Octonion

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In mathematics, the octonions are a normed division algebra over the real numbers. There are only four such algebras, the other three being the quaternions, complex numbers and real numbers. The octonions are the largest such algebra with eight dimensions, double the number of the quaternions which they are an extension of. They are noncommutative and nonassociative, but satisfy a weaker form of associativity, power associativity. The octonion algebra is usually represented by the capital letter O, using boldface O or blackboard bold $\mathbb O$.

Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic.

Contents

History

The octonions were discovered in 1843 by John T. Graves, a friend of William Hamilton, who called them octaves. They were discovered independently by Arthur Cayley (1845). They are sometimes referred to as Cayley numbers or the Cayley algebra.

Definition

The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions {e0, e1, e2, e3, e4, e5, e6, e7 } where e0 is the scalar element. That is, every octonion x can be written in the form $x = x_0 e_0 + x_1\, e_1 + x_2\,e_2 + x_3\,e_3 + x_4\,e_4 + x_5\,e_5 + x_6\,e_6 + x_7\,e_7 \ ,$ with real coefficients {xi}.