Octonion

Octonions
Symbol	${\displaystyle \mathbb {O} }$
Type	Hypercomplex algebra
Units	e0, ..., e7
Multiplicative identity	e0
Main properties	Alternativity Non-commutative Non-associative

In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold ${\displaystyle \mathbb {O} }$. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy weaker forms of associativity; they are alternative and power associative.

Octonions are much less widely studied or used than the quaternions and complex numbers. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic.

The Cayley–Dickson construction builds the octonions from the quaternions and in turn builds the sedenions from the octonions.