Quaternions

The quaternions are an extension of the complex numbers. Instead of having just one square root of minus one: i, why not have 3: i, j and k?

We will need a way of multiplying them together. It turns out that the following works:

ij = k = -ji
jk = i = -kj
ki = j = -ki
ijk = -1

The quaternions are not commutative, which is quite strange. The order in which you multiply them matters! If you have two quaternions, x and y, then:

x × y = - y × x

Why have 3 square roots of -1 and not 2 square roots? Because with 2 the division doesn't work properly. Why not 4, 5, 6 roots etc? Well it works with 7 roots of -1. They are called the octonians. And with 15 roots you get the sedenions. The octonians are not commutative and not associative either. The sedenions are not commutative, associative or alternative.

The quaternions are given the symbol: ℍ