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# Chapter ᚕ(J): The second landing

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T22, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [Its glossary cards](ch22-cards.md) · [Its references](ch22-refs.md) · [Previous chapter](ch21.md) · [Next chapter](ch23.md)

*A footnote such as [r-22.3] is reference entry 22.3 and links to it. Section 22 of 28; chapters are named by their Ogham letter, never by number.*

ᚕ – [Ébad](ch22-cards.md#g-og-ebad): associated with the elecampane, Section 22

## Chapter ᚕ(J): The second landing

> “An electric circuit seemed to close; and a spark flashed forth, the herald (as I foresaw, immediately) of many long years to come of definitely directed thought and work.”

> — William Rowan Hamilton, on the moment at Brougham Bridge, 16 October 1843 · letter to his son Archibald, 5 August 1865 [r-22.1](ch22-refs.md#r-22.1)

Formula: ([ijk + 1](ch22-cards.md#g-ijk) + ([q i q<sup>−1</sup> − j, q = (1 + k)/√2](ch22-cards.md#g-qrot)))/([‖1+i+j+k‖](ch22-cards.md#g-qnorm)·([ij − ji](ch22-cards.md#g-comm))·([(1 + i + j + k)(1 − i − j − k)](ch22-cards.md#g-qconj))·([|Q₈|](ch22-cards.md#g-q8))) 

#### Type of number

Non-real object ·

Key concepts: [Quaternions](ch22-cards.md#g-quaternions): four real components and a multiplication where order matters. It includes symbols you may not have seen before, such as: [i, j, k](ch22-cards.md#g-ijk), and [Q<sub>8</sub>](ch22-cards.md#g-q8). Zero here comes from Hamilton’s ijk = −1, and from a rotation that carries i exactly onto j. The rule lost at this landing: ij and ji differ.

There is room to pause at the four-component landing. Before we climb any farther, imagine turning a book about one axis and then another. Now make the same two turns in the other order. The book’s final orientation can change, and that small experiment is the difficulty our next arithmetic has to respect. William Rowan Hamilton spent years looking for a way to make multiplication remember it.

He was seeking a spatial counterpart to complex arithmetic. His sons would ask at breakfast whether he could multiply triples yet. On 16 October 1843, walking with his wife along Dublin’s Royal Canal, he saw how to proceed with four components and a multiplication that need not commute. He carved [i² = j² = k² = ijk = −1](ch22-cards.md#g-ijk) into Brougham Bridge. [r-22.1](ch22-refs.md#r-22.1)

Field notes

[A quaternion](ch22-cards.md#g-quaternions) has the form a + bi + cj + dk, with four real coefficients. Its units satisfy ij = k, jk = i, and ki = j, and reversing two distinct units changes the sign: ji = −k. Associativity survives, so brackets can still be moved: (ab)c = a(bc). Commutativity does not: ab need not equal ba.

#### Numerator

[ijk + 1](ch22-cards.md#g-ijk) = 0; Hamilton’s carving, completed by one more symbol.

Using ij = k, we get ijk = k² = −1, so [ijk + 1](ch22-cards.md#g-ijk) = 0. [r-22.2](ch22-refs.md#r-22.2)

[q i q<sup>−1</sup> − j, q = (1 + k)/√2](ch22-cards.md#g-qrot) = 0; chosen to show what this arithmetic is for: an exact rotation done by multiplication.

Encode a spatial vector as a purely imaginary quaternion xi + yj + zk. [A unit quaternion](ch22-cards.md#g-versor) q rotates it by v ↦ qvq⁻¹, where q = cos(θ/2) + k sin(θ/2) turns space by θ about the k-axis. Here q = (1 + k)/[√2](ch04-cards.md#g-sqrt2) = cos 45° + k sin 45°, so θ/2 = 45° and θ = 90°: a quarter-turn about the k-axis. Its inverse is q⁻¹ = (1 − k)/[√2](ch04-cards.md#g-sqrt2), and [q i q⁻¹ = (1 + k)i(1 − k)/2 = (i + j)(1 − k)/2 = j](ch22-cards.md#g-qrot). The i direction has turned exactly onto the j direction, like one of the turns we gave the book, so subtracting j leaves zero. [r-22.4](ch22-refs.md#r-22.4)

#### Denominator

A length, a disagreement about order, a conjugate product, and an eight-unit group: four views of [Hamilton’s arithmetic](ch22-cards.md#g-quaternions).

[‖1+i+j+k‖](ch22-cards.md#g-qnorm) = 2; the simplest quaternion with every component present.

The norm of a + bi + cj + dk is √(a² + b² + c² + d²), so here it is [√4 = 2](ch22-cards.md#g-qnorm).

[ij − ji](ch22-cards.md#g-comm) = 2k; the commutator, chosen because it refuses to be zero.

[ij − ji](ch22-cards.md#g-comm) = k − (−k) = 2k. It is nonzero precisely because order matters, and this is the book experiment done in arithmetic. Take r = (1 + i)/[√2](ch04-cards.md#g-sqrt2), a quarter-turn about the i-axis, and s = (1 + j)/[√2](ch04-cards.md#g-sqrt2), a quarter-turn about the j-axis. Then rs = (1 + i + j + k)/2 but sr = (1 + i + j − k)/2: the same two turns, made in the other order, give a different rotation. [r-22.12](ch22-refs.md#r-22.12)

[(1 + i + j + k)(1 − i − j − k)](ch22-cards.md#g-qconj) = 4; a quaternion times its conjugate, the tool that makes division possible.

Conjugation reverses the signs of i, j, and k, and q·q̄ = ‖q‖² always, so [this product is 2² = 4](ch22-cards.md#g-qconj).

[|Q₈|](ch22-cards.md#g-q8) = 8; the quaternion group, the finite group generated by Hamilton’s units, where his minus sign lives.

Multiply any two of [the eight units {±1, ±i, ±j, ±k}](ch22-cards.md#g-q8) and you land on one of the eight again. Any group that obeys Hamilton’s carving, with −1 different from 1, has to hold all eight. [r-22.5](ch22-refs.md#r-22.5) [r-22.16](ch22-refs.md#r-22.16)

The only fraction is (0 + 0)/128k: 2 × 2k × 4 × 8 = 128k, a quaternion rather than a positive real, but nonzero, with inverse −k/128. [r-22.13](ch22-refs.md#r-22.13)

**Fine print.** [Every nonzero quaternion has an inverse](ch22-cards.md#g-quaternions): its conjugate divided by its norm squared. Because multiplication does not commute, dividing on the left or the right can give different answers; with a zero numerator both give zero. The rotation rule is two-sided, qvq⁻¹, and the half-angle is part of the encoding. q and −q encode the same rotation. The unit [quaternions](ch22-cards.md#g-quaternions) form [a group whose multiplication composes spatial rotations in order](ch22-cards.md#g-s3su2), and navigation, computer graphics, and spacecraft attitude control all use them to handle rotations. [r-22.10](ch22-refs.md#r-22.10) [r-22.6](ch22-refs.md#r-22.6) [r-22.14](ch22-refs.md#r-22.14)

What this chapter adds to the formula: (0 + 0) / 128k = 0

Four components let multiplication remember the order of two turns; the commutator below the line is the price tag. The next landing is marked 8. Follow me up, and keep an eye on the brackets this time. They have been rather easy to move around so far; up there, we will need to be more particular about where we put them.

Watch and read

- [Fantastic Quaternions - Numberphile](https://www.youtube.com/watch?v=3BR8tK-LuB0) James Grime tells the bridge story and shows why i, j, k had to come as three, not two

- [Visualizing the 4d numbers Quaternions](https://www.youtube.com/watch?v=d4EgbgTm0Bg) 3Blue1Brown: how to see a four-dimensional number system from inside a three-dimensional world

- [Quaternion](https://en.wikipedia.org/wiki/Quaternion) Wikipedia

- [Quaternions and spatial rotation](https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation) Wikipedia