Octonions are a set of hypercomplex number. The first set of hypercomplex numbers, quaternions, were discovered by William Rowan Hamilton in 1843. Since the real numbers could be seen as being one-dimensional, the complex numbers as two-dimensional, and the quaternions as four-dimensional, it suggested that the next step would be a set of eight-dimensional numbers, which are octonions. Octonions were discovered independently by John Graves and Arthur Cayley and are sometimes known as Cayley numbers.

In the system of octonions, there are seven imaginary quantities, denoted `i`, `j`, `k`, `p`, `q`, `r`, and `s`, where

Like the quaternions, the commutative law of multiplication does not hold. However, unlike the quaternions, the associative law does not hold either.

One might also wonder whether complex arithmetics in 16, 32, 64, 128, ... dimensions are possible. Cayley showed that this process can be carried on indefinitely. For example, the sixteen-dimensional set of hypercomplex numbers is known as sedenions. However, as this process is continued, it requires giving up increasingly more laws of arithmetic, making these systems less useful. For example, two nonzero sedenions can be multiplied to produce zero, which makes division of sedenions impossible.

Sources used (see bibliography page for titles corresponding to numbers): 47.