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# Glossary cards — Chapter ᚔ(I): The first landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T21, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch21.md)

*Complex numbers ℂ, with the dual and split-complex numbers: two-component arithmetic, Euler’s identity, and alternative rules for a new unit. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Iodhadh — `ᚔ` {#g-og-idad}

*Ogham letter*

Ogham letter name probably made from ibhar, ‘yew’, to pair with Eadhadh; its kennings are all about age.

Idad is of uncertain meaning. McManus believes it is probably an artificial form of ibhar, ‘yew’, coined to make a pair with edad, and notes that the kennings support that sense. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give sinem fedo, “oldest tree”; caínem sen, “fairest of the ancients”; and lúth lobair, “energy of an infirm person,” the last uncertain. The Auraicept glosses ibhar, yew, which is the meaning the arboreal tradition gives it and the one the kennings will bear. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [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)

### Euler’s identity — `e^{iπ} + 1` {#g-euler}

*Integer or rational*

e^(iπ) = −1, so the numerator is 0.

Euler published the formula e^{iθ} = cos θ + i sin θ in 1748 in his Introductio in analysin infinitorum; setting θ = π gives e^{iπ} = −1, so e^{iπ} + 1 = 0. Five fundamental constants appear once each—0, 1, i, π, and e—and the polls keep noticing: a 1990 poll by the Mathematical Intelligencer named it the most beautiful theorem in mathematics, and a 2004 Physics World poll tied it with Maxwell’s equations as the greatest equation ever. It is exactly zero, a rational number, so it carries the integer color: the arithmetic happens in ℂ but the answer is 0.

Cites: [21.1](ch21-refs.md#r-21.1), [21.2](ch21-refs.md#r-21.2)

See also: [Euler’s formula](ch21-cards.md#g-eulerformula), [Maxwell’s equations](ch11-cards.md#g-maxwell)

### i times its conjugate — `i·ī` {#g-ibar}

*Integer or rational*

1: i times its conjugate −i is −i² = 1, the first of three ways this denominator makes i disappear.

The conjugate z̄ of z = a + bi is a − bi, the reflection across the real axis. For any z the product z·z̄ = a² + b² = |z|² is the square of the modulus, so i·ī = i·(−i) = −i² = 1, the first of three ways this denominator makes i vanish. Value 1, an integer.

Cites: [21.3](ch21-refs.md#r-21.3)

### norm of the Gaussian integer 1 + i — `(1+i)(1−i)` {#g-gaussint}

*Integer or rational*

2, which is why 2 is not prime among the Gaussian integers.

Gauss introduced the Gaussian integers ℤ[i] = {a + bi : a, b ∈ ℤ} in his second memoir on biquadratic reciprocity (1832). The norm N(a + bi) = (a + bi)(a − bi) = a² + b² is multiplicative, and N(1 + i) = 2, so 2 = −i(1 + i)² is no longer prime in ℤ[i]. The Gaussian primes are 1 + i and its associates, the ordinary primes p ≡ 3 (mod 4) such as 3, 7, 11, and the factors a ± bi of the primes p ≡ 1 (mod 4), which split because they are sums of two squares (5 = (2 + i)(2 − i)). Value 2, an integer.

Cites: [21.4](ch21-refs.md#r-21.4)

### modulus of a point on the unit circle — `|e^{iθ}|` {#g-unitcirc}

*Integer or rational*

1 for every real θ.

For any real θ, |e^{iθ}| = √(cos²θ + sin²θ) = 1: the point e^{iθ} sits on the unit circle at angle θ. This is Euler’s formula of 1748 read as geometry, and it is why multiplying by e^{iθ} is a rotation with no stretching. Value 1 for every θ, an integer.

Cites: [21.1](ch21-refs.md#r-21.1), [21.5](ch21-refs.md#r-21.5)

See also: [Euler’s formula](ch21-cards.md#g-eulerformula)

### the split-complex unit squared — `j²` {#g-splitj}

*Integer or rational*

In the split-complex numbers j² = 1 with j ≠ ±1, the third way to build a two-component number from the reals; this j is not the quaternion one.

The split-complex numbers a + bj, with j² = +1 and j ≠ ±1, go back to James Cockle’s tessarines of 1848, of which they are the subalgebra spanned by 1 and j; Clifford used them in the 1870s as coefficients for his split-biquaternions. They are the third and last way to build a two-component number from the real line (the others being i² = −1 and ε² = 0), and their unit hyperbola plays the role the unit circle plays for ℂ, which is why, since the late twentieth century, their multiplication has been read as the Lorentz boost of a spacetime plane. This j is unrelated to the quaternion j of Chapter ᚕ(J). Value j² = 1, an integer.

Cites: [21.6](ch21-refs.md#r-21.6)

See also: [the dual-number zero](ch21-cards.md#g-dual), [the real numbers](ch01-cards.md#g-reals)

### the dual-number zero — `ε²` {#g-dual}

*Integer or rational*

In the dual numbers ε² = 0 with ε ≠ 0, the second way to build a two-component number from the reals; this ε is not the surreal one.

The dual numbers a + bε with ε² = 0 and ε ≠ 0 were introduced by William Clifford in 1873. Because (a + bε)² = a² + 2abε, a function evaluated at x + ε returns f(x) + f′(x)ε: the derivative comes for free, which is the idea behind forward-mode automatic differentiation. ε² is exactly 0 by definition, so the rider carries the integer color. This ε is not the surreal infinitesimal of Chapter ᚙ(X).

Cites: [21.7](ch21-refs.md#r-21.7)

See also: [surreal infinitesimal](ch26-cards.md#g-epsS), [the real numbers](ch01-cards.md#g-reals)

### the real part of i — `Re(i)` {#g-re_i}

*Integer or rational*

0: a complex number z = x + iy has real part Re z = x and imaginary part Im z = y, and i sits entirely on the imaginary axis.

Re and Im are the two coordinates of the complex plane, the picture Caspar Wessel presented in 1797 (printed 1799), Argand drew in 1806, and Gauss made respectable in 1831. Re(i) = 0 and Im(1) = 0 say that 1 and i lie on different axes: a real number has no imaginary part and i has no real part. Exactly 0, an integer, riding beside Euler’s identity.

Cites: [21.8](ch21-refs.md#r-21.8)

See also: [the complex plane](ch21-cards.md#g-cplane), [Euler’s identity](ch21-cards.md#g-euler)

### de Moivre’s formula, at a sixth root of unity — `(cos π/3 + i sin π/3)^{6}` {#g-demoivre}

*Integer or rational*

1: de Moivre’s formula (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ, with θ = π/3 and n = 6, lands on cos 2π + i sin 2π = 1.

The formula (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ goes back to Abraham de Moivre’s work on roots of equations (1707) and on cos nθ (1722, 1730), and it follows in one line from Euler’s e^{iθ} = cos θ + i sin θ. Raising a point on the unit circle to the nth power multiplies its angle by n, so the sixth power of the point at 60° returns to 1; the six sixth roots of unity are the corners of a regular hexagon, and this is one of them. Value 1, an integer.

Cites: [21.1](ch21-refs.md#r-21.1), [21.9](ch21-refs.md#r-21.9)

See also: [Euler’s formula](ch21-cards.md#g-eulerformula)

### the fundamental theorem of algebra, counted — `#{z : z^{2} + 1 = 0}` {#g-fta}

*Integer or rational*

2: every degree-n polynomial with complex coefficients has exactly n complex roots counted with multiplicity, and z² + 1 has the two roots i and −i.

The fundamental theorem of algebra says that every nonconstant polynomial with complex coefficients has a root in ℂ, so that a degree-n polynomial has n roots counted with multiplicity. d’Alembert attempted it in 1746; Gauss’s doctoral proof of 1799 had a topological gap that Ostrowski filled only in 1920; Argand gave the first rigorous proof in 1806, and was the first to state the theorem for complex rather than real coefficients. Here it counts the roots of z² + 1: exactly two, i and −i. Value 2, an integer.

Cites: [21.10](ch21-refs.md#r-21.10)

See also: [Liouville’s theorem](ch16-cards.md#g-liouville)

### the argument of i — `arg i` {#g-argi}

*Proven transcendental*

π/2 is the principal argument of i; an argument is an angle from the positive real axis and is defined modulo 2π.

The polar form z = |z|e^{i arg z} writes a complex number by its modulus and its angle; arg i = π/2 because i is one unit up the imaginary axis. The argument is defined only up to multiples of 2π, and the principal value is taken in (−π, π]. Since π is transcendental (Lindemann, 1882), so is π/2, hence the transcendental color.

Cites: [21.11](ch21-refs.md#r-21.11), [21.18](ch21-refs.md#r-21.18)

See also: [pi](ch04-cards.md#g-pi)

### imaginary unit — `i` {#g-i}

*Non-real object*

A square root of −1: i² = −1.

Cardano computed with such roots in the Ars Magna of 1545 and called the result “as subtle as it is useless”; Rafael Bombelli gave the multiplication rules that made them work in L’algebra of 1572, and used them to reach the three real roots of particular irreducible cubics; Descartes coined the word imaginary for them in La Géométrie (1637), to stress their unreal nature, and Newton was using the word by 1670; Euler introduced the letter i (a paper of 1777, printed 1794); Wessel (1797), Argand (1806), and Gauss (1831) drew it as the point one unit above the origin, and Gauss gave the name complex number, and suggested lateral in place of imaginary. Its powers cycle with period four: i, −1, −i, 1. It is a non-real object, a tool of this chapter’s framework rather than a value the formula hands home, so it is tinted as an object rather than a number.

Cites: [21.12](ch21-refs.md#r-21.12), [21.16](ch21-refs.md#r-21.16), [21.18](ch21-refs.md#r-21.18), [21.8](ch21-refs.md#r-21.8)

### the Riemann sphere — `ℂ ∪ {∞}` {#g-riemannsphere}

*Non-real object*

The complex plane with one point at infinity added, which makes it a sphere; 1/0 = ∞ is legal there, and the formula does not go there.

Riemann introduced the extended complex plane in 1857: adjoin a single point ∞ and the plane closes up into a sphere, via stereographic projection. Möbius transformations z ↦ (az + b)/(cz + d) become bijections of the sphere, and 1/0 = ∞ becomes a legitimate value. It is a non-real object and a tool of this chapter only; the formula’s rule against dividing by zero is a rule of the plane, not the sphere, which is why the sphere is listed here as a tool but plays no part in the chapter’s fraction.

Cites: [21.13](ch21-refs.md#r-21.13)

See also: [the complex plane](ch21-cards.md#g-cplane), [a Möbius transformation](ch16-cards.md#g-mobiustrans)

### Euler’s formula — `e^{iθ} = cos θ + i sin θ` {#g-eulerformula}

*Function or operator*

The engine of the chapter, from Euler’s Introductio of 1748: the exponential of an imaginary angle is the point on the unit circle at that angle, and θ = π gives the identity in the numerator.

Euler published e^{iθ} = cos θ + i sin θ in the Introductio in analysin infinitorum of 1748; Roger Cotes had the logarithmic form ln(cos θ + i sin θ) = iθ in 1714. It puts e^{iθ} on the unit circle, so |e^{iθ}| = 1, and it turns multiplication of complex numbers into addition of angles, which is why de Moivre’s formula follows in one line, and why the polar form z = |z|e^{i arg z} exists. Euler’s identity is the case θ = π. A formula rather than a value, hence the tool tag.

Cites: [21.1](ch21-refs.md#r-21.1), [21.5](ch21-refs.md#r-21.5)

See also: [de Moivre’s formula, at a sixth root of unity](ch21-cards.md#g-demoivre), [Euler’s identity](ch21-cards.md#g-euler), [modulus of a point on the unit circle](ch21-cards.md#g-unitcirc)

### the complex plane — `ℂ ≅ ℝ²` {#g-cplane}

*Non-real object*

Every complex number is a point: a + bi sits at (a, b). Wessel drew it in 1797, Argand in 1806, and Gauss’s announcement of 1831 made it respectable; it is also called the Argand diagram.

Identifying a + bi with the point (a, b) makes addition a translation and multiplication a rotation combined with a scaling, which is the geometric content of Euler’s formula. Caspar Wessel presented the picture to the Danish Academy in 1797 (printed 1799, and unnoticed), Jean-Robert Argand published it in 1806, and Gauss’s 1831 announcement of his second memoir on biquadratic residues, printed in 1832, fixed the modern view and the name. The plane cannot be ordered in a way compatible with its arithmetic, which is the property the first rung of the ladder gives up. A non-real object and the chapter’s picture.

Cites: [21.5](ch21-refs.md#r-21.5), [21.8](ch21-refs.md#r-21.8)

See also: [Euler’s formula](ch21-cards.md#g-eulerformula)

### the Brahmagupta–Fibonacci two-square identity — `(a² + b²)(c² + d²) = (ac − bd)² + (ad + bc)²` {#g-twosq}

*Function or operator*

The complex norm is multiplicative, |zw| = |z||w|, written out in integers: Diophantus stated it in the third century, Brahmagupta c. 628, Fibonacci in 1225.

The identity appears in Diophantus’s Arithmetica (III, 19), was rediscovered by Brahmagupta c. 628 and generalized to his identity for Pell’s equation, and reached Europe in Fibonacci’s Liber Quadratorum of 1225, which is why it often carries his name. It says the norm of a product of Gaussian integers is the product of the norms, and it is the n = 2 case of the ladder’s chain of norm identities: Euler’s four squares (1748) for the quaternions, Degen’s eight squares (c. 1818) for the octonions, and, by Hurwitz’s theorem of 1898, no more. A formula rather than a value, hence the tool tag.

Cites: [21.14](ch21-refs.md#r-21.14)

See also: [Hurwitz’s theorem](ch23-cards.md#g-hurwitz), [the octonions](ch23-cards.md#g-octonions), [the quaternions](ch22-cards.md#g-quaternions)

### Fermat’s theorem on sums of two squares — `p ≡ 1 (mod 4)` {#g-fermat2sq}

*Function or operator*

Fermat’s theorem on sums of two squares says an odd prime is expressible as a sum of two squares exactly when it leaves remainder 1 on division by 4, and 13 = 2² + 3² is the smallest example after 5.

Pierre de Fermat stated the theorem around 1640 without a surviving proof; Leonhard Euler found the first proof after much effort, announcing it to Goldbach in letters of 1747 and 1749 and publishing it between 1752 and 1755. A prime p ≡ 1 (mod 4) is always expressible as a² + b², uniquely up to order and sign, while a prime p ≡ 3 (mod 4) never is; the theorem reflects the fact that such primes split, rather than stay prime, when factored inside the Gaussian integers, where a² + b² = (a+bi)(a−bi). It connects to two other cards of this chapter: the Brahmagupta–Fibonacci identity, which says a product of two sums of two squares is again a sum of two squares, and the norm of the Gaussian integer 1 + i.

Cites: [21.20](ch21-refs.md#r-21.20)

See also: [norm of the Gaussian integer 1 + i](ch21-cards.md#g-gaussint)