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

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.

The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Hamilton's discovery of quaternions. Graves called his discovery **octaves**. They were discovered independently by Arthur Cayley and are sometimes referred to as **Cayley numbers** or the **Cayley algebra**.

Read more about Octonions: Definition, Properties

### Other articles related to "octonions, octonion":

**Octonions**- Properties - Automorphisms

... An automorphism, A, of the

**octonions**is an invertible linear transformation of O which satisfies The set of all automorphisms of O forms a group called G2 ...

... The split-

**octonions**, like the

**octonions**, are noncommutative and nonassociative ... Also like the

**octonions**, they form a composition algebra since the quadratic form N is multiplicative ... That is, The split-

**octonions**satisfy the Moufang identities and so form an alternative algebra ...

**Octonions**

... Circular quaternions and

**octonions**from the Musean hypernumbers are identical to quaternions and

**octonions**from Cayley–Dickson construction ...

... In mathematics, the split-

**octonions**are a nonassociative extension of the quaternions (or the split-quaternions) ... They differ from the

**octonions**in the signature of quadratic form the split-

**octonions**have a split-signature (4,4) whereas the

**octonions**have a positive-definite signature (8,0) ... The split-

**octonions**form the unique split

**octonion**algebra over the real numbers ...

**Octonions**

...

**Octonions**were developed independently by Arthur Cayley in 1845 and John T ... of double quaternion that is now days often called an

**octonion**, and showing that they were what we now call normed division algebra Graves called them octaves ... of Graves, but inspired by Hamilton's publication of his own work, published on

**octonions**in March 1845 – as an appendix to a paper on a different ...