On a new Species of Imaginary Quantities connected with a theory of Quaternions
Add a fourth dimension, and directions in space can be multiplied — and rotated.
What does a 19th-century walk along a Dublin canal have to do with the way your phone knows which way it's facing? Everything.
Numbers you can turn
Ordinary numbers live on a line. Complex numbers live on a flat plane, and multiplying by one can spin a point around — which makes them perfect for describing rotations on paper. Hamilton wanted the same magic for three-dimensional space, so things could be turned in the round. After years stuck, he realised the trick needed a fourth dimension and one strange sacrifice: with his new numbers, a × b no longer equals b × a. Order matters — exactly as it does when you rotate an object: turn it left then up, and you don't land where you would turning up then left.
The formula on the bridge
By 1843 William Rowan Hamilton, Ireland's leading mathematician, had been wrestling for years with how to multiply triples of numbers. On 16 October, walking with his wife along the Royal Canal to a meeting of the Royal Irish Academy, the answer struck him in a flash — as he later told his son, 'a spark flashed forth.' Afraid he would forget, he took out a pocketknife and carved the key formula into the stone of Brougham Bridge as he passed: i² = j² = k² = ijk = −1. The carving wore away long ago, but a plaque marks the spot, and mathematicians still make a yearly 'Hamilton Walk' to it.
Why it mattered
Hamilton had done something nobody thought allowed: he built a complete, sensible number system in which the order of multiplication changes the answer. That single act of permission opened modern algebra — it told mathematicians they were free to invent new systems with new rules, so long as the rules were consistent. And his four-numbered objects, quaternions, turned out to be the cleanest way ever found to describe and combine rotations in space, a use that would only become essential a century later.
Turning a book
Hold a book flat. Rotate it 90° about a vertical axis, then 90° about a horizontal one, and note which way the cover faces. Now start over and do the two turns in the opposite order — the book ends up pointing somewhere else entirely. Rotations don't commute: order changes the outcome. Quaternions are numbers that behave the same way, which is exactly why they can stand in for rotations. Multiply the quaternions in one order and you get one final orientation; swap the order and you get another.
Where it sits
Hamilton's quaternions are a sequel to the complex numbers that Gauss and others had made respectable — one dimension becomes four — and a prequel to much of modern mathematics: the matrices, vectors and group theory that fill later chapters of this Library grew partly out of them. The vector arrows physicists draw, and the dot and cross products they use, were carved out of Hamilton's algebra a generation later. After a century in the shadows, quaternions came roaring back with the computer, which needed exactly their gift for rotation.
i² = j² = k² = ijk = −1
ij = k, jk = i, ki = j; ji = −k, kj = −i, ik = −j.
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…
Nor could I resist the impulse — unphilosophical as it may have been — to cut with a knife on a stone of Brougham Bridge, as we passed it, the fundamental formula with the symbols, i, j, k; namely, i² = j² = k² = ijk = −1, which contains the Solution of the Problem, but of course, as an inscription, has long since mouldered away.