/wiki/Function_(mathematics)", "own_work": false, "note": "states that the empty function, with empty domain, is the unique function from the empty set to any set.", "fetched": [ "2026-09-22" ] }, { "title": "Empty set, Wikipedia", "url": "https://en.wikipedia.org/wiki/Empty_set", "own_work": false, "note": "carries the empty set as the unique initial object of Set: “As a result, the empty set is the unique initial object of the category of sets and functions.” It calls the empty topological space, in the category of topological spaces, a strict initial object: “only the empty set has a function to the empty set.”", "fetched": [ "2026-09-22" ] }, { "title": "Initial and terminal objects, Wikipedia", "url": "https://en.wikipedia.org/wiki/Initial_and_terminal_objects", "own_work": false, "note": "carries that Set has no zero object: “The empty set is the unique initial object in Set, the category of sets. Every one-element set (singleton) is a terminal object in this category; there are no zero objects.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.1", "url": "https://www.teclamairme.com/#r-28.1", "chapter": 28, "status": "confirmed", "claim": "The Millennium Prize Problems were announced in Paris, at the Collège de France, on 24 May 2000, a century after Hilbert’s 1900 Paris address, with seven prizes of one million dollars each; Poincare is the solved problem and the other six remain open.", "sources": [ { "title": "Clay Mathematics Institute, The Millennium Prize Problems", "url": "https://www.claymath.org/millennium-problems/", "own_work": false, "note": "Clay states that the problems were announced on 24 May 2000, that the fund was $7 million with $1 million per problem, and identifies Poincare as solved while listing the remaining problems as unsolved; “The prizes were announced at a meeting in Paris, held on May 24, 2000 at the Collège de France,” and the Riemann hypothesis “also appears in the list of twenty-three problems discussed in the address given in Paris by David Hilbert on August 9, 1900.”", "fetched": [ "2026-09-09" ] } ], "editorial_note": "" }, { "id": "r-28.2", "url": "https://www.teclamairme.com/#r-28.2", "chapter": 28, "status": "judgement", "claim": "The Hilbert lecture is dated 8 August 1900 in the congress timetable, while Clay gives 9 August; the chapter epigraph reproduces Hilbert’s opening sentence. Sources differ: 8 August (the congress timetable; Wikipedia, “speaking on August 8 at the Sorbonne”) or 9 August (Clay); corrections welcome.", "sources": [ { "title": "MacTutor, 1900 ICM Paris", "url": "https://mathshistory.st-andrews.ac.uk/ICM/ICM_Paris_1900/", "own_work": false, "note": "the timetable places Hilbert’s communication in the Wednesday 8 August session.", "fetched": [ "2026-09-09" ] }, { "title": "Clay Mathematics Institute, The Millennium Prize Problems", "url": "https://www.claymath.org/millennium-problems/", "own_work": false, "note": "Clay’s historical note says Hilbert’s address was on August 9, 1900.", "fetched": [ "2026-09-09" ] }, { "title": "MacTutor, Hilbert’s Mathematical Problems speech", "url": "https://mathshistory.st-andrews.ac.uk/Extras/Hilbert_Problems_speech/", "own_work": false, "note": "the transcription opens with the sentence used as the epigraph; “The address Mathematische Probleme appeared in Göttinger Nachrichten in 1900, before the Proceedings of the Congress were published.”", "fetched": [ "2026-09-10" ] } ], "editorial_note": "We follow the congress timetable, which places the address on 8 August; Clay’s 9 August is recorded as the outlier." }, { "id": "r-28.3", "url": "https://www.teclamairme.com/#r-28.3", "chapter": 28, "status": "confirmed", "claim": "The Riemann Hypothesis puts the nontrivial zeta zeros on the critical line and controls deviations in the distribution of primes; the de Bruijn–Newman constant is a definite real with RH equivalent to Λ ≤ 0, while Rodgers and Tao proved Λ ≥ 0 in a paper posted to arXiv in January 2018.", "sources": [ { "title": "Rodgers and Tao, The De Bruijn–Newman constant is non-negative", "url": "https://arxiv.org/abs/1801.05914", "own_work": false, "note": "the abstract defines Λ by the heat-flow deformation, states RH is equivalent to Λ ≤ 0, and proves Newman’s conjecture Λ ≥ 0; the page is headed “Submitted on 18 Jan 2018.”", "fetched": [ "2026-09-09" ] }, { "title": "Clay Mathematics Institute, Riemann Hypothesis", "url": "https://www.claymath.org/millennium/Riemann-Hypothesis/", "own_work": false, "note": "Clay states that the prime number theorem gives the average distribution of primes and RH controls the deviation from that average.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.4", "url": "https://www.teclamairme.com/#r-28.4", "chapter": 28, "status": "confirmed", "claim": "Nicolaas de Bruijn showed in 1950 that the deformed function has only real zeros for parameters at least one half, in The Roots of Trigonometric Integrals, Duke Mathematical Journal 17 (1950), 197–226, and Charles Newman gave the constant its second half in Fourier Transforms with only Real Zeros, Proceedings of the American Mathematical Society 61 (1976), 245–251. Platt and Trudgian rigorously verified RH up to height 3·10^12, yielding the upper bound Lambda <= 0.2 used here; the paper is Bulletin of the London Mathematical Society 53 (2021), pp. 792–797, published online 22 January 2021, doi:10.1112/blms.12460, from the preprint arXiv:2004.09765 of 21 April 2020. Platt and Trudgian’s bound Λ ≤ 0.2 appeared as a preprint in April 2020 and in print in 2021. Newman remarked that his conjecture is “a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.”", "sources": [ { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12", "url": "https://arxiv.org/abs/2004.09765", "own_work": false, "note": "the abstract states that interval arithmetic verifies all zeros with 0 < gamma <= 3·10^12 lie on the critical line.", "fetched": [ "2026-09-09" ] }, { "title": "de Bruijn-Newman constant, reference summary", "url": "https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant", "own_work": false, "note": "the history records the 2020 Platt-Trudgian improvement Lambda <= 0.2. Re-read raw on 17 September 2026 for the two founding dates: “De Bruijn showed in 1950 that H has only real zeros if λ ≥ 1/2, and moreover, that if H has only real zeros for some λ, H also has only real zeros if λ is replaced by any larger value,” and the two papers are given as de Bruijn, N. G. (1950), The Roots of Trigonometric Integrals, Duke Math. J. 17 (3), 197–226, and Newman, C. M. (1976), Fourier Transforms with only Real Zeros, Proc. Amer. Math. Soc. 61 (2), 245–251.", "fetched": [ "2026-09-09" ] }, { "title": "J. Andrade, A. Chang, S. J. Miller, Newman’s conjecture in various settings, arXiv:1310.3477", "url": "https://arxiv.org/abs/1310.3477", "own_work": false, "note": "read from the PDF: “Newman made the conjecture that Λ ≥ 0 along with the remark that “the new conjecture is a quantitative version of the dictum that the Riemann hypothesis, if true, is only barely so.””", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The verification is dated 2021 throughout the book, which is its publication year. Checked 17 September 2026 and not to be reopened: the arXiv record reads “[Submitted on 21 Apr 2020]” and the Crossref record for doi:10.1112/blms.12460 gives Bulletin of the London Mathematical Society 53 (3), pp. 792–797, published online 22 January 2021 and in print June 2021. 2020 is the preprint and 2021 is the paper; where a source summary here says “the 2020 Platt-Trudgian improvement” it is dating the preprint." }, { "id": "r-28.5", "url": "https://www.teclamairme.com/#r-28.5", "chapter": 28, "status": "confirmed", "claim": "For the conductor-37 elliptic curve y²+y=x³−x, the root number is −1, L(E, 1)=0, and LMFDB records analytic rank 1 and algebraic rank 1. The L-function of 37.a1 has sign −1, analytic rank 1 and L(1) = 0.", "sources": [ { "title": "LMFDB, elliptic curve 37.a1", "url": "https://www.lmfdb.org/EllipticCurve/Q/37/a/1", "own_work": false, "note": "the record identifies the curve and gives conductor 37, analytic rank 1, and rank 1 (the sign −1 is on the L-function line below); its BSD formula line gives the leading coefficient as L′(E, 1) = #Ш(E/Q)·Ω_E·Reg(E/Q)·∏c_p / #E(Q)_tor² (as typeset on the page).", "fetched": [ "2026-09-09" ] }, { "title": "Clay Mathematics Institute, Birch and Swinnerton-Dyer official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "the official description states the conjectural equality between analytic order of vanishing and algebraic rank and describes the known analytic-rank-zero and analytic-rank-one results.", "fetched": [ "2026-09-10" ] }, { "title": "LMFDB, L-function of elliptic curve isogeny class 37.a", "url": "https://www.lmfdb.org/L/EllipticCurve/Q/37/a/", "own_work": false, "note": "“Sign −1,” “Analytic rank 1,” and among the special values L(1) = 0.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.6", "url": "https://www.teclamairme.com/#r-28.6", "chapter": 28, "status": "confirmed", "claim": "Poincare posed the three-dimensional conjecture in 1904 and Perelman proved it using Ricci flow in work posted in 2002-2003 and never submitted to a refereed journal; the corresponding bracket is 1. Perelman declined the Fields Medal in 2006 and the Millennium Prize in 2010, having also declined the European Mathematical Society prize in 1996.", "sources": [ { "title": "Clay Mathematics Institute, Poincare Conjecture", "url": "https://www.claymath.org/millennium-problems/poincare-conjecture/", "own_work": false, "note": "Clay describes Poincare’s 1904 question and Perelman’s proof as the solved Millennium problem.", "fetched": [ "2026-09-09" ] }, { "title": "Clay annual report 2010, The Millennium Prize", "url": "https://www.claymath.org/library/annual_report/ar2010/ar2010.pdf", "own_work": false, "note": "the report states that Perelman’s three arXiv articles appeared in 2002 and 2003 and solved the Poincare conjecture, and that “Perelman did not take the usual route of submission of his work to a refereed journal.”", "fetched": [ "2026-09-09" ] }, { "title": "Grigori Perelman, Wikipedia", "url": "https://en.wikipedia.org/wiki/Grigori_Perelman", "own_work": false, "note": "read as raw HTML: the awards list gives “Saint Petersburg Mathematical Society Prize (1991),” “EMS Prize (1996, declined),” “Fields Medal (2006, declined)” and “Millennium Prize (2010, declined).” Added because #clay’s quick line ends “one has been solved and the prize declined,” which no entry carried, and a quick line is repeated in every popup.", "fetched": [ "2026-09-17" ] }, { "title": "Poincaré conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture", "own_work": false, "note": "read as raw wikitext: “after publication he found his announced theorem to be incorrect. In his fifth and final supplement, published in 1904, he proved this with the counterexample of the Poincaré homology sphere”; “Hamilton’s program was started in his 1982 paper in which he introduced the Ricci flow”; “The Ricci flow usually deforms the manifold towards a rounder shape.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.7", "url": "https://www.teclamairme.com/#r-28.7", "chapter": 28, "status": "confirmed", "claim": "P versus NP asks whether every efficiently checkable problem is efficiently solvable; Cook formulated the problem in 1971, Levin did so independently in work published in 1973, and most complexity theorists believe P ≠ NP.", "sources": [ { "title": "Clay Mathematics Institute, P vs NP", "url": "https://www.claymath.org/millennium/p-vs-np/", "own_work": false, "note": "Clay defines the easy-to-check versus easy-to-find question.", "fetched": [ "2026-09-09" ] }, { "title": "Stephen Cook, The P versus NP Problem", "url": "https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf", "own_work": false, "note": "the official problem description gives the Cook, Levin and Karp history and says that most complexity theorists believe P is not equal to NP.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.8", "url": "https://www.teclamairme.com/#r-28.8", "chapter": 28, "status": "confirmed", "claim": "The Hodge conjecture concerns rational (p, p)-classes on smooth projective complex varieties, and Atiyah and Hirzebruch’s work published in 1962 shows that the integer-coefficient analogue is false.", "sources": [ { "title": "Clay Mathematics Institute, Hodge Conjecture", "url": "https://www.claymath.org/millennium/hodge-conjecture/", "own_work": false, "note": "Clay’s landing page gives only the informal wording, that Hodge cycles are ‘(rational linear) combinations of geometric pieces called algebraic cycles’. It does not use the phrase type (p, p), and it names the variety class only as “particularly nice types of spaces called projective algebraic varieties,” without saying smooth. The type-(p, p) and smooth-projective halves of this entry’s claim are carried by Deligne below.", "fetched": [ "2026-09-09", "2026-09-11" ] }, { "title": "Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "printed p. 46: ‘Rational (p, p)-classes are called Hodge classes’, and ‘On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles’. This is the official statement, and the source for both halves of the claim above. The same document is cited at r-28.21 for the 1950 date.", "fetched": [ "2026-09-11" ] }, { "title": "Atiyah and Hirzebruch, Analytic cycles on complex manifolds", "url": "https://doi.org/10.1016/0040-9383(62)90094-0", "own_work": false, "note": "the Topology article gives the torsion counterexample underlying the failure of the integral version; the journal metadata dates the article to 1962.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.9", "url": "https://www.teclamairme.com/#r-28.9", "chapter": 28, "status": "confirmed", "claim": "Navier and Stokes introduced equations for viscous flow in the nineteenth century, Leray proved global weak solutions in three dimensions in 1934, and the Clay problem asks whether three-dimensional solutions remain smooth and unique. For incompressible flow the enstrophy is the integral of the squared vorticity; Leray’s strong solution lasts at least C/‖∇u0‖⁴ (unit viscosity) and ‖∇u‖ must blow up at a finite blow-up time; Leray’s weak solutions exist for all time and satisfy the equations in mean value; smooth solutions are known for small data or for a short time.", "sources": [ { "title": "Clay Mathematics Institute, Navier–Stokes Equation", "url": "https://www.claymath.org/millennium/Navier-Stokes-Equation/", "own_work": false, "note": "Clay describes the fluid equations, their nineteenth-century origin, and the unresolved existence and uniqueness questions.", "fetched": [ "2026-09-09" ] }, { "title": "Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace", "url": "https://doi.org/10.1007/BF02547354", "own_work": false, "note": "the 1934 Acta Mathematica paper is Leray’s foundational work on global weak solutions in three-dimensional space.", "fetched": [ "2026-09-10" ] }, { "title": "Enstrophy, Wikipedia", "url": "https://en.wikipedia.org/wiki/Enstrophy", "own_work": false, "note": "read as raw wikitext: “In the case that the flow is incompressible, or equivalently that ∇·u = 0, the enstrophy can be described as the integral of the square of the vorticity.”", "fetched": [ "2026-09-22" ] }, { "title": "W. S. Ożański and B. C. Pooley, Leray’s fundamental work on the Navier–Stokes equations: a modern review, arXiv:1708.09787", "url": "https://arxiv.org/abs/1708.09787", "own_work": false, "note": "read from the PDF: Corollary 3.11, the maximal existence time of the strong solution satisfies “T0 > C/‖∇u0‖⁴” (ν = 1), and Corollary 3.12 gives blow-up rates, ‖∇u(t)‖ ≥ C/(T0 − t)^(1/4); Leray “constructs a sequence of solutions … converging to a global-in-time weak solution”; the terms “Leray weak solution (or Leray-Hopf weak solution)” recognise the 1934 paper.", "fetched": [ "2026-09-22" ] }, { "title": "Navier–Stokes existence and smoothness, Wikipedia", "url": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness", "own_work": false, "note": "read as raw wikitext: Leray “also proved the existence of so-called weak solutions to the Navier–Stokes equations, which may not satisfy the equations pointwise but do satisfy them in mean value.”", "fetched": [ "2026-09-22" ] }, { "title": "Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "read from the PDF (Fefferman): “it is known that (A) and (B) hold provided the initial velocity u° satisfies a smallness condition,” and otherwise “if the time interval [0, ∞) is replaced by a small time interval [0, T).”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.10", "url": "https://www.teclamairme.com/#r-28.10", "chapter": 28, "status": "judgement", "claim": "The normalized supremum of reciprocal lifetimes is finite because local existence gives a universal positive lower bound for the normalized data class; it is zero exactly when no smooth solution breaks down. The equivalence of 1/T∗ = 0 with the Clay statement on ℝ³ is this book’s own derivation from two sourced facts: Leray’s lower bound on the existence time in terms of the initial enstrophy, and the scale invariance of the unforced equations.", "sources": [ { "title": "Clay Mathematics Institute, Navier-Stokes official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "the official problem is existence and smoothness for three-dimensional Navier-Stokes solutions.", "fetched": [ "2026-09-09" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "For the declared class, T* = inf T(u0) and 1/T* = sup 1/T(u0), with 1/infinity defined as 0. A positive uniform local-existence time makes the supremum finite; if every smooth solution is global, all reciprocals are 0, while any finite-time breakdown makes the supremum positive.", "fetched": [] }, { "title": "W. S. Ożański and B. C. Pooley, Leray’s fundamental work on the Navier–Stokes equations: a modern review, arXiv:1708.09787", "url": "https://arxiv.org/abs/1708.09787", "own_work": false, "note": "read from the PDF: Corollary 3.11, the maximal existence time of the strong solution satisfies “T0 > C/‖∇u0‖⁴” (ν = 1), and Corollary 3.12 gives blow-up rates, ‖∇u(t)‖ ≥ C/(T0 − t)^(1/4); Leray “constructs a sequence of solutions … converging to a global-in-time weak solution”; the terms “Leray weak solution (or Leray-Hopf weak solution)” recognise the 1934 paper.", "fetched": [ "2026-09-22" ] }, { "title": "Terence Tao, Why global regularity for Navier–Stokes is hard (author’s blog, 2007)", "url": "https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/", "own_work": false, "note": "“the Navier-Stokes equation … obeys a scale invariance: specifically, given any scaling parameter λ, and any smooth velocity field u solving the Navier-Stokes equations for some time T, one can form a new velocity field u^(λ) to the Navier-Stokes equation up to time λ²T,” in the absence of an external force.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "The book’s exact normalized invariant is a construction rather than a standard Clay notation. We mark the mathematical definition and its equivalence to the smoothness statement as a judgement call, while the underlying Clay question is sourced." }, { "id": "r-28.11", "url": "https://www.teclamairme.com/#r-28.11", "chapter": 28, "status": "confirmed", "claim": "The Yang-Mills Millennium problem asks for a rigorous four-dimensional theory with a positive mass gap; no proof of the mass-gap property is known and simulations provide evidence only.", "sources": [ { "title": "Clay Mathematics Institute, Yang-Mills & the Mass Gap", "url": "https://www.claymath.org/millennium/yang-mills-the-maths-gap/", "own_work": false, "note": "Clay: “Experiment and computer simulations suggest the existence of a “mass gap” in the solution to the quantum versions of the Yang-Mills equations. But no proof of this property is known.” The four-dimensional statement is on the Jaffe–Witten line below.", "fetched": [ "2026-09-09" ] }, { "title": "Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "(p. 142): carries the exact wording the bracket quotes — a non-trivial quantum Yang–Mills theory for every compact simple gauge group on ℝ⁴, satisfying the rigorous axioms, with a mass gap Δ > 0. The Clay landing page above carries the mass-gap blurb but not this wording.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.12", "url": "https://www.teclamairme.com/#r-28.12", "chapter": 28, "status": "judgement", "claim": "The Yang-Mills bracket is total: it records both existence of the theory and positivity of the mass gap, so it has value 0 or 1 even before the theory is constructed.", "sources": [ { "title": "Clay Mathematics Institute, Yang-Mills & the Mass Gap", "url": "https://www.claymath.org/millennium/yang-mills-the-maths-gap/", "own_work": false, "note": "Clay formulates the problem as the mathematical foundation of quantum Yang-Mills theory together with a positive mass gap.", "fetched": [ "2026-09-09" ] }, { "title": "Arthur Jaffe and Edward Witten, Quantum Yang–Mills Theory, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "(p. 142): the official statement the bracket encodes, requiring both that the theory exist and that Δ > 0.", "fetched": [ "2026-09-10" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "An Iverson bracket of the complete proposition ‘the theory exists and Delta > 0’ is defined as 1 when the proposition is true and 0 otherwise; unlike a raw Delta, it is total by construction.", "fetched": [] } ], "editorial_note": "This is the chapter’s encoding choice, not a standard physical observable. We leave it as a judgement call and explain why it avoids an undefined mass quotient." }, { "id": "r-28.13", "url": "https://www.teclamairme.com/#r-28.13", "chapter": 28, "status": "confirmed", "claim": "The CODATA 2022 proton rest mass is 938.27208943(29) MeV/c^2.", "sources": [ { "title": "CODATA 2022 recommended values", "url": "https://physics.nist.gov/cuu/Constants/", "own_work": false, "note": "the front page of NIST’s CODATA 2022 constants database, whose search and lists lead to the proton mass; the value itself is on the next line.", "fetched": [ "2026-09-09" ] }, { "title": "NIST, proton mass energy equivalent in MeV (CODATA 2022)", "url": "https://physics.nist.gov/cgi-bin/cuu/Value?mpc2mev", "own_work": false, "note": "numerical value 938.272 089 43 MeV, standard uncertainty 0.000 000 29 MeV; source: 2022 CODATA recommended values.", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.14", "url": "https://www.teclamairme.com/#r-28.14", "chapter": 28, "status": "confirmed", "claim": "More than 10^12 nontrivial zeta zeros have been verified on the critical line, with the cited rigorous computation reaching height 3·10^12.", "sources": [ { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12", "url": "https://arxiv.org/abs/2004.09765", "own_work": false, "note": "the paper verifies every zero through height 3·10^12 on the critical line.", "fetched": [ "2026-09-09" ] }, { "title": "Clay Mathematics Institute, Riemann Hypothesis", "url": "https://www.claymath.org/millennium/Riemann-Hypothesis/", "own_work": false, "note": "“This has been checked for the first 10,000,000,000,000 solutions.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.15", "url": "https://www.teclamairme.com/#r-28.15", "chapter": 28, "status": "confirmed", "claim": "The Mordell-Weil rank counts independent rational points of infinite order, and the conductor-37 curve has rank one.", "sources": [ { "title": "LMFDB, elliptic curve 37.a1", "url": "https://www.lmfdb.org/EllipticCurve/Q/37/a/1", "own_work": false, "note": "the record gives rank 1 for 37.a1; its properties table gives “Conductor 37,” “Discriminant 37,” “Rank 1,” and “Torsion structure” “trivial”; its Mordell-Weil generators table lists the point (0, 0), of infinite order.", "fetched": [ "2026-09-09" ] }, { "title": "Birch and Swinnerton-Dyer conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture", "own_work": false, "note": "read as raw wikitext: “The number of independent basis points with infinite order is called the rank of the curve”; “it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s=1.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.16", "url": "https://www.teclamairme.com/#r-28.16", "chapter": 28, "status": "confirmed", "claim": "Hamilton introduced Ricci flow in 1982, and Perelman controlled its singularities to prove Poincare and geometrization.", "sources": [ { "title": "Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255–306", "url": "https://doi.org/10.4310/jdg/1214436922", "own_work": false, "note": "Hamilton’s foundational Ricci-flow paper. (The earlier DOI given here resolved to Cannon, Shrinking Cell-Like Decompositions of Manifolds, Annals 110 (1979); corrected.)", "fetched": [ "2026-09-10" ] }, { "title": "Clay Mathematics Institute, Poincare Conjecture", "url": "https://www.claymath.org/millennium-problems/poincare-conjecture/", "own_work": false, "note": "Clay describes Perelman’s proof through Ricci flow and its relation to geometrization; its list of Perelman’s articles gives “The entropy formula for the Ricci flow and its geometric applications” and “Ricci flow with surgery on three-manifolds.”", "fetched": [ "2026-09-09" ] }, { "title": "Ricci flow, Wikipedia", "url": "https://en.wikipedia.org/wiki/Ricci_flow", "own_work": false, "note": "read as raw wikitext: a Ricci flow assigns to each t a metric gₜ “such that ∂gₜ/∂t = −2 Ric^(gₜ)”; “It is often said to be analogous to the diffusion of heat and the heat equation”; “For a 3-dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.”", "fetched": [ "2026-09-22" ] }, { "title": "Poincaré conjecture, Wikipedia (Hamilton’s program)", "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture", "own_work": false, "note": "read as raw wikitext: “Hamilton’s program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture”; “The Ricci flow usually deforms the manifold towards a rounder shape.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.17", "url": "https://www.teclamairme.com/#r-28.17", "chapter": 28, "status": "confirmed", "claim": "NP-complete problems are in NP and every NP problem reduces to them in polynomial time; Cook established satisfiability as NP-complete and Karp used that result to show twenty other natural problems NP-complete in 1972.", "sources": [ { "title": "Stephen Cook, The P versus NP Problem", "url": "https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf", "own_work": false, "note": "the official description gives the P/NP framework and states that Karp used Cook’s result to show twenty other natural problems NP-complete. It does not name them.", "fetched": [ "2026-09-10" ] }, { "title": "Karp’s 21 NP-complete problems, Wikipedia", "url": "https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems", "own_work": false, "note": "lists all twenty-one. Chromatic number and knapsack are on it; the travelling salesman problem is not — Karp has Directed Hamilton Circuit and Undirected Hamilton Circuit. The card named traveling salesman until this was checked.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.18", "url": "https://www.teclamairme.com/#r-28.18", "chapter": 28, "status": "confirmed", "claim": "A Hodge class is a rational cohomology class of type (p, p), and algebraic subvarieties supply such classes.", "sources": [ { "title": "Clay Mathematics Institute, Hodge Conjecture", "url": "https://www.claymath.org/millennium/hodge-conjecture/", "own_work": false, "note": "states that Hodge cycles are rational linear combinations of algebraic cycles. It does not use the phrase “type (p, p).”", "fetched": [ "2026-09-09" ] }, { "title": "Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "the official description, which is where the rational type-(p, p) formulation is stated.", "fetched": [ "2026-09-10" ] } ], "editorial_note": "" }, { "id": "r-28.19", "url": "https://www.teclamairme.com/#r-28.19", "chapter": 28, "status": "confirmed", "claim": "A glueball is a bound state of gluons; lattice calculations place the lightest scalar glueball near 1.7 GeV, while experiment has not isolated an unambiguous glueball.", "sources": [ { "title": "Morningstar and Peardon, The glueball spectrum from an anisotropic lattice study", "url": "https://doi.org/10.1103/PhysRevD.60.034509", "own_work": false, "note": "the paper reports the lattice glueball spectrum, including the scalar ground state near 1.7 GeV. The APS page refuses automated readers; the arXiv full text (hep-lat/9901004) gives “m(0++) = 1730(50)(80) MeV.”", "fetched": [ "2026-09-09" ] }, { "title": "Chen et al., Glueball spectrum and matrix elements on anisotropic lattices", "url": "https://doi.org/10.1103/PhysRevD.73.014516", "own_work": false, "note": "the paper reports the scalar glueball mass estimate and lattice uncertainties.", "fetched": [ "2026-09-09" ] }, { "title": "Morningstar, Update on Glueballs", "url": "https://arxiv.org/abs/2502.02547", "own_work": false, "note": "the 2025 review describes the continuing difficulty of identifying glueballs experimentally and the mixing of glueball candidates with ordinary hadrons.", "fetched": [ "2026-09-10" ] }, { "title": "Glueball, Wikipedia", "url": "https://en.wikipedia.org/wiki/Glueball", "own_work": false, "note": "read as raw wikitext: “It consists solely of gluons, without valence quarks.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.20", "url": "https://www.teclamairme.com/#r-28.20", "chapter": 28, "status": "confirmed", "claim": "Hilbert’s list contained 23 problems; the first was the continuum hypothesis, the eighth included the Riemann Hypothesis, the seventh concerned 2^(√2), and the tenth was answered no in 1970 by Yuri Matiyasevich, completing twenty-one years of work with Martin Davis, Hilary Putnam and Julia Robinson — the result is also called the MRDP theorem for that reason.", "sources": [ { "title": "MacTutor, 1900 ICM Paris", "url": "https://mathshistory.st-andrews.ac.uk/ICM/ICM_Paris_1900/", "own_work": false, "note": "the congress record identifies Hilbert’s mathematical-problems address and the 23-problem context.", "fetched": [ "2026-09-09" ] }, { "title": "Wikipedia, Hilbert’s problems", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_problems", "own_work": false, "note": "the problem list identifies the eighth, seventh and tenth problems and their standard historical resolutions; “Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later”; its table gives the 1st as “The continuum hypothesis: there is no set whose cardinality is strictly between that of the integers and that of the real numbers.” (spacing as normalized here).", "fetched": [ "2026-09-09" ] }, { "title": "Gelfond–Schneider constant, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant", "own_work": false, "note": "read as raw wikitext: on the seventh problem, “In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2^√2.”", "fetched": [ "2026-09-22" ] }, { "title": "Own reading", "url": null, "own_work": true, "note": "The seven Millennium Prize Problems (28.1) are the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP, the Poincaré conjecture, the Riemann Hypothesis, and Yang–Mills existence and the mass gap. Set beside Hilbert’s twenty-three (the Hilbert’s problems line above), only the Riemann Hypothesis is on both lists, as part of the eighth problem; Hilbert’s sixth (the axioms of physics) and his nineteenth and twentieth (on variational and boundary-value problems) lie near Yang–Mills and Navier–Stokes but state neither. Clay says only that the Riemann hypothesis “also appears” in Hilbert’s list, so “only” rests on this comparison.", "fetched": [] } ], "editorial_note": "" }, { "id": "r-28.21", "url": "https://www.teclamairme.com/#r-28.21", "chapter": 28, "status": "confirmed", "claim": "Hodge posed the conjecture in his address to the 1950 International Congress of Mathematicians, and the official Clay problem description states it for rational cohomology classes of type (p, p) on a projective non-singular algebraic variety over the complex numbers.", "sources": [ { "title": "Pierre Deligne, The Hodge Conjecture, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "read at printed pp. 45-53. Page 46: “Rational (p, p)-classes are called Hodge classes,” and “In [6], Hodge posed the Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.” Bibliography entry [6], page 53, is W. V. D. Hodge, The topological invariants of algebraic varieties, in Proceedings ICM 1950, AMS, Providence, RI, 1952, 181-192, which is what dates the conjecture to 1950. The Clay landing page cited at r-28.8 and r-28.18 gives only the informal “(rational linear) combinations of algebraic cycles” wording, with no year and no type (p, p).", "fetched": [ "2026-09-11" ] } ], "editorial_note": "" }, { "id": "r-28.22", "url": "https://www.teclamairme.com/#r-28.22", "chapter": 28, "status": "judgement", "claim": "In September 2026 a finite-time singularity was announced for a Navier–Stokes flow driven by an external force. The Clay Mathematics Institute removed the problem from its list of unsolved problems, marked its page Active, and by 17 September 2026 had given it a heading of its own: the problem list now reads Active problems, holding Navier–Stokes alone, then Unsolved problems, holding the other five, then Solved problems, holding Poincaré. The official problem is stated by Fefferman as four separate statements; a forced blow-up bears on the forced statements (C) and (D), while the number this book builds is defined over unforced flows, which is statement (A). Clay has published no adjudication, and the book continues to treat the problem as open.", "sources": [ { "title": "Clay Mathematics Institute, The Millennium Prize Problems", "url": "https://www.claymath.org/millennium-problems/", "own_work": false, "note": "Read as raw HTML rather than through a summarising fetcher. The page now carries three group headings.

Active problems

is followed by
holding one tile,

Navier-Stokes Equation

.

Unsolved problems

is followed by
holding five — Birch and Swinnerton-Dyer, Hodge, P vs NP, Riemann, and Yang-Mills & the Mass Gap.

Solved problems

is followed by
holding one, the Poincaré conjecture. On 11 September 2026 there was no Active group at all and Navier-Stokes appeared in the page’s navigation but in neither list.", "fetched": [ "2026-09-17" ] }, { "title": "Clay Mathematics Institute, Navier-Stokes Equation", "url": "https://www.claymath.org/millennium/navier-stokes-equation/", "own_work": false, "note": "The page carries

Active

, the status label that now heads a group of its own on the problem list, and its metadata gives dateModified 2026-09-11T07:43:31+00:00. It states no result, names no claimant, and awards nothing. Re-read raw on 18 September 2026: unchanged, as is the problem list.", "fetched": [ "2026-09-17" ] }, { "title": "Navier–Stokes existence and smoothness, Wikipedia", "url": "https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness", "own_work": false, "note": "read as raw wikitext: records the 8 September 2026 announcement, the priority dispute that followed it, and that as of 18 September 2026 the construction “has not yet been independently verified”.", "fetched": [ "2026-09-18" ] }, { "title": "C. Fefferman, Existence and smoothness of the Navier–Stokes equation, in The Millennium Prize Problems (Clay Mathematics Institute)", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "“we ask for a proof of one of the following four statements”; statement (C), “Breakdown of Navier–Stokes solutions on R3,” is a breakdown for some smooth force f(x, t).", "fetched": [ "2026-10-01" ] } ], "editorial_note": "Added 11 September 2026 and filed as a judgement rather than a confirmation, because what is confirmed is only what Clay’s own pages show: a status label and a removal from one list. The announced result is unrefereed, publicly disputed, and priority-contested, the claimants decline the prize, and Clay has adjudicated nothing. Under the book’s recency rule a newer source supersedes when it is a retraction, a correction, or a re-measurement; an unrefereed announcement is none of those. The author ruled on 11 September 2026 that the problem stays open in the book and that the chapter should say the news. The forced/unforced distinction is the reason the chapter’s own number is untouched: 1/T∗ is a supremum over flows with no force term. Re-read raw on 17 September 2026: the problem page is still Active with no result, no claimant, and no award, and the only change is that the problem list has grown an explicit Active problems group holding Navier-Stokes alone. Nothing supersedes the entry’s claim; only its description of the list page had gone stale, and that is what was rewritten. On 18 September 2026 the chapter’s Navier–Stokes paragraph was given the sentence the ruling asked for, and the entry gained the announcement’s own page and an independent record of it, since until then it had cited only Clay’s pages, which describe a status and never the announcement." }, { "id": "r-28.23", "url": "https://www.teclamairme.com/#r-28.23", "chapter": 28, "status": "confirmed", "claim": "A 3-manifold is a topological space in which every point has a neighborhood homeomorphic to Euclidean 3-space; closed means compact and without boundary. The Poincaré conjecture, the Clay problem solved by Grigori Perelman, asks whether every simply connected closed 3-manifold is homeomorphic to the 3-sphere.", "sources": [ { "title": "3-manifold, Wikipedia", "url": "https://en.wikipedia.org/wiki/3-manifold", "own_work": false, "note": "read as raw HTML: defines a 3-manifold by local homeomorphism with Euclidean 3-space, and closed as compact without boundary, and states the Poincaré conjecture in the simply-connected form.", "fetched": [ "2026-09-17" ] }, { "title": "Poincaré Conjecture, Clay Mathematics Institute", "url": "https://www.claymath.org/millennium/poincare-conjecture/", "own_work": false, "note": "the prize body’s own statement of the problem Perelman settled.", "fetched": [ "2026-09-17" ] } ], "editorial_note": "" }, { "id": "r-28.24", "url": "https://www.teclamairme.com/#r-28.24", "chapter": 28, "status": "confirmed", "claim": "A projective variety is the zero set in projective space of a family of homogeneous polynomials; over the complex numbers a smooth projective variety is a nonsingular such set, and it is compact, which is what makes its cohomology well behaved. The Hodge conjecture is stated for smooth complex projective varieties.", "sources": [ { "title": "Projective variety, Wikipedia", "url": "https://en.wikipedia.org/wiki/Projective_variety", "own_work": false, "note": "read as raw HTML: a projective variety as the zero locus in projective space of a family of homogeneous polynomials, and the complex-analytic consequences of projectivity, in particular compactness.", "fetched": [ "2026-09-17" ] }, { "title": "Hodge Conjecture, Clay Mathematics Institute", "url": "https://www.claymath.org/millennium/hodge-conjecture/", "own_work": false, "note": "the prize page; read again on 23 September 2026 it says only “projective algebraic varieties,” and the non-singular complex setting is carried by the official description (the MPPc line below).", "fetched": [ "2026-09-17" ] }, { "title": "Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "read from the PDF (Deligne): “Hodge Conjecture. On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.25", "url": "https://www.teclamairme.com/#r-28.25", "chapter": 28, "status": "confirmed", "claim": "A gauge theory is a field theory whose Lagrangian is invariant under a group of local transformations. The concept and the name come from Hermann Weyl’s work of 1918, where Eichinvarianz meant invariance under a local change of scale; after quantum mechanics, Weyl with Vladimir Fock and Fritz London replaced the scale factor by a phase, making it a U(1) gauge symmetry. The Clay problem asks, for every compact simple gauge group, for a nontrivial quantum Yang–Mills theory on ℝ⁴ with a mass gap.", "sources": [ { "title": "Gauge theory, Wikipedia", "url": "https://en.wikipedia.org/wiki/Gauge_theory", "own_work": false, "note": "read as raw HTML: “The concept and the name of gauge theory derives from the work of Hermann Weyl in 1918… conjectured that Eichinvarianz or invariance under the change of scale might also be a local symmetry… Weyl used the word ‘gauge’ to describe this concept, after a Latin word referring to standard cask sizes… Weyl, Vladimir Fock and Fritz London replaced the simple scale factor with a complex quantity and turned the scale transformation into a change of phase, which is a U(1) gauge symmetry”; “For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons.”", "fetched": [ "2026-09-17" ] }, { "title": "Yang–Mills and the Mass Gap, Clay Mathematics Institute", "url": "https://www.claymath.org/millennium/yang-mills-the-maths-gap/", "own_work": false, "note": "the prize body’s own page for the problem: “Progress in establishing the existence of the Yang-Mills theory and a mass gap will require the introduction of fundamental new ideas both in physics and in mathematics.” The wording for any compact simple gauge group is in the Jaffe–Witten official description, on the line below.", "fetched": [ "2026-09-17" ] }, { "title": "Hermann Weyl, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hermann_Weyl", "own_work": false, "note": "read as raw wikitext: “In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl’s gauge theory was an unsuccessful attempt to model the electromagnetic field and the gravitational field as geometrical properties of spacetime.”", "fetched": [ "2026-09-22" ] }, { "title": "Clay Mathematics Institute, The Millennium Prize Problems (official problem descriptions, 2006)", "url": "https://www.claymath.org/wp-content/uploads/2022/02/MPPc.pdf", "own_work": false, "note": "read from the PDF (Jaffe–Witten, p. 142): “Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has a mass gap Δ > 0.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.26", "url": "https://www.teclamairme.com/#r-28.26", "chapter": 28, "status": "confirmed", "claim": "Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Platt and Trudgian rigorously verified in 2021 that every zero up to height 3*10^12 lies on the critical line; Morningstar and Peardon’s lattice glueball spectrum, 1999; Chen et al.’s lattice glueball masses, 2006; Cook’s 1971 result established Boolean satisfiability as NP-complete; Claude-Louis Navier wrote the equations of viscous flow in 1822; George Stokes wrote the equations of viscous flow in 1845; Perelman’s three papers of 2002-2003 completed Hamilton’s Ricci-flow program; Chen Ning Yang and Robert Mills wrote the non-abelian gauge theory in 1954. The lightest glueball’s lattice mass is about 1.8 times the proton’s, a ratio of the book’s own.", "sources": [ { "title": "Platt and Trudgian, The Riemann hypothesis is true up to 3*10^12, Bulletin of the London Mathematical Society 53(3), 792-797 - Crossref record", "url": "https://api.crossref.org/works/10.1112/blms.12460", "own_work": false, "note": "read raw: “the publisher’s Crossref record gives the year of publication as 2021.”", "fetched": [ "2026-09-18" ] }, { "title": "Morningstar and Peardon, The glueball spectrum from an anisotropic lattice study, arXiv abstract page", "url": "https://arxiv.org/abs/hep-lat/9901004", "own_work": false, "note": "read raw: “Phys.Rev.D60:034509,1999”; the full text gives “m(0++) = 1730(50)(80) MeV and m(2++) = 2400(25)(120) MeV.”", "fetched": [ "2026-09-18" ] }, { "title": "Chen et al., Glueball spectrum and matrix elements on anisotropic lattices, arXiv abstract page", "url": "https://arxiv.org/abs/hep-lat/0510074", "own_work": false, "note": "read raw: “Phys.Rev.D73:014516,2006”; the full text’s table of masses gives 0++ at 1710(50)(80) MeV.", "fetched": [ "2026-09-18" ] }, { "title": "Stephen Cook, The P versus NP Problem, Clay Mathematics Institute official problem description", "url": "https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf", "own_work": false, "note": "read raw: “in 1971 the present author [9] introduced a notion of NP-completeness as a polynomial-time analog of c.e.-completeness.”", "fetched": [ "2026-09-18" ] }, { "title": "Claude Louis Marie Henri Navier, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Navier/", "own_work": false, "note": "read raw: “He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids..”", "fetched": [ "2026-09-18" ] }, { "title": "George Gabriel Stokes, MacTutor", "url": "https://mathshistory.st-andrews.ac.uk/Biographies/Stokes/", "own_work": false, "note": "read raw: “he published On the theories of the internal friction of fluids in motion in 1845..”", "fetched": [ "2026-09-18" ] }, { "title": "Poincare Conjecture, Clay Mathematics Institute", "url": "https://www.claymath.org/millennium/poincare-conjecture/", "own_work": false, "note": "read raw: “Nearly a century passed between its formulation in 1904 by Henri Poincaré and its solution by Grigoriy Perelman, announced in preprints posted on ArXiv.org in 2002 and 2003..”", "fetched": [ "2026-09-18" ] }, { "title": "Robert Mills (physicist), Wikipedia", "url": "https://en.wikipedia.org/w/index.php?title=Robert_Mills_(physicist)", "own_work": false, "note": "read raw: “C.N. Yang and Robert Mills formulated in 1954 a theory now known as the Yang–Mills theory.”", "fetched": [ "2026-09-18" ] }, { "title": "Own computation", "url": null, "own_work": true, "note": "with the CODATA 2022 proton rest mass 938.27208943 MeV/c² (28.13), 1730/938.27 ≈ 1.84 and 1710/938.27 ≈ 1.82 for the two lattice masses above, so the lightest glueball is about 1.8 times the proton.", "fetched": [] } ], "editorial_note": "Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim." }, { "id": "r-28.27", "url": "https://www.teclamairme.com/#r-28.27", "chapter": 28, "status": "confirmed", "claim": "States the Birch and Swinnerton-Dyer conjecture, that an elliptic curve’s rank equals the order of vanishing of its L-function at s=1, formulated in the early 1960s. It was formulated by Bryan Birch and Peter Swinnerton-Dyer; the conductor-37 curve 37.a1 has rank 1 and analytic rank 1. It is one of the seven Millennium Prize Problems, and no proof yet covers curves of rank greater than 1.", "sources": [ { "title": "Birch and Swinnerton-Dyer conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture", "own_work": false, "note": "states the conjecture and its formulation by Birch and Swinnerton-Dyer from numerical computations in the early 1960s; “The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute” and “There are currently no proofs involving curves with a rank greater than 1.” Also: “An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p,” and “In the early 1960s, Swinnerton-Dyer used the EDSAC-2 computer at the University of Cambridge Computer Laboratory to calculate the number N_{p} of points modulo p for a large number of primes p on elliptic curves whose rank was known. From these numerical results, he and his colleague Bryan John Birch conjectured ….”", "fetched": [ "2026-09-22" ] }, { "title": "Elliptic curve 37.a1, LMFDB", "url": "https://www.lmfdb.org/EllipticCurve/Q/37.a1/", "own_work": false, "note": "lists Mordell–Weil rank 1 and analytic rank 1 for the curve: “This is the elliptic curve of minimal conductor with positive rank.”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" }, { "id": "r-28.30", "url": "https://www.teclamairme.com/#r-28.30", "chapter": 28, "status": "confirmed", "claim": "States Goldbach’s conjecture, that every even integer greater than 2 is the sum of two primes, in Goldbach’s 1742 letter to Euler, and its verification by computer up to large bounds. Euler’s reply recalled Goldbach’s own even-number form. It is verified up to 4×10¹⁸, sits within Hilbert’s eighth problem of 1900 with the Riemann Hypothesis, and Chen’s 1973 theorem is the closest proven result.", "sources": [ { "title": "Goldbach’s conjecture, Wikipedia", "url": "https://en.wikipedia.org/wiki/Goldbach%27s_conjecture", "own_work": false, "note": "states the conjecture, dates Goldbach’s 1742 letter to Euler, and gives the scale of computer verification.", "fetched": [ "2026-09-22" ] }, { "title": "Hilbert’s eighth problem, Wikipedia", "url": "https://en.wikipedia.org/wiki/Hilbert%27s_eighth_problem", "own_work": false, "note": "carries Goldbach and the Riemann hypothesis within the eighth problem: “Hilbert’s eighth problem is one of David Hilbert’s list of open mathematical problems posed in 1900.”; “(where the twin prime conjecture and Goldbach conjecture are special cases of this equation)”", "fetched": [ "2026-09-22" ] } ], "editorial_note": "" } ] }