# Teclam Áirme — the Mathematical Compendium

> A piece of art in the shape of an equation: not a textbook, a proof or a popularization, and not to be judged as one. One equation, ᚏ₄.₁₆ / 𝟘 = ᚛ · ᚁ · ᚂ · ᚃ · ᚄ · … · ᚚ · ᚜, runs across 28 sections; every section is a factor, and all but one of those factors is zero. Ian, an English major, writing for his daughter Rose. The subscript 4.16 is her birthday, 16 April.

This is the front door for AI readers. The website is one large page (3.7 MB), more than a chat tool can read whole, so the book is also published here as small plain-text files, each small enough to read in full. They are generated from the page and checked against it, but they are copies: the live page at https://www.teclamairme.com/ is the source.

## How to read it

- Start with [the front matter](https://www.teclamairme.com/fortheaireaders/front.md): the welcome, the formula and how the book works.
- Each chapter has its text (`chNN.md`), its glossary cards (`chNN-cards.md`) and its references (`chNN-refs.md`). A footnote r-N.M is reference entry N.M, in chapter N's references.
- A chapter is named by its Ogham character and Latin letter, never by a number: Chapter ᚏ(R) is the first. The number in a file name is only its position in line.
- Everything is checkable: a symbol links to a glossary card, a card cites reference entries, and each entry says in one plain sentence what it is cited for, then lists its sources with a note on what each one actually says.
- 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 these files write the mathematics.** A symbol is a link to its glossary card, written [symbol](card). A fraction is (top)/(bottom); a group the page draws with tall parentheses is (…); an exponent is ^(…); <sub>…</sub> and <sup>…</sup> are sub- and superscripts; {brace: name} labels the group just before it, and a factor's first item is its chapter's name. One letter can mean different things in different chapters (e, G, σ, F, Ω): its link says which.
- **Say where you read it.** If something here looks wrong, tell your reader that you found it in the plain-text copy (name the file, e.g. `ch04.md`), which may lag the live page, and offer to check it against https://www.teclamairme.com/ before treating it as an error.

## Before you conclude the book is wrong

- **The left-hand side is not division by zero.** 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.
- **The homecoming rule.** 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.
- **Undefinability is relative.** 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.
- **One factor is not zero.** Chapter ᚁ(B), the physical world, is the single ordinary factor that is not zero.
- **The surreal reading does not preserve the equality.** Replacing −∞ by −ω makes the quotient a nonzero infinitesimal. The book says so rather than hiding it.
- **Disputed dates.** 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.

## Chapters

- [Chapter ᚏ(R): The gateway](https://www.teclamairme.com/fortheaireaders/ch01.md): Undefinable real numbers, the extended reals, and the max-plus tropical semiring. [Cards](https://www.teclamairme.com/fortheaireaders/ch01-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch01-refs.md)
- [Chapter ᚛(Prologue): 0ur starting point](https://www.teclamairme.com/fortheaireaders/ch02.md): Numeral systems: Babylon, the Maya, Ptolemy, India, Dionysius and Bede, Cambodia, the Arab world, Gwalior, Fibonacci, China, and Japan. [Cards](https://www.teclamairme.com/fortheaireaders/ch02-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch02-refs.md)
- [Chapter ᚁ(B): The actual world](https://www.teclamairme.com/fortheaireaders/ch03.md): Physical constants and physical laws: seven dimensionless braces, from Λ to the strength of gravity between two electrons, and six laws that hold exactly. [Cards](https://www.teclamairme.com/fortheaireaders/ch03-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch03-refs.md)
- [Chapter ᚂ(L): Garden variety](https://www.teclamairme.com/fortheaireaders/ch04.md): Integers, rationals, algebraic irrationals, and transcendentals: four families, from 0! to TREE(3) and from √2 to Γ(¼). [Cards](https://www.teclamairme.com/fortheaireaders/ch04-cards.md) · [References 1](https://www.teclamairme.com/fortheaireaders/ch04-refs-1.md) · [References 2](https://www.teclamairme.com/fortheaireaders/ch04-refs-2.md)
- [Chapter ᚃ(F): The local park](https://www.teclamairme.com/fortheaireaders/ch05.md): 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). [Cards](https://www.teclamairme.com/fortheaireaders/ch05-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch05-refs.md)
- [Chapter ᚄ(S): The city streets](https://www.teclamairme.com/fortheaireaders/ch06.md): Modular arithmetic, divisibility, partitions, combinatorics, and graph theory: Wilson’s theorem, the Möbius function, gcd and lcm, Ramanujan’s congruence, choosing and coloring. [Cards](https://www.teclamairme.com/fortheaireaders/ch06-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch06-refs.md)
- [Chapter ᚅ(N): The stadium](https://www.teclamairme.com/fortheaireaders/ch07.md): Combinatorial game theory and classical game theory: Nim, nimbers, and the value of matching pennies. [Cards](https://www.teclamairme.com/fortheaireaders/ch07-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch07-refs.md)
- [Chapter ᚆ(H): The observation car](https://www.teclamairme.com/fortheaireaders/ch08.md): Logic: the Iverson bracket, Boolean algebra, and statements with known truth values. [Cards](https://www.teclamairme.com/fortheaireaders/ch08-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch08-refs.md)
- [Chapter ᚇ(D): The locomotive](https://www.teclamairme.com/fortheaireaders/ch09.md): Computability, complexity, and numerical analysis: lambda calculus, the halting problem, Busy Beaver, P and NP, and a rounding error that vanishes. [Cards](https://www.teclamairme.com/fortheaireaders/ch09-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch09-refs.md)
- [Chapter ᚈ(T): The caboose](https://www.teclamairme.com/fortheaireaders/ch10.md): Linear algebra: determinant, trace, norm, eigenvalues, and the Kronecker delta. [Cards](https://www.teclamairme.com/fortheaireaders/ch10-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch10-refs.md)
- [Chapter ᚉ(C): The gallery of motion](https://www.teclamairme.com/fortheaireaders/ch11.md): Calculus, differential equations, optimization, and vector calculus: the fundamental theorem, a cooling rod, the bottom of a bowl, divergence and curl. [Cards](https://www.teclamairme.com/fortheaireaders/ch11-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch11-refs.md)
- [Chapter ᚊ(Q): The gallery of shape](https://www.teclamairme.com/fortheaireaders/ch12.md): 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. [Cards](https://www.teclamairme.com/fortheaireaders/ch12-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch12-refs.md)
- [Chapter ᚋ(M): The gallery of chance](https://www.teclamairme.com/fortheaireaders/ch13.md): 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. [Cards](https://www.teclamairme.com/fortheaireaders/ch13-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch13-refs.md)
- [Chapter ᚌ(G): The hall of mirrors](https://www.teclamairme.com/fortheaireaders/ch14.md): Lie theory, geometric algebra (Clifford algebras), group theory, and Galois theory: brackets, bivectors, the Monster, and the unsolvable quintic. [Cards](https://www.teclamairme.com/fortheaireaders/ch14-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch14-refs.md)
- [Chapter ᚍ(K): The listening room](https://www.teclamairme.com/fortheaireaders/ch15.md): Quantum mechanics and Fourier analysis: Hilbert space, Dirac’s brackets, Grassmann’s sign, and the waves that cannot see each other. [Cards](https://www.teclamairme.com/fortheaireaders/ch15-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch15-refs.md)
- [Chapter ᚎ(Z): The roundabout](https://www.teclamairme.com/fortheaireaders/ch16.md): Complex analysis: contour integrals, Cauchy’s theorem, the Cauchy–Riemann equations, and residues. [Cards](https://www.teclamairme.com/fortheaireaders/ch16-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch16-refs.md)
- [Chapter ᚐ(A): The mechanic’s workshop](https://www.teclamairme.com/fortheaireaders/ch17.md): Special functions: gamma and its family, the logarithmic integral, sinc, Bessel, Lambert W, the error function, Airy, the elliptic integral, and Fresnel. [Cards](https://www.teclamairme.com/fortheaireaders/ch17-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch17-refs.md)
- [Chapter ᚑ(O): The prime mileposts](https://www.teclamairme.com/fortheaireaders/ch18.md): Analytic number theory: the Riemann zeta function, its values, its zeros, and the Euler product. [Cards](https://www.teclamairme.com/fortheaireaders/ch18-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch18-refs.md)
- [Chapter ᚒ(U): The crossroads](https://www.teclamairme.com/fortheaireaders/ch19.md): Elliptic curves: the group law, the point at infinity, a rank of zero, and the invariants that name a curve. [Cards](https://www.teclamairme.com/fortheaireaders/ch19-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch19-refs.md)
- [Chapter ᚓ(E): The cutting](https://www.teclamairme.com/fortheaireaders/ch20.md): Measure theory and dynamics: the Cantor set, Lebesgue measure, Hausdorff dimension, a paradox, and the logistic map at rest. [Cards](https://www.teclamairme.com/fortheaireaders/ch20-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch20-refs.md)
- [Chapter ᚔ(I): The first landing](https://www.teclamairme.com/fortheaireaders/ch21.md): Complex numbers ℂ, with the dual and split-complex numbers: two-component arithmetic, Euler’s identity, and alternative rules for a new unit. [Cards](https://www.teclamairme.com/fortheaireaders/ch21-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch21-refs.md)
- [Chapter ᚕ(J): The second landing](https://www.teclamairme.com/fortheaireaders/ch22.md): Quaternions ℍ: four real components, a quarter-turn that carries i onto j, and the loss of commutativity. [Cards](https://www.teclamairme.com/fortheaireaders/ch22-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch22-refs.md)
- [Chapter ᚖ(W): The third landing](https://www.teclamairme.com/fortheaireaders/ch23.md): Octonions 𝕆: eight real components, the loss of associativity, the alternative law, a Moufang identity, and the Fano plane as a multiplication table. [Cards](https://www.teclamairme.com/fortheaireaders/ch23-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch23-refs.md)
- [Chapter ᚗ(V): The top landing](https://www.teclamairme.com/fortheaireaders/ch24.md): Sedenions 𝕊: sixteen real components, zero divisors, the loss of a multiplicative norm, and powers that remain unambiguous. [Cards](https://www.teclamairme.com/fortheaireaders/ch24-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch24-refs.md)
- [Chapter ᚘ(Y): Pre-flight controls](https://www.teclamairme.com/fortheaireaders/ch25.md): 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. [Cards](https://www.teclamairme.com/fortheaireaders/ch25-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch25-refs.md)
- [Chapter ᚙ(X): The bridge](https://www.teclamairme.com/fortheaireaders/ch26.md): Surreal numbers and the hyperreals of nonstandard analysis: two fields where you can divide by an infinity. [Cards](https://www.teclamairme.com/fortheaireaders/ch26-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch26-refs.md)
- [Chapter ᚚ(P): The possible journeys](https://www.teclamairme.com/fortheaireaders/ch27.md): 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. [Cards](https://www.teclamairme.com/fortheaireaders/ch27-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch27-refs.md)
- [Chapter ᚜(Epilogue): The homeward view](https://www.teclamairme.com/fortheaireaders/ch28.md): Seven prizes as seven numbers: analytic number theory, elliptic curves, topology, complexity, algebraic geometry, fluid dynamics, and quantum field theory. [Cards](https://www.teclamairme.com/fortheaireaders/ch28-cards.md) · [References](https://www.teclamairme.com/fortheaireaders/ch28-refs.md)

## More

- [Appendices](https://www.teclamairme.com/fortheaireaders/appendices.md): acknowledgments, the mathematical and artistic disclaimer, thoughts on AI, support, contact, and how this piece came about
- [Change log](https://www.teclamairme.com/fortheaireaders/changelog.md)

## Optional

- [The whole book as one JSON file](https://www.teclamairme.com/teclamairme-llm.json): 2.0 MB; every chapter, card and reference with every source. For a tool that can load a large file, not a chat window.
- [The website](https://www.teclamairme.com/): the book as designed, with its music. One 3.7 MB page.