Octonions

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 OctonionsDefinition, 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 ...
Split-octonion - Properties
... 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 ...
Musean Hypernumber - Conic Sedenions / "16 Dimensional M-algebra" - List of Number Types and Their Isomorphisms - Circular Quaternions and Octonions
... Circular quaternions and octonions from the Musean hypernumbers are identical to quaternions and octonions from Cayley–Dickson construction ...
Split-octonion
... 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 ...
History Of Quaternions - 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 ...