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Concept

Division algebra

An algebra in which every non-zero element can be divided by. Over the real numbers there are exactly four with a multiplicative length — dimensions one, two, four and eight — and each step up the list gives up a law the one below it kept.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

A multiplication that remembers the order

Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.

algebra · Quaternions The four-square identity, and how few squares a number needs. A product of two whole quaternions with both sides of the four-square identity evaluated, above a strip colouring every number by the fewest squares that add to it.

The identity that multiplies sums of squares

A sum of two squares times a sum of two squares is a sum of two squares, and the same is true of four and of eight. It is true of three of nothing: 3 and 5 are each a sum of three squares, 15 is not, and one small pair of numbers rules the case out forever.

algebra · Quaternions The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair.

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

algebra · Quaternions

Named alongside it

The objects these essays reach for when they reach for this one.

QuaternionAssociativityComplex numbersDimensionNormConjugateFinite geometryGroupIdentityMultiplicationMultiplicative functionPrimes

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