# Chapter ᚂ(L): Garden variety *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T4, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch04-cards.md) · [Its references (part 1)](ch04-refs-1.md) · [Its references (part 2)](ch04-refs-2.md) · [Previous chapter](ch03.md) · [Next chapter](ch05.md) *A footnote such as [r-4.3] is reference entry 4.3 and links to it. Section 4 of 28; chapters are named by their Ogham letter, never by number.* ## Close to home ᚂ – [Luis](ch04-cards.md#g-og-luis): associated with the rowan tree, Section 4 ## Chapter ᚂ(L): Garden variety > “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. (God made the integers; all else is the work of man.)” > — Leopold Kronecker · said at the Berlin meeting of natural scientists, 1886; reported by Heinrich Weber in the *Jahresbericht der Deutschen Mathematiker-Vereinigung*, 1893 [r-4.25](ch04-refs-1.md#r-4.25) Formula: (([(9³ + 10³) − (1³ + 12³)](ch04-cards.md#g-taxicab)))/(([0!](ch04-cards.md#g-zerofact) + [4 + 9](ch04-cards.md#g-squares) + [20](ch04-cards.md#g-twenty) + [42](ch04-cards.md#g-fortytwo) + [60](ch04-cards.md#g-sixty) + [6 + 28 + 496 + 8128](ch04-cards.md#g-perfect) + [142857](ch04-cards.md#g-cyclic) + [6174](ch04-cards.md#g-kaprekar) + [Belphegor’s](ch04-cards.md#g-belphegor) + [220 + 284](ch04-cards.md#g-amicable) + [561](ch04-cards.md#g-carmichael) + [F₁₂](ch04-cards.md#g-fib144) + [7!](ch04-cards.md#g-fact7) + [5#](ch04-cards.md#g-primorial5) + [65537](ch04-cards.md#g-fermat4)) {brace: small} · ([M₃₁](ch04-cards.md#g-m31)·[3↑↑3](ch04-cards.md#g-knuth3)·[10⁶³](ch04-cards.md#g-sand)·[無量大数](ch04-cards.md#g-muryo)·[136·2²⁵⁶](ch04-cards.md#g-eddington)·[10¹⁰⁰](ch04-cards.md#g-googol)·[10¹²⁰](ch04-cards.md#g-shannon)·[M_136279841](ch04-cards.md#g-lkp)·[10¹⁰¹⁰⁰](ch04-cards.md#g-googolplex)·[Skewes’](ch04-cards.md#g-skewes)·[Moser’s](ch04-cards.md#g-moser)·[G₆₄](ch04-cards.md#g-g64)·[TREE(3)](ch04-cards.md#g-tree3)·[SSCG(3)](ch04-cards.md#g-sscg3)) {brace: GARGANTUAN} ) {brace: [integers](ch04.md#ch-T4-integers)} · ((3·[⅓](ch04-cards.md#g-third) − 1) + ([0.999…](ch04-cards.md#g-nines) − 1))/(([22/7 + 355/113](ch04-cards.md#g-piapprox) + [B₂](ch04-cards.md#g-bern2) + [H₄](ch04-cards.md#g-h4))) {brace: [rationals](ch04.md#ch-T4-rationals)} · (([φ² − φ − 1](ch04-cards.md#g-phipoly)))/(([φ](ch04-cards.md#g-phi) + [√2](ch04-cards.md#g-sqrt2) + [δ_S](ch04-cards.md#g-silver) + [cos(2π/17)](ch04-cards.md#g-cos17)) {brace: [constructible](ch04-cards.md#g-constructible)} · ([ρ](ch04-cards.md#g-plastic) + [∛2](ch04-cards.md#g-cbrt2) + [2cos(2π/7)](ch04-cards.md#g-cos7) + [τ₃](ch04-cards.md#g-tribonacci) + [λ_L](ch04-cards.md#g-lehmer) + [Conway’s](ch04-cards.md#g-conway)) {brace: [not constructible](ch04-cards.md#g-nonconstructible)} ) {brace: [algebraic](ch04.md#ch-T4-algebraic)} · ([sin π](ch04-cards.md#g-sinpi)·([√2](ch04-cards.md#g-sqrt2)·[2^√2](ch04-cards.md#g-gelfond)·([Champernowne](ch04-cards.md#g-champ))/([Copeland–Erdős](ch05-cards.md#g-cope)))^([∛2](ch04-cards.md#g-cbrt2)))/(([e](ch04-cards.md#g-e) + [π](ch04-cards.md#g-pi) + [2/π](ch04-cards.md#g-twoverpi) + [ln 2](ch04-cards.md#g-ln2) + [sin 1 + cos 1](ch04-cards.md#g-sincos1) + [Laplace limit](ch04-cards.md#g-laplacelim)) {brace: [Hermite, Lindemann](ch04-cards.md#g-lindweier)} · ([2^√2](ch04-cards.md#g-gelfond) + [e^π](ch04-cards.md#g-epi) + [iⁱ](ch04-cards.md#g-ii) + [e^π√163](ch04-cards.md#g-ram) + [log₂ 3](ch04-cards.md#g-log23) + [√2^√2](ch04-cards.md#g-sqrt2sqrt2)) {brace: [Gelfond, Schneider](ch04-cards.md#g-gelfschn)} · ([Liouville’s](ch04-cards.md#g-liou) + [Champernowne](ch04-cards.md#g-champ) + [Thue–Morse](ch04-cards.md#g-tm) + [q_KL](ch04-cards.md#g-komornik) + [P_fold](ch04-cards.md#g-paperfold)) {brace: built by hand} · ([ϖ](ch04-cards.md#g-lemn) + [Dottie](ch04-cards.md#g-dottie) + [Gauss’s](ch04-cards.md#g-gaussc) + [parabolic P](ch04-cards.md#g-parab) + [Cahen’s](ch04-cards.md#g-cahen) + [Γ(¼)](ch04-cards.md#g-gamma14)) {brace: other theorems} ) {brace: [transcendental](ch04.md#ch-T4-transcendental)} #### Type of number Algebraic irrational · Proven transcendental · Status open · Key concepts: Integers, rationals, algebraic irrationals, and transcendentals. It includes symbols you may not have seen before, such as: [0!](ch04-cards.md#g-zerofact), [↑↑](ch04-cards.md#g-knuth3), [G₆₄](ch04-cards.md#g-g64), and [TREE(3)](ch04-cards.md#g-tree3). Here each family builds its own zero out of its own members, from [1729](ch04-cards.md#g-taxicab) written two ways to [sin π](ch04-cards.md#g-sinpi). Our backyard contains a wonderful garden of the staple ingredients for many common recipes. Let’s stroll among four common species: integers, rationals, algebraic irrationals, and transcendentals. [The integers](ch01-cards.md#g-ints) are the [counting numbers](ch01-cards.md#g-nats) with their negatives. [The rationals](ch01-cards.md#g-rats) are fractions of integers. The algebraic numbers solve equations with integer coefficients: [√2](ch04-cards.md#g-sqrt2) solves x² = 2, the [golden ratio](ch04-cards.md#g-phi) solves x² = x + 1. The transcendentals are everything real that is left, and they are [almost all of the real line](ch04-cards.md#g-cantorcount), though nobody could exhibit one until 1844, or write one out as a decimal until 1851. [r-4.21](ch04-refs-1.md#r-4.21) The garden did not arrive fully planted. The discovery that [√2](ch04-cards.md#g-sqrt2) is not a fraction forced a new view of number. Liouville built [the first transcendental](ch04-cards.md#g-liou) by hand. Hermite showed in 1873 that [e](ch04-cards.md#g-e) is transcendental, Lindemann did the same for [π](ch04-cards.md#g-pi) in 1882 and ended two thousand years of trying to square the circle, [r-4.30](ch04-refs-1.md#r-4.30) [r-4.9](ch04-refs-1.md#r-4.9) and Cantor showed in 1874 that transcendentals are the rule, not the exception. [r-4.21](ch04-refs-1.md#r-4.21) Each of our four species gets a brace in the formula to serve as a basket for its varieties, with its own way of saying nothing woven into the numerator. Field notes ### [integers](ch04.md#ch-T4-integers) {#ch-T4-integers} The [counting numbers](ch01-cards.md#g-nats) and their negatives. The numerator makes a zero out of a famous taxi ride; the denominator shows off the basket twice, once small and once GARGANTUAN. #### Numerator [(9³ + 10³) − (1³ + 12³)](ch04-cards.md#g-taxicab) = 0; the famous taxicab number canceling itself, one of mathematics’ best-loved stories. G. H. Hardy took a taxi to visit Ramanujan, who was ill at Putney, and remarked that its number, [1729](ch04-cards.md#g-taxicab), seemed dull. Ramanujan knew better: it is the smallest positive integer that is a sum of two positive cubes in two different ways, 1³ + 12³ = 9³ + 10³. Subtracting one representation from the other leaves 1729 − 1729 = 0. [r-4.2](ch04-refs-1.md#r-4.2) [r-4.34](ch04-refs-2.md#r-4.34) #### Denominator Small vs Large: a sum of tiny famous integers multiplied by the product of GARGANTUAN ones. Tiny [0!](ch04-cards.md#g-zerofact) = 1; the smallest member, and a convention everything else leans on. There is exactly one way to arrange nothing. The empty product is 1, the convention that makes every [binomial formula](ch06-cards.md#g-binomthm) and every Taylor series come out right. [42](ch04-cards.md#g-fortytwo) = 42; the Answer to one of life’s greatest questions. This number is here because not every famous number is famous for its mathematics. I debated including 007 as well, lol. [r-4.3](ch04-refs-1.md#r-4.3) [142857](ch04-cards.md#g-cyclic) = 142857; a cyclic number, the repeating period of 1/7. Multiplying it by 1 through 6 only permutes its digits. [561](ch04-cards.md#g-carmichael) = 561; the smallest Carmichael number, a composite that fools Fermat’s test. 561 = 3 · 11 · 17, yet it passes Fermat’s primality test for every base coprime to it. [Fermat’s little theorem](ch06-cards.md#g-fermatlittle), in Chapter ᚄ(S), is the test it fools. [r-4.16](ch04-refs-1.md#r-4.16) [65537](ch04-cards.md#g-fermat4) = 65537; the largest known Fermat prime, and a polygon you can draw. [65537](ch04-cards.md#g-fermat4) = 2¹⁶ + 1 is prime, so by Gauss’s criterion the 65537-gon is [constructible](ch04-cards.md#g-constructible) with compass and straightedge. The same criterion, in the algebraic basket below, is why [cos(2π/17)](ch04-cards.md#g-cos17) is [constructible](ch04-cards.md#g-constructible): 17 and 65537 are both Fermat primes. [r-4.6](ch04-refs-1.md#r-4.6) [4 + 9](ch04-cards.md#g-squares), [20](ch04-cards.md#g-twenty), [60](ch04-cards.md#g-sixty), [6 + 28 + 496 + 8128](ch04-cards.md#g-perfect), [6174](ch04-cards.md#g-kaprekar), [Belphegor’s](ch04-cards.md#g-belphegor), [220 + 284](ch04-cards.md#g-amicable), [F₁₂](ch04-cards.md#g-fib144), [7!](ch04-cards.md#g-fact7), [5#](ch04-cards.md#g-primorial5) = themselves; the rest of the small, famous crowd. Perfect squares [4 and 9](ch04-cards.md#g-squares); [20](ch04-cards.md#g-twenty) and [60](ch04-cards.md#g-sixty), the bases a Mayan and a Babylonian child counted in; [the four perfect numbers](ch04-cards.md#g-perfect) the ancients knew; Kaprekar’s [6174](ch04-cards.md#g-kaprekar); [Belphegor’s prime](ch04-cards.md#g-belphegor), written as a name because thirty-one digits would wreck the line; the amicable pair [220 and 284](ch04-cards.md#g-amicable); 144, [the only square Fibonacci number](ch04-cards.md#g-fib144) after 1; [7! = 5040](ch04-cards.md#g-fact7); and the primorial [5# = 30](ch04-cards.md#g-primorial5). Every one is a positive integer, so the whole sum is positive. GARGANTUAN Multiplied in increasing order and written in the notations that reach them. The smallest has ten digits, and every step after the [googolplex](ch04-cards.md#g-googolplex) is larger than the last by an amount with no name. [M₃₁](ch04-cards.md#g-m31) = 2,147,483,647; Euler’s [Mersenne prime](ch05-cards.md#g-mersnum) 2³¹ − 1, the smallest giant by an absurd margin. Euler proved it prime in 1772; [r-4.35](ch04-refs-2.md#r-4.35) it is also the largest signed 32-bit integer. [3↑↑3](ch04-cards.md#g-knuth3) = 7,625,597,484,987; [Knuth’s arrows](ch04-cards.md#g-uparrow) at their smallest. A tower of three threes: [3^(3^3) = 3²⁷](ch04-cards.md#g-knuth3). One arrow expression on the page is enough; what happens when the arrows multiply is explained on the glossary cards for [Conway’s chains](ch04-cards.md#g-chain), the [hyperoperations](ch04-cards.md#g-hyperop), and the [fast-growing hierarchy](ch04-cards.md#g-fgh). [10¹⁰¹⁰⁰](ch04-cards.md#g-googolplex) = 10¹⁰¹⁰⁰; ten to the [googol](ch04-cards.md#g-googol), the point past which the steps have no names. It was named in 1920, at the same time as the [googol](ch04-cards.md#g-googol), and it has more digits than the universe has particles. [r-4.31](ch04-refs-1.md#r-4.31) [G₆₄](ch04-cards.md#g-g64) finite and positive; Graham’s number, famous as the largest number ever used in a serious proof. Ronald Graham’s bound for a problem in [Ramsey theory](ch06-cards.md#g-ramsey33), published by Martin Gardner in 1977. [r-4.36](ch04-refs-2.md#r-4.36) It is an exact integer; its size is the only thing stopping us from writing it. [TREE(3)](ch04-cards.md#g-tree3) finite and positive; incomparably larger than [Graham’s number](ch04-cards.md#g-g64), and back again in Chapter ᚃ(F) on purpose. It comes from [Kruskal’s tree theorem](ch04-cards.md#g-kruskal) and Friedman’s TREE function, and it is finite. It returns next chapter beside Rayo’s number; the [sixth Busy Beaver number](ch05-cards.md#g-bb6) there is so far known only from below. [10⁶³](ch04-cards.md#g-sand), [無量大数](ch04-cards.md#g-muryo), [136·2²⁵⁶](ch04-cards.md#g-eddington), [10¹⁰⁰](ch04-cards.md#g-googol), [10¹²⁰](ch04-cards.md#g-shannon), [M_136279841](ch04-cards.md#g-lkp), [Skewes’](ch04-cards.md#g-skewes), [Moser’s](ch04-cards.md#g-moser), [SSCG(3)](ch04-cards.md#g-sscg3) = themselves; the rest of the giants, in order of size. Archimedes’ 10⁶³ grains of sand; Japan’s [無量大数](ch04-cards.md#g-muryo), the immeasurable 10⁶⁸; Eddington’s count of the protons in the universe; the [googol](ch04-cards.md#g-googol); Shannon’s 10¹²⁰ chess games; the largest prime anyone had found when this was written, forty-one million digits long; [Skewes’ number](ch04-cards.md#g-skewes); [Moser’s number](ch04-cards.md#g-moser); and [Friedman’s SSCG(3)](ch04-cards.md#g-sscg3), larger than [TREE(3)](ch04-cards.md#g-tree3). Each is a finite positive integer, so their product is too. The first fraction is 0/(an integer too long to write): a positive sum times a positive product is positive. ### [rationals](ch04.md#ch-T4-rationals) {#ch-T4-rationals} Fractions of integers. The column is small and honest. #### Numerator 3 · [⅓](ch04-cards.md#g-third) − 1 = 0; the plainest way a fraction says nothing. Three thirds make exactly one in [the rationals](ch01-cards.md#g-rats), with no decimal expansion needed. [0.999…](ch04-cards.md#g-nines) − 1 = 0; chosen because it surprises people. [0.999…](ch04-cards.md#g-nines) is not creeping up on 1; it is the rational number 1 written a second way, as ½ and 0.5 are one number written twice. Call it x, notice that 10x − x is 9, and solve. #### Denominator [22/7 + 355/113](ch04-cards.md#g-piapprox) ≈ 3.1429 + 3.1416; the two fractions used for [π](ch04-cards.md#g-pi) for two thousand years. These fractions show how well rational numbers can imitate an irrational one. Archimedes’ 22/7 is an upper bound, and Zu Chongzhi’s 355/113 rounds to the same six decimal places as [π](ch04-cards.md#g-pi); neither is π itself. They are the second and fourth convergents of [π](ch04-cards.md#g-pi)’s [continued fraction](ch04-cards.md#g-contfrac). [r-4.7](ch04-refs-1.md#r-4.7) [B₂](ch04-cards.md#g-bern2) = ⅙; one of the fractions hiding inside every sum of powers and every even zeta value. The [Bernoulli numbers](ch04-cards.md#g-bernoulli) run 1, −½, ⅙, 0, −1/30, …; the one used here is positive. [H₄](ch04-cards.md#g-h4) = 25/12; a reminder that no [harmonic number](ch04-cards.md#g-harmonicnum) after the first is a whole number. [1 + ½ + ⅓ + ¼ = 25/12](ch04-cards.md#g-h4). Add as many more fractions as you like and the sum never lands on a whole number again; Leopold Theisinger proved it in 1915. [r-4.31](ch04-refs-1.md#r-4.31) The second fraction is 0/(27003/3164): [22/7 + 355/113](ch04-cards.md#g-piapprox) + [⅙](ch04-cards.md#g-bern2) + 25/12 = 27003/3164. ### [algebraic](ch04.md#ch-T4-algebraic) {#ch-T4-algebraic} Roots of polynomial equations with integer coefficients. The denominator splits by what compass and straightedge can do. #### Numerator [φ² − φ − 1](ch04-cards.md#g-phipoly) = 0; the [golden ratio](ch04-cards.md#g-phi)’s own defining equation. [φ](ch04-cards.md#g-phi) is the positive root of x² − x − 1, so substituting it gives exactly zero. #### Denominator Constructible [Constructible](ch04-cards.md#g-constructible) numbers can be drawn with compass and straightedge, starting from a segment of length 1. Every such number is algebraic, of a degree that is a power of 2 (necessary, though not enough on its own), and everything else is [non-constructible](ch04-cards.md#g-nonconstructible). [r-4.20](ch04-refs-1.md#r-4.20) [φ](ch04-cards.md#g-phi) ≈ 1.618; the [golden ratio](ch04-cards.md#g-phi), the root the numerator just used. The positive root of x² − x − 1. [√2](ch04-cards.md#g-sqrt2) ≈ 1.414; the Pythagoreans’ scandal, by tradition the first number ever proven irrational. It solves x² = 2, and the diagonal of a unit square draws it. [cos(2π/17)](ch04-cards.md#g-cos17) ≈ 0.9325; the number Gauss found at eighteen, in 1796. [r-4.35](ch04-refs-2.md#r-4.35) [r-4.8](ch04-refs-1.md#r-4.8) It is algebraic of degree 8 and [constructible](ch04-cards.md#g-constructible), which is why the regular 17-gon can be drawn. [r-4.8](ch04-refs-1.md#r-4.8) [δ_S](ch04-cards.md#g-silver) ≈ 2.414; the silver ratio, the [golden ratio](ch04-cards.md#g-phi)’s sibling. 1 + [√2](ch04-cards.md#g-sqrt2), the positive root of x² − 2x − 1. Not constructible [Not constructible](ch04-cards.md#g-nonconstructible): still algebraic, but out of the compass’s reach. [∛2](ch04-cards.md#g-cbrt2) ≈ 1.260; the classic impossibility. Wantzel proved in 1837 that it cannot be constructed, and with it the doubling of the cube. [r-4.35](ch04-refs-2.md#r-4.35) [2cos(2π/7)](ch04-cards.md#g-cos7) ≈ 1.2470; the reason the heptagon cannot be drawn. It is a root of x³ + x² − 2x − 1, of degree 3, which compass and straightedge cannot reach. [λ_L](ch04-cards.md#g-lehmer) ≈ 1.17628; Lehmer’s number, with [an open question](ch04-cards.md#g-lehmermahler) attached. It is the largest root of Lehmer’s degree-10 polynomial and the smallest known [Salem number](ch04-cards.md#g-salem); whether anything smaller exists is still open. [Conway’s](ch04-cards.md#g-conway) ≈ 1.3036; Conway’s constant, from [the look-and-say sequence](ch04-cards.md#g-lookandsay). It is the sequence’s growth rate, the root of a degree-71 polynomial Conway wrote out. [ρ](ch04-cards.md#g-plastic), [τ₃](ch04-cards.md#g-tribonacci) ≈ 1.3247, 1.8393; two more cubic roots. The [plastic number](ch04-cards.md#g-plastic) is the real root of x³ = x + 1; [the tribonacci constant](ch04-cards.md#g-tribonacci), the real root of x³ = x² + x + 1, is the growth rate of the sequence in which each term is the sum of the three before. The third fraction is 0/(about 52): every root listed is positive, so both sums are positive and their product is about 52. ### [transcendental](ch04.md#ch-T4-transcendental) {#ch-T4-transcendental} [Real numbers](ch01-cards.md#g-reals) that solve no such equation. The denominator groups its members by who proved them. #### Numerator [sin π](ch04-cards.md#g-sinpi) · ([√2](ch04-cards.md#g-sqrt2) · [2^√2](ch04-cards.md#g-gelfond) · [Champernowne](ch04-cards.md#g-champ) / [Copeland–Erdős](ch05-cards.md#g-cope))^{[∛2](ch04-cards.md#g-cbrt2)} = 0; a little factory: parts the compass can draw, parts it never can, two numbers manufactured digit by digit, and one sin π that turns the whole output into nothing. [sin π](ch04-cards.md#g-sinpi) = 0, because π is the first positive point where the sine wave crosses the axis; Lindemann proved that number transcendental, which is why the circle cannot be squared. Beside the sine, the ingredients fall into groups. [√2](ch04-cards.md#g-sqrt2) is [constructible](ch04-cards.md#g-constructible): it is the diagonal of a unit square. The exponent [∛2](ch04-cards.md#g-cbrt2) is algebraic but [non-constructible](ch04-cards.md#g-nonconstructible), which is why the cube cannot be doubled. [r-4.20](ch04-refs-1.md#r-4.20) [2^√2](ch04-cards.md#g-gelfond) and [Champernowne](ch04-cards.md#g-champ)’s 0.123456789101112… are transcendental, so no construction will ever reach them. The [Copeland–Erdős constant](ch05-cards.md#g-cope), 0.235711131719…, [Champernowne](ch04-cards.md#g-champ)’s cousin made of primes, has never been proved transcendental, and we will meet it again in the park; like [Champernowne](ch04-cards.md#g-champ)’s, it is *manufactured*, written out digit by digit by a rule, which is a kind of construction if you like poetry. The parenthesis raised to that power is about 2.4, and multiplying by [sin π](ch04-cards.md#g-sinpi) makes the numerator exactly zero. Hilariously, after all that manufacture, we have built another way to say nothing. The joke hides a lesson: knowing what kind of number each ingredient is does not tell you what kind of number the parenthesis is, and the zero in front settles the numerator without our ever having to find out. [r-4.13](ch04-refs-1.md#r-4.13) #### Denominator Familiar functions, powers, and numbers built digit by digit: the groups below show several ways transcendental numbers enter mathematics. Hermite, Lindemann [e](ch04-cards.md#g-e) + [π](ch04-cards.md#g-pi) ≈ 2.71828 + 3.14159; the two most famous transcendentals. Hermite proved [e](ch04-cards.md#g-e) transcendental in 1873; Lindemann proved it for [π](ch04-cards.md#g-pi) in 1882, ending the quest to square the circle. [r-4.9](ch04-refs-1.md#r-4.9) [r-4.35](ch04-refs-2.md#r-4.35) [2/π](ch04-cards.md#g-twoverpi), [ln 2](ch04-cards.md#g-ln2), [sin 1 + cos 1](ch04-cards.md#g-sincos1), [Laplace limit](ch04-cards.md#g-laplacelim) each positive; the rest of this group, all settled by the same theorems. [2/π](ch04-cards.md#g-twoverpi), [ln 2](ch04-cards.md#g-ln2), and [sin 1 + cos 1](ch04-cards.md#g-sincos1) fall to the [Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier). [The Laplace limit](ch04-cards.md#g-laplacelim) of celestial mechanics, about 0.6627, is settled by Hermite–Lindemann in three lines from its own defining equation. [r-4.15](ch04-refs-1.md#r-4.15) Gelfond, Schneider [Gelfond and Schneider](ch04-cards.md#g-gelfschn) independently settled [Hilbert’s seventh problem](ch04-cards.md#g-hilbert7) in 1934: an algebraic number other than 0 or 1, raised to an irrational algebraic power, is always transcendental. [r-4.9](ch04-refs-1.md#r-4.9) [r-4.19](ch04-refs-1.md#r-4.19) [iⁱ](ch04-cards.md#g-ii) ≈ 0.2079; an imaginary raised to an imaginary, which comes out real. [iⁱ](ch04-cards.md#g-ii) = [e](ch04-cards.md#g-e)^−π/2 (its principal value), because i itself is e^iπ/2. [r-4.9](ch04-refs-1.md#r-4.9) [e^π√163](ch04-cards.md#g-ram) ≈ 262537412640768743.99999999999925; a famous almost-integer. Its near miss is no accident. 163 is the last of just nine numbers d for which ℚ(√−d) has [class number](ch04-cards.md#g-classnum) 1, meaning that factoring in that number system works as neatly as it does for ordinary whole numbers, and that is what pulls the number within a hair of an integer. Almost an integer is still not one: the theorem makes it transcendental all the same. [r-4.28](ch04-refs-1.md#r-4.28) [2^√2](ch04-cards.md#g-gelfond), [e^π](ch04-cards.md#g-epi), [log₂ 3](ch04-cards.md#g-log23), [√2^√2](ch04-cards.md#g-sqrt2sqrt2) each positive; the rest of this group. [2^√2](ch04-cards.md#g-gelfond); [e^π](ch04-cards.md#g-epi), about 23.14; [log₂ 3](ch04-cards.md#g-log23), about 1.585, the number of bits in a trit; and [√2](ch04-cards.md#g-sqrt2sqrt2)^√2, about 1.6325, [the kind of number Hilbert asked about in 1900](ch04-cards.md#g-hilbert7). [r-4.31](ch04-refs-1.md#r-4.31) Built by hand [Liouville’s](ch04-cards.md#g-liou) ≈ 0.110001; Liouville’s constant, the first decimal constant ever proven transcendental. [Σ 10^−n!](ch04-cards.md#g-liou), constructed in 1851 precisely so that it could be proved transcendental. [r-4.35](ch04-refs-2.md#r-4.35) [Champernowne](ch04-cards.md#g-champ), [Thue–Morse](ch04-cards.md#g-tm), [q_KL](ch04-cards.md#g-komornik), [P_fold](ch04-cards.md#g-paperfold) each positive; more numbers built digit by digit. Mahler proved [Champernowne](ch04-cards.md#g-champ)’s and the [Thue–Morse](ch04-cards.md#g-tm) constant transcendental; [the Komornik–Loreti constant](ch04-cards.md#g-komornik), built from the same Thue–Morse digits, and [the paperfolding constant](ch04-cards.md#g-paperfold), whose binary digits are the creases of a strip folded in half again and again, complete the group. [r-4.10](ch04-refs-1.md#r-4.10) [r-4.34](ch04-refs-2.md#r-4.34) Other theorems [Dottie](ch04-cards.md#g-dottie) ≈ 0.7391; the Dottie number, which anyone can find. It is the fixed point of cosine: press the cos key over and over and the display settles on it (in radians). Strictly, it belongs with the first group, since the [Lindemann–Weierstrass theorem](ch04-cards.md#g-lindweier) is what makes it transcendental; it sits here for the calculator trick. [r-4.9](ch04-refs-1.md#r-4.9) [Γ(¼)](ch04-cards.md#g-gamma14) ≈ 3.6256; the value that ties [the gamma function](ch17-cards.md#g-gammafn) to the lemniscate. Proved transcendental by Gregory Chudnovsky around 1976. [r-4.36](ch04-refs-2.md#r-4.36) [ϖ](ch04-cards.md#g-lemn), [Gauss’s](ch04-cards.md#g-gaussc), [parabolic P](ch04-cards.md#g-parab), [Cahen’s](ch04-cards.md#g-cahen) each positive; the rest, most with a theorem of their own. [The lemniscate constant ϖ](ch04-cards.md#g-lemn), [π](ch04-cards.md#g-pi)’s twin for the lemniscate curve; [Gauss’s constant](ch04-cards.md#g-gaussc), which is ϖ divided by π; the [universal parabolic constant](ch04-cards.md#g-parab), the parabola’s answer to π (another that [Lindemann–Weierstrass](ch04-cards.md#g-lindweier) settles, like the [Dottie number](ch04-cards.md#g-dottie)); and [Cahen’s constant](ch04-cards.md#g-cahen), an alternating sum built from Sylvester’s sequence. [r-4.11](ch04-refs-1.md#r-4.11) [r-4.9](ch04-refs-1.md#r-4.9) The fourth and final fraction is 0/(about 8.6 × 10¹⁹): four positive sums multiplied give a positive denominator of about 8.6 × 10¹⁹. ### [Left on the workbench](ch04.md#ch-T4-left-on-the-workbench) {#ch-T4-left-on-the-workbench} Numbers I wrote about and then took off the page, because a brace can only be so wide: [3² − 2³ − 1 = 0](ch04-cards.md#g-mihailescu), the only pair of consecutive perfect powers, Catalan’s conjecture of 1844 and Mihăilescu’s theorem of 2002; [r-4.22](ch04-refs-1.md#r-4.22) [153](ch04-cards.md#g-narc153), the sum of the cubes of its own digits; [2520](ch04-cards.md#g-lcm2520), the smallest number that 1 through 10 all divide; [4900](ch04-cards.md#g-cannonball), the only square after 1 that is also a square pyramid of cannonballs; [r-4.22](ch04-refs-1.md#r-4.22) the [2⁶⁴ − 1](ch04-cards.md#g-wheat) grains of wheat on the chessboard; the [centillion](ch04-cards.md#g-centillion), the highest of the -illions the major dictionaries list; [3↑↑4](ch04-cards.md#g-knuth34) and [3↑↑↑3](ch04-cards.md#g-knuth5), which is Conway’s chain 3→3→3; [Loader’s number](ch04-cards.md#g-loader), among the largest numbers anyone has produced from a short computable definition; [r-4.5](ch04-refs-1.md#r-4.5) Theodorus’ [√3](ch04-cards.md#g-sqrt3); the [supergolden ratio](ch04-cards.md#g-supergolden); the [twelfth root of 2](ch04-cards.md#g-semitone), the semitone of every piano since equal temperament; [τ = 2π](ch04-cards.md#g-tau); the [golden angle](ch04-cards.md#g-goldenangle) that leaves sunflower seeds room to grow; and the [magic angle](ch04-cards.md#g-magicangle) of the chemist’s spinning sample. [r-4.23](ch04-refs-1.md#r-4.23) All real, all belonging to one of our four families, all still welcome in the margin. The four braces multiply to 0 · 0 · 0 · 0, the integer zero this chapter hands home: every denominator is a sum or product of positive numbers, so none can be zero, and each numerator is zero. [r-4.14](ch04-refs-1.md#r-4.14) What this chapter adds to the formula: 0 · 0 · 0 · 0 = 0 Some of these have familiar names and settled identities. Others we can recognize on sight while they keep something from us. The park is just beyond our fence gate and will provide fertile ground for showing us more exotic numbers. Bring your binoculars; they are hard to see. Ready for a brisk walk? Watch and read - [TREE vs Graham’s Number - Numberphile](https://www.youtube.com/watch?v=0X9DYRLmTNY) Tony Padilla on the giants in the integers column, and why TREE(3) makes Graham’s number look like a rounding error. - [List of mathematical constants](https://en.wikipedia.org/wiki/List_of_mathematical_constants) Wikipedia - [Knuth’s up-arrow notation](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation) Wikipedia