# Chapter ᚔ(I): The first landing *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T21, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch21-cards.md) · [Its references](ch21-refs.md) · [Previous chapter](ch20.md) · [Next chapter](ch22.md) *A footnote such as [r-21.3] is reference entry 21.3 and links to it. Section 21 of 28; chapters are named by their Ogham letter, never by number.* ## The launch-tower ladder ᚔ – [Iodhadh](ch21-cards.md#g-og-idad): associated with the yew tree, Section 21 ## Chapter ᚔ(I): The first landing > “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.” > — Benjamin Peirce, to his Harvard class, after deriving e^iπ = −1 · from notes by his student W. E. Byerly, published in R. C. Archibald’s *Benjamin Peirce, 1809–1880* (1925); also in Kasner and Newman, *Mathematics and the Imagination*, 1940 [r-21.17](ch21-refs.md#r-21.17) Formula: ([e^iπ + 1](ch21-cards.md#g-euler) + ([ε²](ch21-cards.md#g-dual)) + ([Re(i)](ch21-cards.md#g-re_i)))/([i·ī](ch21-cards.md#g-ibar) + [(1+i)(1−i)](ch21-cards.md#g-gaussint) + [|e^iθ|](ch21-cards.md#g-unitcirc) + [j²](ch21-cards.md#g-splitj) + [(cos π/3 + i sin π/3)⁶](ch21-cards.md#g-demoivre) + [#{z : z² + 1 = 0}](ch21-cards.md#g-fta) + [arg i](ch21-cards.md#g-argi)) #### Type of number Proven transcendental · Key concepts: Complex numbers, with the dual and split-complex numbers. It includes symbols you may not have seen before, such as: [i](ch21-cards.md#g-i), [ε](ch21-cards.md#g-dual), and [j](ch21-cards.md#g-splitj). Zero here comes from Euler’s [e^iπ + 1](ch21-cards.md#g-euler), and from an ε whose square vanishes though ε does not. The first landing of a ladder where each doubling gains a trick and costs a rule. You won! As we slowly climb to board the spaceship, notice we can learn even at this pace. The ladder stops at four landings, marked 2, 4, 8, and 16. Those are not floor numbers. The ordinary [real numbers](ch01-cards.md#g-reals) are our ground level, and at each landing [the Cayley–Dickson construction](ch24-cards.md#g-cdrule) doubles the number of real components a number carries: complex numbers, then [quaternions](ch22-cards.md#g-quaternions), [octonions](ch23-cards.md#g-octonions), and [sedenions](ch24-cards.md#g-sedenions). Every doubling buys something new and costs a familiar rule. We have already used the first of these in waves, contours, and zeta. Here at the landing marked 2, let me show you how its arithmetic is put together, and which rules it still keeps. Field notes A complex number a + bi has two real components and [i](ch21-cards.md#g-i)² = −1. The need for them arose in sixteenth-century cubic equations: a formula could pass through square roots of negative numbers on its way to a real answer. Cardano ran into them in 1545; Bombelli, in his *L’algebra* of 1572, gave the rules for multiplying them, and when they were treated consistently the imaginary parts canceled. The unfamiliar arithmetic was doing useful work before its geometry became familiar. This landing also shows two other ways to build a two-component number, the [dual](ch21-cards.md#g-dual) and [split-complex](ch21-cards.md#g-splitj) numbers. [r-21.16](ch21-refs.md#r-21.16) #### Numerator [e^iπ + 1](ch21-cards.md#g-euler) = 0; Euler’s identity, the celebrated equation. Picture each complex number a + bi as the point (a, b) of the [complex plane](ch21-cards.md#g-cplane). [r-21.8](ch21-refs.md#r-21.8) Multiplying by [i](ch21-cards.md#g-i) turns the whole plane a quarter-turn about the origin: 1 goes to i, and i goes to −1. [Euler’s formula](ch21-cards.md#g-eulerformula) e^iθ = cos θ + i sin θ gives the point on the unit circle at angle θ, measured in radians, so that π is a half-turn; at θ = π it is −1. [The identity](ch21-cards.md#g-euler) is a half-turn followed by a one-unit step back to the origin: multiplication has acquired a direction as well as a size. [r-21.1](ch21-refs.md#r-21.1) [r-21.5](ch21-refs.md#r-21.5) The price is order. The complex numbers cannot be lined up, smaller to larger, in a way that still works with addition and multiplication: in any such ordering every square is positive or zero, and i² = −1. They keep the familiar laws of arithmetic: multiplication still commutes, and we can divide by any number but zero. [r-21.8](ch21-refs.md#r-21.8) [ε²](ch21-cards.md#g-dual) = 0; the dual numbers, a second way to build a two-component number from [the reals](ch01-cards.md#g-reals), chosen for a unit that squares to zero. Their unit ε satisfies [ε²](ch21-cards.md#g-dual) = 0 with ε ≠ 0, so the term is zero by the multiplication rule. The arithmetic records first-order change: (x + ε)² = x² + 2xε, where the ordinary part is the function value and the ε coefficient is its derivative. Software uses exactly this trick, called automatic differentiation: the derivative comes straight out of the algebra, with no need to nudge x by a tiny step and measure the change. [r-21.7](ch21-refs.md#r-21.7) [Re(i)](ch21-cards.md#g-re_i) = 0; a simple coordinate reading, chosen to show where i sits. i = 0 + 1i lies entirely on the imaginary axis, so its real part is 0. Taking a real part extracts one coordinate; it does not claim that i itself vanishes. #### Denominator Conjugates measure squared length and reveal a factorization; multiplication makes turns, and complex numbers supply roots the real line lacks. The [split-complex](ch21-cards.md#g-splitj) entry changes one rule and gets a different geometry. [i·ī](ch21-cards.md#g-ibar) = 1; the first of three ways i disappears when a number meets its conjugate. Conjugation changes a + bi to a − bi, and a number times its conjugate is a² + b², its squared length. So [i·ī](ch21-cards.md#g-ibar) = i(−i) = 1. [r-21.3](ch21-refs.md#r-21.3) [(1+i)(1−i)](ch21-cards.md#g-gaussint) = 2; chosen because it shows 2 is not prime among the Gaussian integers. (1 + i)(1 − i) = 1² + 1² = 2, the second: 1 + i times its conjugate. Among [the Gaussian integers](ch21-cards.md#g-gaussint), the numbers a + bi with whole-number a and b, the only ones with a Gaussian-integer inverse are ±1 and ±i, and like ±1 among the whole numbers they never count as factors. Neither 1 + i nor 1 − i is one of them, so this is a genuine factorization: 2, prime among the ordinary whole numbers, is not prime here. [r-21.4](ch21-refs.md#r-21.4) [|e^iθ|](ch21-cards.md#g-unitcirc) = 1; every point [Euler’s formula](ch21-cards.md#g-eulerformula) reaches. For real θ, e^iθ = cos θ + i sin θ, and times its conjugate cos θ − i sin θ it gives cos²θ + sin²θ = 1, the third. So every point Euler’s formula reaches [lies on the unit circle](ch21-cards.md#g-unitcirc), at distance 1 from the origin. [r-21.3](ch21-refs.md#r-21.3) [j²](ch21-cards.md#g-splitj) = 1; the split-complex numbers, the third two-component design. Their unit j satisfies [j²](ch21-cards.md#g-splitj) = +1 with j ≠ ±1. Where a complex number’s squared length is a² + b², theirs is a² − b². Multiplying by one of them with a² − b² = 1 slides points along hyperbolas, as multiplying by e^iθ turns them round circles: a difference of squares is preserved rather than a sum. [r-21.6](ch21-refs.md#r-21.6) [(cos π/3 + i sin π/3)⁶](ch21-cards.md#g-demoivre) = 1; de Moivre’s formula, landing on a full turn. [(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ](ch21-cards.md#g-demoivre). The angle π/3 is one sixth of a full turn, so six multiplications give cos 2π + i sin 2π = 1. [r-21.9](ch21-refs.md#r-21.9) [#{z : z² + 1 = 0}](ch21-cards.md#g-fta) = 2; the fundamental theorem of algebra, counted on its simplest interesting case. [z² + 1 has the two complex roots i and −i](ch21-cards.md#g-fta). In general every complex polynomial of degree n has exactly n roots, provided a repeated root is counted as often as it repeats: (z − 1)² has the single root 1, counted twice. We counted the same two roots from the roundabout by walking around them; here the theorem hands them over without a walk. [r-21.10](ch21-refs.md#r-21.10) [arg i](ch21-cards.md#g-argi) = π/2; the angle of i: a size alone does not fix a direction, and this one happens to bring a transcendental into the sum. An argument is the angle from the positive real axis. i points straight up, so [its principal argument, in radians, is π/2](ch21-cards.md#g-argi). [r-21.11](ch21-refs.md#r-21.11) The only fraction is (0 + 0 + 0)/(8 + π/2): 1 + 2 + 1 + 1 + 1 + 2 + π/2 = 8 + π/2, positive and finite. [r-21.15](ch21-refs.md#r-21.15) **Fine print.** The numerator’s zeros come from two algebras and a real-coordinate operation, each read as the scalar 0, as always. The ε here is not [the surreal infinitesimal](ch26-cards.md#g-epsS) we will meet later, and this j is not Hamilton’s quaternion j of the next landing. Dual and [split-complex](ch21-cards.md#g-splitj) numbers are alternative two-component designs; our doubling sequence goes directly from complex numbers to [quaternions](ch22-cards.md#g-quaternions). [r-21.6](ch21-refs.md#r-21.6) arg is defined modulo 2π; the principal convention is used here. [r-21.11](ch21-refs.md#r-21.11) What this chapter adds to the formula: (0 + 0 + 0) / (8 + π/2) = 0 Two components gave multiplication a direction: multiplying by [i](ch21-cards.md#g-i) turns the plane. Hamilton wanted the same trick for space, where the order of two turns can matter. Come up to the landing marked 4. I have a story for our pause there: it begins at Hamilton’s breakfast table and ends beside a Dublin canal, with a discovery he could not wait to write down. Watch and read - [e to the pi i for dummies](https://www.youtube.com/watch?v=-dhHrg-KbJ0) Mathologer, 2015: the identity built from multiplication as rotation, for someone who knows arithmetic and nothing else - [Euler’s formula with introductory group theory](https://www.youtube.com/watch?v=mvmuCPvRoWQ) 3Blue1Brown: why e^iπ = −1 is a statement about turning, not a coincidence - [Complex number](https://en.wikipedia.org/wiki/Complex_number) Wikipedia - [Euler’s identity](https://en.wikipedia.org/wiki/Euler%27s_identity) Wikipedia