# Glossary cards — Chapter ᚂ(L): Garden variety *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T4, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch04.md) *Integers, rationals, algebraic irrationals, and transcendentals: four families, from 0! to TREE(3) and from √2 to Γ(¼). Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.* ### the letter Luis — `ᚂ` {#g-og-luis} *Ogham letter* Ogham letter name meaning either ‘blaze’ or ‘herb’; the medieval glossators read it as the rowan. 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: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19) ### Hardy–Ramanujan taxicab number — `1729` {#g-taxicab} *Integer or rational* The smallest number that is a sum of two positive cubes in two ways, 1³ + 12³ = 9³ + 10³. 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: [4.2](ch04-refs-1.md#r-4.2), [4.34](ch04-refs-2.md#r-4.34) See also: [the integers](ch01-cards.md#g-ints) ### the first two square numbers after 1 — `4, 9` {#g-squares} *Integer or rational* The squares of 2 and 3, the smallest curiosities a reader can check by hand. 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: [4.1](ch04-refs-1.md#r-4.1) ### the Answer — `42` {#g-fortytwo} *Integer or rational* 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). “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: [4.1](ch04-refs-1.md#r-4.1), [4.3](ch04-refs-1.md#r-4.3) ### the first four perfect numbers — `496, 8128` {#g-perfect} *Integer or rational* Each equals the sum of its proper divisors: 6, 28, 496, and 8128, all known to the ancient Greeks. 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: [4.3](ch04-refs-1.md#r-4.3), [4.33](ch04-refs-2.md#r-4.33) ### the cyclic number — `142857` {#g-cyclic} *Integer or rational* The repeating period of 1/7; multiplying it by 1 through 6 permutes its digits. 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: [4.3](ch04-refs-1.md#r-4.3) ### Kaprekar’s constant — `6174` {#g-kaprekar} *Integer or rational* Any four-digit number with at least two distinct digits reaches 6174 within seven steps of Kaprekar’s routine, if leading zeros are kept. 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: [4.3](ch04-refs-1.md#r-4.3), [4.31](ch04-refs-1.md#r-4.31) ### the Mersenne prime 2³¹ − 1 — `M_{31}` {#g-m31} *Integer or rational* 2147483647, Euler’s prime of 1772 and the largest signed 32-bit integer; the smallest giant by an absurd margin. 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: [4.4](ch04-refs-1.md#r-4.4) See also: [the largest known prime](ch04-cards.md#g-lkp) ### the Sand Reckoner — `10^{63}` {#g-sand} *Integer or rational* Archimedes’ count of grains of sand that would fill the universe, third century BC. 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: [4.4](ch04-refs-1.md#r-4.4) ### muryōtaisū — `無量大数` {#g-muryo} *Integer or rational* 10⁶⁸, the immeasurable large number, the top of the traditional Japanese number names. 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: [4.32](ch04-refs-1.md#r-4.32), [4.33](ch04-refs-2.md#r-4.33), [4.4](ch04-refs-1.md#r-4.4) ### googol — `10^{100}` {#g-googol} *Integer or rational* 10¹⁰⁰, a one followed by a hundred zeros; named by Kasner’s nine-year-old nephew in 1920 and published in 1940. 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: [4.31](ch04-refs-1.md#r-4.31), [4.4](ch04-refs-1.md#r-4.4) ### googolplex — `10^{10^{100}}` {#g-googolplex} *Integer or rational* Ten to the googol, named at the same time in 1920; it has more digits than the universe has particles. 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: [4.31](ch04-refs-1.md#r-4.31), [4.4](ch04-refs-1.md#r-4.4) See also: [googol](ch04-cards.md#g-googol) ### Skewes’ number — `Skewes’` {#g-skewes} *Integer or rational* 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. 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: [4.34](ch04-refs-2.md#r-4.34), [4.4](ch04-refs-1.md#r-4.4) See also: [logarithmic integral](ch17-cards.md#g-li), [the prime-counting function](ch18-cards.md#g-primecount), [the Riemann Hypothesis](ch18-cards.md#g-rh) ### Moser’s number — `Moser’s` {#g-moser} *Integer or rational* A 2 inside a mega-gon in Steinhaus–Moser polygon notation. 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: [4.4](ch04-refs-1.md#r-4.4) See also: [Skewes’ number](ch04-cards.md#g-skewes) ### Graham’s number — `G_{64}` {#g-g64} *Integer or rational* 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. 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: [4.5](ch04-refs-1.md#r-4.5) See also: [Conway’s chained-arrow notation](ch04-cards.md#g-chain), [the fast-growing hierarchy](ch04-cards.md#g-fgh) ### TREE(3) — `TREE(3)` {#g-tree3} *Integer or rational* From Kruskal’s tree theorem of 1960 and Friedman’s TREE function of the 2000s; incomparably larger than Graham’s number. 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: [4.34](ch04-refs-2.md#r-4.34), [4.35](ch04-refs-2.md#r-4.35), [4.5](ch04-refs-1.md#r-4.5) See also: [the fast-growing hierarchy](ch04-cards.md#g-fgh), [Graham’s number](ch04-cards.md#g-g64), [Kruskal’s tree theorem](ch04-cards.md#g-kruskal), [Rayo’s number](ch05-cards.md#g-rayo) ### Belphegor’s prime — `Belphegor’s` {#g-belphegor} *Integer or rational* 1, thirteen zeros, 666, thirteen zeros, 1: a 31-digit palindromic prime, written as a name because the digits wreck the line. 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: [4.3](ch04-refs-1.md#r-4.3) ### the Mayan base — `20` {#g-twenty} *Integer or rational* Vigesimal, fingers and toes. 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: [2.4](ch02-refs.md#r-2.4) ### the Babylonian base — `60` {#g-sixty} *Integer or rational* Sexagesimal, the reason there are 60 minutes and 360 degrees. 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: [2.3](ch02-refs.md#r-2.3), [4.1](ch04-refs-1.md#r-4.1) ### the first amicable pair — `220 + 284` {#g-amicable} *Integer or rational* Each is the sum of the other’s proper divisors: 220 → 284 and 284 → 220, known to the Pythagoreans. 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: [4.29](ch04-refs-1.md#r-4.29), [4.3](ch04-refs-1.md#r-4.3) ### the first Carmichael number — `561` {#g-carmichael} *Integer or rational* 3 · 11 · 17, a composite that passes Fermat’s primality test for every base coprime to it, the smallest such liar. 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: [4.16](ch04-refs-1.md#r-4.16) See also: [Fermat’s little theorem](ch06-cards.md#g-fermatlittle) ### the only square Fibonacci number — `F_{12} = 144` {#g-fib144} *Integer or rational* The twelfth Fibonacci number, 144 = 12², and apart from 0 and 1 the only Fibonacci number that is a perfect square (Cohn, 1964). 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: [4.17](ch04-refs-1.md#r-4.17) See also: [golden ratio](ch04-cards.md#g-phi) ### seven factorial — `7! = 5040` {#g-fact7} *Integer or rational* 1 · 2 · 3 · 4 · 5 · 6 · 7 = 5040, the factorial notation’s small famous value, Plato’s number of citizens in the Laws. 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: [4.1](ch04-refs-1.md#r-4.1), [4.24](ch04-refs-1.md#r-4.24) See also: [counting notation](ch06-cards.md#g-permfact), [Wilson’s theorem](ch06-cards.md#g-wilson) ### the primorial of five — `5# = 30` {#g-primorial5} *Integer or rational* 2 · 3 · 5 = 30, the product of the primes up to 5, in the primorial notation p#. 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: [4.1](ch04-refs-1.md#r-4.1), [4.24](ch04-refs-1.md#r-4.24) ### the largest known Fermat prime — `65537` {#g-fermat4} *Integer or rational* 2¹⁶ + 1, the fifth Fermat prime and the largest known; by Gauss’s criterion the 65537-gon can be drawn with compass and straightedge. 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: [4.30](ch04-refs-1.md#r-4.30), [4.6](ch04-refs-1.md#r-4.6) ### three tetrated to three, in Knuth’s arrows — `3↑↑3` {#g-knuth3} *Integer or rational* 3^(3^3) = 3²⁷ = 7,625,597,484,987: the smallest giant written in Knuth’s up-arrow notation of 1976. 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: [4.31](ch04-refs-1.md#r-4.31), [4.5](ch04-refs-1.md#r-4.5) See also: [Graham’s number](ch04-cards.md#g-g64), [the hyperoperation sequence](ch04-cards.md#g-hyperop), [three tetrated to four](ch04-cards.md#g-knuth34), [Knuth’s up-arrow notation](ch04-cards.md#g-uparrow) ### the Eddington number — `136·2^{256}` {#g-eddington} *Integer or rational* Arthur Eddington’s 1938 count of the protons in the observable universe, 136 × 2²⁵⁶ ≈ 1.57 × 10⁷⁹. 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: [4.30](ch04-refs-1.md#r-4.30), [4.4](ch04-refs-1.md#r-4.4) ### the Shannon number — `10^{120}` {#g-shannon} *Integer or rational* Claude Shannon’s 1950 lower bound for the number of possible chess games, 10¹²⁰. 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: [4.34](ch04-refs-2.md#r-4.34), [4.4](ch04-refs-1.md#r-4.4) See also: [googol](ch04-cards.md#g-googol) ### the largest known prime — `M_{136279841}` {#g-lkp} *Integer or rational* 2¹³⁶²⁷⁹⁸⁴¹ − 1, found on 12 October 2024 by Luke Durant’s GPU search for GIMPS; 41,024,320 digits. 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: [4.18](ch04-refs-1.md#r-4.18) ### Friedman’s SSCG(3) — `SSCG(3)` {#g-sscg3} *Integer or rational* 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). 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: [4.5](ch04-refs-1.md#r-4.5) See also: [TREE(3)](ch04-cards.md#g-tree3) ### zero factorial — `0!` {#g-zerofact} *Integer or rational* 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. 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: [4.1](ch04-refs-1.md#r-4.1) See also: [the binomial theorem](ch06-cards.md#g-binomthm), [counting notation](ch06-cards.md#g-permfact) ### the look-and-say sequence — `1, 11, 21, 1211, 111221, ...` {#g-lookandsay} *Function or operator* The integer sequence built by reading the previous term aloud, run by run: 1, 11, 21, 1211, 111221, and onward. 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: [4.49](ch04-refs-2.md#r-4.49) See also: [Conway’s constant](ch04-cards.md#g-conway) ### one third — `⅓` {#g-third} *Integer or rational* Three of them make exactly one in the rationals, with no decimal expansion needed. 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: [4.1](ch04-refs-1.md#r-4.1) ### zero point nine repeating — `0.999…` {#g-nines} *Integer or rational* Exactly equal to 1, not approximately: let x = 0.999…, then 10x − x = 9. 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: [4.1](ch04-refs-1.md#r-4.1) ### rational approximations of π — `22/7, 355/113` {#g-piapprox} *Integer or rational* Archimedes’ upper bound and Zu Chongzhi’s fraction, which is correct to six decimal places. 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: [4.33](ch04-refs-2.md#r-4.33), [4.7](ch04-refs-1.md#r-4.7) ### the Bernoulli number B₂ — `B_{2} = ⅙` {#g-bern2} *Integer or rational* 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. 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: [4.29](ch04-refs-1.md#r-4.29), [4.7](ch04-refs-1.md#r-4.7) See also: [Bernoulli numbers](ch04-cards.md#g-bernoulli), [Brun’s constant](ch05-cards.md#g-brun) ### the fourth harmonic number — `H_{4} = 25/12` {#g-h4} *Integer or rational* 1 + ½ + ⅓ + ¼ = 25/12; no harmonic number after the first is a whole number (Theisinger, 1915). 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: [4.31](ch04-refs-1.md#r-4.31), [4.7](ch04-refs-1.md#r-4.7) See also: [Euler–Mascheroni constant](ch05-cards.md#g-gamma), [harmonic numbers](ch04-cards.md#g-harmonicnum), [counting notation](ch06-cards.md#g-permfact) ### continued fractions and convergents — `[a₀; a₁, a₂, …]` {#g-contfrac} *Function or operator* Peel off the integer part, invert the remainder, repeat. The truncations are the convergents, giving a sequence of rational approximations. 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: [4.26](ch04-refs-1.md#r-4.26) See also: [pi](ch04-cards.md#g-pi) ### Bernoulli numbers — `Bₙ` {#g-bernoulli} *Function or operator* 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. 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: [4.41](ch04-refs-2.md#r-4.41) See also: [the rational numbers](ch01-cards.md#g-rats), [Riemann zeta function](ch18-cards.md#g-zeta) ### harmonic numbers — `Hₙ = Σ 1/k` {#g-harmonicnum} *Function or operator* The partial sums of the harmonic series, 1, 3/2, 11/6, 25/12, ..., always rational and, after the first, never a whole number. 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: [4.44](ch04-refs-2.md#r-4.44) ### golden ratio — `φ` {#g-phi} *Algebraic irrational* The positive root of x² − x − 1, about 1.618. 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: [4.1](ch04-refs-1.md#r-4.1), [4.24](ch04-refs-1.md#r-4.24) See also: [a constructible number](ch04-cards.md#g-constructible) ### square root of two — `√2` {#g-sqrt2} *Algebraic irrational* The Pythagoreans’ scandal, by tradition the first number shown not to be a fraction. √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: [4.8](ch04-refs-1.md#r-4.8), [8.9](ch08-refs.md#r-8.9) See also: [a constructible number](ch04-cards.md#g-constructible) ### plastic number — `ρ` {#g-plastic} *Algebraic irrational* The real root of x³ = x + 1, about 1.3247, van der Laan’s ratio. 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: [4.33](ch04-refs-2.md#r-4.33), [4.8](ch04-refs-1.md#r-4.8) See also: [golden ratio](ch04-cards.md#g-phi) ### silver ratio — `δ_{S}` {#g-silver} *Algebraic irrational* 1 + √2, the positive root of x² − 2x − 1, about 2.414. 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: [4.1](ch04-refs-1.md#r-4.1) See also: [a constructible number](ch04-cards.md#g-constructible), [square root of two](ch04-cards.md#g-sqrt2) ### Conway’s constant — `Conway’s` {#g-conway} *Algebraic irrational* The growth rate of the look-and-say sequence, about 1.3036, algebraic of degree 71. 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: [4.29](ch04-refs-1.md#r-4.29), [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible), [the look-and-say sequence](ch04-cards.md#g-lookandsay) ### cube root of two — `∛2` {#g-cbrt2} *Algebraic irrational* The classical non-constructible number: doubling the cube with compass and straightedge is impossible (Wantzel 1837). ∛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: [4.29](ch04-refs-1.md#r-4.29), [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible), [a non-constructible number](ch04-cards.md#g-nonconstructible) ### the golden ratio’s own equation — `φ² − φ − 1` {#g-phipoly} *Integer or rational* φ is defined as the positive root of x² − x − 1, so the polynomial vanishes at φ exactly. 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: [4.1](ch04-refs-1.md#r-4.1) See also: [golden ratio](ch04-cards.md#g-phi) ### the heptadecagon’s cosine — `cos(2π/17)` {#g-cos17} *Algebraic irrational* About 0.9325, algebraic of degree 8 and constructible: Gauss found in 1796 that the regular 17-gon can be drawn. 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: [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible) ### the heptagon’s cosine — `2cos(2π/7)` {#g-cos7} *Algebraic irrational* About 1.2470, a root of x³ + x² − 2x − 1: degree 3, so the regular heptagon cannot be drawn with compass and straightedge. 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: [4.29](ch04-refs-1.md#r-4.29), [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible) ### the tribonacci constant — `τ_{3}` {#g-tribonacci} *Algebraic irrational* 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. 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: [4.35](ch04-refs-2.md#r-4.35), [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible), [golden ratio](ch04-cards.md#g-phi) ### Lehmer’s number — `λ_{L}` {#g-lehmer} *Algebraic irrational* 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. 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: [4.32](ch04-refs-1.md#r-4.32), [4.8](ch04-refs-1.md#r-4.8) See also: [a constructible number](ch04-cards.md#g-constructible), [a Salem number](ch04-cards.md#g-salem) ### a Salem number — `θ > 1, |θⁱ| ≤ 1` {#g-salem} *Function or operator* 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. 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: [4.27](ch04-refs-1.md#r-4.27) ### the class number — `h(ℚ(√−d))` {#g-classnum} *Function or operator* 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). 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: [4.28](ch04-refs-1.md#r-4.28) See also: [the j-invariant](ch14-cards.md#g-jinv), [Ramanujan’s constant](ch04-cards.md#g-ram), [unique factorization](ch18-cards.md#g-ufd) ### Lehmer’s Mahler measure problem — `M(θ) < 1.176…?` {#g-lehmermahler} *Status open* 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. 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: [4.45](ch04-refs-2.md#r-4.45) See also: [Lehmer’s number](ch04-cards.md#g-lehmer) ### Champernowne’s constant — `Champernowne` {#g-champ} *Proven transcendental* 0.123456789101112…, normal in base 10 and transcendental (Mahler 1937). 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: [4.10](ch04-refs-1.md#r-4.10), [4.29](ch04-refs-1.md#r-4.29) See also: [a normal number](ch05-cards.md#g-normalnum) ### Gelfond–Schneider constant — `2^{√2}` {#g-gelfond} *Proven transcendental* Transcendental by the 1934 theorem that settled Hilbert’s seventh problem. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn), [Hilbert’s seventh problem](ch04-cards.md#g-hilbert7), [square root of two](ch04-cards.md#g-sqrt2) ### sine of pi — `sin π` {#g-sinpi} *Integer or rational* Exactly 0: π is where sine first vanishes after zero, so the transcendental column has a zero of its own. 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: [4.13](ch04-refs-1.md#r-4.13), [4.9](ch04-refs-1.md#r-4.9) ### Euler’s number — `e` {#g-e} *Proven transcendental* The base of natural logarithms, proven transcendental by Hermite in 1873. 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: [4.29](ch04-refs-1.md#r-4.29), [4.30](ch04-refs-1.md#r-4.30), [4.9](ch04-refs-1.md#r-4.9) ### pi — `π` {#g-pi} *Proven transcendental* Proven transcendental by Lindemann in 1882, which ended the quest to square the circle. 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: [4.33](ch04-refs-2.md#r-4.33), [4.9](ch04-refs-1.md#r-4.9) See also: [a constructible number](ch04-cards.md#g-constructible) ### Gelfond’s constant — `e^{π}` {#g-epi} *Proven transcendental* About 23.14, transcendental by Gelfond–Schneider. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn) ### i to the i — `i^{i}` {#g-ii} *Proven transcendental* An imaginary raised to an imaginary is real: e^(−π/2), about 0.2079. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [Euler’s formula](ch21-cards.md#g-eulerformula), [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn) ### two over pi — `2/π` {#g-twoverpi} *Proven transcendental* Transcendental because π is; Buffon’s needle probability and the average of |cos| over a period. 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: [13.6](ch13-refs.md#r-13.6), [4.9](ch04-refs-1.md#r-4.9) ### Ramanujan’s constant — `e^{π√163}` {#g-ram} *Proven transcendental* 262537412640768743.99999999999925, an almost-integer explained by the class number of ℚ(√−163). 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: [4.33](ch04-refs-2.md#r-4.33), [4.34](ch04-refs-2.md#r-4.34), [4.9](ch04-refs-1.md#r-4.9) See also: [the class number](ch04-cards.md#g-classnum), [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn), [the j-invariant](ch14-cards.md#g-jinv) ### lemniscate constant — `ϖ` {#g-lemn} *Proven transcendental* π’s twin for the lemniscate curve, about 2.622, with its own rare symbol. 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: [4.11](ch04-refs-1.md#r-4.11), [4.12](ch04-refs-1.md#r-4.12), [4.32](ch04-refs-1.md#r-4.32) See also: [square root of two](ch04-cards.md#g-sqrt2) ### natural logarithm of 2 — `ln 2` {#g-ln2} *Proven transcendental* Transcendental by Lindemann–Weierstrass, because e^(ln 2) = 2 is algebraic. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier) ### sine and cosine of one radian — `sin 1, cos 1` {#g-sincos1} *Proven transcendental* Two numbers on every calculator, both transcendental by Lindemann–Weierstrass. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier) ### the Laplace limit — `Laplace limit` {#g-laplacelim} *Proven transcendental* 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. 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: [4.15](ch04-refs-1.md#r-4.15) ### Liouville’s constant — `Liouville’s` {#g-liou} *Proven transcendental* Σ 10^(−n!) = 0.110001000…, the first decimal constant ever proven transcendental; Liouville announced the method in 1844 and gave this number in 1851. 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: [4.32](ch04-refs-1.md#r-4.32) See also: [counting notation](ch06-cards.md#g-permfact) ### Prouhet–Thue–Morse constant — `Thue–Morse` {#g-tm} *Proven transcendental* Its binary digits follow the Thue–Morse sequence; transcendental (Mahler 1929). 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: [4.10](ch04-refs-1.md#r-4.10), [4.34](ch04-refs-2.md#r-4.34) See also: [Champernowne’s constant](ch04-cards.md#g-champ), [Liouville’s constant](ch04-cards.md#g-liou) ### Dottie number — `Dottie` {#g-dottie} *Proven transcendental* The fixed point of cosine, 0.7390…, found by anyone who presses cos repeatedly; transcendental. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier) ### Gauss’s constant — `Gauss’s` {#g-gaussc} *Proven transcendental* The reciprocal of the arithmetic–geometric mean of 1 and √2, about 0.8346, proven transcendental. 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: [4.11](ch04-refs-1.md#r-4.11), [4.12](ch04-refs-1.md#r-4.12), [4.30](ch04-refs-1.md#r-4.30) See also: [lemniscate constant](ch04-cards.md#g-lemn), [square root of two](ch04-cards.md#g-sqrt2) ### universal parabolic constant — `parabolic P` {#g-parab} *Proven transcendental* ln(1 + √2) + √2, about 2.2955, the parabola’s answer to π; transcendental by Lindemann–Weierstrass. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier), [square root of two](ch04-cards.md#g-sqrt2) ### Cahen’s constant — `Cahen’s` {#g-cahen} *Proven transcendental* About 0.6434, an alternating sum over Sylvester’s sequence, transcendental. 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: [4.11](ch04-refs-1.md#r-4.11) See also: [continued fractions and convergents](ch04-cards.md#g-contfrac) ### the logarithm of three in base two — `log_{2} 3` {#g-log23} *Proven transcendental* About 1.585, the number of bits in a trit; transcendental by Gelfond–Schneider, since 2 raised to it is 3. 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: [4.9](ch04-refs-1.md#r-4.9) See also: [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn) ### root two to the root two — `√2^{√2}` {#g-sqrt2sqrt2} *Proven transcendental* 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. √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: [4.19](ch04-refs-1.md#r-4.19), [4.9](ch04-refs-1.md#r-4.9) See also: [Gelfond’s constant](ch04-cards.md#g-epi), [Gelfond–Schneider constant](ch04-cards.md#g-gelfond), [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn), [Hilbert’s seventh problem](ch04-cards.md#g-hilbert7), [square root of two](ch04-cards.md#g-sqrt2) ### the Komornik–Loreti constant — `q_{KL}` {#g-komornik} *Proven transcendental* About 1.7872, the smallest base in which 1 has only one expansion using the digits 0 and 1; transcendental (Allouche and Cosnard, 2000). 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: [4.10](ch04-refs-1.md#r-4.10) See also: [Prouhet–Thue–Morse constant](ch04-cards.md#g-tm) ### the paperfolding constant — `P_{fold}` {#g-paperfold} *Proven transcendental* About 0.8507, the number whose binary digits are the creases of a strip of paper folded in half again and again; transcendental. 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: [4.10](ch04-refs-1.md#r-4.10), [4.33](ch04-refs-2.md#r-4.33) ### gamma of one quarter — `Γ(¼)` {#g-gamma14} *Proven transcendental* About 3.6256, the value that ties the gamma function to the lemniscate; transcendental (Chudnovsky, ~1976). 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: [4.12](ch04-refs-1.md#r-4.12), [4.54](ch04-refs-2.md#r-4.54) See also: [the gamma function](ch17-cards.md#g-gammafn), [gamma of one half](ch17-cards.md#g-gammahalf), [Gauss’s constant](ch04-cards.md#g-gaussc), [lemniscate constant](ch04-cards.md#g-lemn), [pi](ch04-cards.md#g-pi) ### three tetrated to four — `3↑↑4` {#g-knuth34} *Integer or rational* 3^(3^(3^3)) = 3^7625597484987, a number with about 3.6 trillion digits. 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: [4.5](ch04-refs-1.md#r-4.5) See also: [googolplex](ch04-cards.md#g-googolplex), [three tetrated to three, in Knuth’s arrows](ch04-cards.md#g-knuth3), [three pentated to three, Conway’s 3→3→3](ch04-cards.md#g-knuth5) ### three pentated to three, Conway’s 3→3→3 — `3↑↑↑3` {#g-knuth5} *Integer or rational* A tower of 7,625,597,484,987 threes; in Conway’s chained-arrow notation it is 3→3→3. 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: [4.32](ch04-refs-1.md#r-4.32), [4.5](ch04-refs-1.md#r-4.5) See also: [Ackermann’s function](ch09-cards.md#g-ackfn), [Conway’s chained-arrow notation](ch04-cards.md#g-chain), [Graham’s number](ch04-cards.md#g-g64), [three tetrated to three, in Knuth’s arrows](ch04-cards.md#g-knuth3), [Skewes’ number](ch04-cards.md#g-skewes), [Knuth’s up-arrow notation](ch04-cards.md#g-uparrow) ### Loader’s number — `Loader’s` {#g-loader} *Integer or rational* 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. 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: [4.32](ch04-refs-1.md#r-4.32), [4.5](ch04-refs-1.md#r-4.5) See also: [Friedman’s SSCG(3)](ch04-cards.md#g-sscg3), [TREE(3)](ch04-cards.md#g-tree3) ### Knuth’s up-arrow notation — `a↑ⁿb` {#g-uparrow} *Function or operator* 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. 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: [4.35](ch04-refs-2.md#r-4.35), [4.5](ch04-refs-1.md#r-4.5) See also: [Graham’s number](ch04-cards.md#g-g64), [the hyperoperation sequence](ch04-cards.md#g-hyperop), [three tetrated to three, in Knuth’s arrows](ch04-cards.md#g-knuth3), [three pentated to three, Conway’s 3→3→3](ch04-cards.md#g-knuth5) ### Conway’s chained-arrow notation — `a→b→c` {#g-chain} *Function or operator* 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. 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: [4.29](ch04-refs-1.md#r-4.29), [4.5](ch04-refs-1.md#r-4.5) See also: [Graham’s number](ch04-cards.md#g-g64), [three pentated to three, Conway’s 3→3→3](ch04-cards.md#g-knuth5), [Knuth’s up-arrow notation](ch04-cards.md#g-uparrow) ### the hyperoperation sequence — `H_{n}(a, b)` {#g-hyperop} *Function or operator* Succession, addition, multiplication, exponentiation, tetration, pentation, hexation: each operation repeats the one before, in Goodstein’s 1947 naming. 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: [4.31](ch04-refs-1.md#r-4.31), [4.5](ch04-refs-1.md#r-4.5) See also: [Conway’s chained-arrow notation](ch04-cards.md#g-chain), [Knuth’s up-arrow notation](ch04-cards.md#g-uparrow) ### the fast-growing hierarchy — `f_{α}(n)` {#g-fgh} *Function or operator* 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}. 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: [4.30](ch04-refs-1.md#r-4.30), [4.5](ch04-refs-1.md#r-4.5) See also: [the Busy Beaver function](ch09-cards.md#g-busybeaver), [Graham’s number](ch04-cards.md#g-g64), [TREE(3)](ch04-cards.md#g-tree3) ### a non-constructible number — `∛2, π` {#g-nonconstructible} *Function or operator* A real number that cannot be reached from a unit length using compass and unmarked straightedge. 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: [4.20](ch04-refs-1.md#r-4.20), [4.9](ch04-refs-1.md#r-4.9) See also: [cube root of two](ch04-cards.md#g-cbrt2), [a constructible number](ch04-cards.md#g-constructible), [pi](ch04-cards.md#g-pi), [uncomputable](ch05-cards.md#g-uncomputable) ### a constructible number — `ℚ→√` {#g-constructible} *Function or operator* Reachable from a unit length by compass and unmarked straightedge, using arithmetic and successive square roots. 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: [4.20](ch04-refs-1.md#r-4.20), [4.8](ch04-refs-1.md#r-4.8) See also: [cube root of two](ch04-cards.md#g-cbrt2) ### the Lindemann–Weierstrass theorem — `L–W` {#g-lindweier} *Function or operator* 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. 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: [4.9](ch04-refs-1.md#r-4.9) ### the Gelfond–Schneider theorem — `G–S` {#g-gelfschn} *Function or operator* If a is algebraic and not 0 or 1, and b is algebraic and irrational, then a^b is transcendental. 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: [4.31](ch04-refs-1.md#r-4.31), [4.9](ch04-refs-1.md#r-4.9) See also: [Gelfond’s constant](ch04-cards.md#g-epi), [square root of two](ch04-cards.md#g-sqrt2) ### Cantor’s counting argument — `ℵ₀ < 𝕔` {#g-cantorcount} *Function or operator* The real algebraic numbers are countable and the reals are not, so almost every real number is transcendental. 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: [4.21](ch04-refs-1.md#r-4.21) See also: [a one-to-one correspondence](ch27-cards.md#g-bijection), [the real numbers](ch01-cards.md#g-reals) ### Catalan’s conjecture, Mihăilescu’s theorem — `3² − 2³ − 1` {#g-mihailescu} *Integer or rational* 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. 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: [4.22](ch04-refs-1.md#r-4.22) ### a narcissistic number — `153` {#g-narc153} *Integer or rational* 1³ + 5³ + 3³ = 153: a number equal to the sum of the cubes of its own digits. 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: [4.23](ch04-refs-1.md#r-4.23) ### the least common multiple of 1 through 10 — `2520` {#g-lcm2520} *Integer or rational* The smallest number that each of 1 through 10 divides. 2520 = 2³ · 3² · 5 · 7, each prime taken to the highest power that appears among the numbers 1 to 10. An integer. Cites: [4.23](ch04-refs-1.md#r-4.23) ### the cannonball number — `4900` {#g-cannonball} *Integer or rational* Apart from 1, the only perfect square that is also a square pyramidal number: 1² + 2² + … + 24² = 70². 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: [4.22](ch04-refs-1.md#r-4.22) ### the wheat and chessboard — `2⁶⁴ − 1` {#g-wheat} *Integer or rational* One grain on the first square, doubling on each of the 64: 18,446,744,073,709,551,615 grains in all. 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: [4.23](ch04-refs-1.md#r-4.23) ### centillion — `10³⁰³` {#g-centillion} *Integer or rational* 10³⁰³ on the short scale and 10⁶⁰⁰ on the long: the highest -illion name the major English dictionaries include. Larger -illions can be built by the same naming rules, but the dictionaries stop here. An integer. Cites: [4.23](ch04-refs-1.md#r-4.23) ### Theodorus’ constant — `√3` {#g-sqrt3} *Algebraic irrational* About 1.7320508, the square root of 3, which Theodorus of Cyrene proved irrational, as Plato reports in the Theaetetus. 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: [4.23](ch04-refs-1.md#r-4.23) See also: [a constructible number](ch04-cards.md#g-constructible) ### supergolden ratio — `ψ` {#g-supergolden} *Algebraic irrational* About 1.4655712, the real root of x³ = x² + 1. 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: [4.20](ch04-refs-1.md#r-4.20), [4.23](ch04-refs-1.md#r-4.23) See also: [golden ratio](ch04-cards.md#g-phi) ### the twelfth root of two — `¹²√2` {#g-semitone} *Algebraic irrational* About 1.059463, the frequency ratio of one semitone in twelve-tone equal temperament. 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: [4.23](ch04-refs-1.md#r-4.23) ### tau — `τ` {#g-tau} *Proven transcendental* τ = 2π ≈ 6.283185, one full turn in radians; transcendental because π is. 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: [4.23](ch04-refs-1.md#r-4.23) See also: [pi](ch04-cards.md#g-pi), [the tribonacci constant](ch04-cards.md#g-tribonacci) ### golden angle — `2π/φ²` {#g-goldenangle} *Proven transcendental* About 137.5077°, or 2π/φ² radians: the angle that separates successive florets on a sunflower head. 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: [4.23](ch04-refs-1.md#r-4.23) See also: [golden ratio](ch04-cards.md#g-phi) ### the magic angle — `θₘ` {#g-magicangle} *Proven transcendental* arccos(1/√3) ≈ 54.7356°, the tilt used in magic-angle spinning NMR. 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: [4.23](ch04-refs-1.md#r-4.23) See also: [the Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier), [Theodorus’ constant](ch04-cards.md#g-sqrt3) ### Kruskal’s tree theorem — `Tᵢ ≤ Tⱼ` {#g-kruskal} *Function or operator* 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). 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: [4.37](ch04-refs-2.md#r-4.37) See also: [TREE(3)](ch04-cards.md#g-tree3) ### Hilbert’s seventh problem — `a^{b}` {#g-hilbert7} *Function or operator* 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. 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: [4.38](ch04-refs-2.md#r-4.38) See also: [Gelfond’s constant](ch04-cards.md#g-epi), [Gelfond–Schneider constant](ch04-cards.md#g-gelfond)