Octonions

Octonions are a non-commutative and non-associative extension of the real numbers. They were were first discovered by John Graves, a friend of Sir William Rowan Hamilton who first described the related quaternions. Although Hamilton offered to publicize Graves discovery, it took Arthur Cayley to rediscover and publish in 1845, for this reason octonions are also known as Cayley Numbers.

Definition & basic operations

The octinions, $\mathbb {O}$, are a eight-dimensional normed division algebra over the real numbers.

$\mathbb {O} =\left\lbrace a_{0}+\sum _{i=7}^{7}a_{i}e_{i}|a_{0},\dots ,a_{7}\in {\mathbb {R} }\right\rbrace$