octonion
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 or blackboard bold \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 a weaker form of associativity; namely, they are alternative. They are also power associative.
Key facts
- Number system.official_name
- Octonions
- Number system.symbol
- \mathbb O
- Number system.type
- Hypercomplex algebra
- Number system.units
- e0, ..., e7
- Number system.identity
- e0
- Number system.hide_common
- true
via Wikipedia infobox
Article
20 sectionsContents
- History
- Definition
- Cayley–Dickson construction
- Arithmetic and operations
- Addition and subtraction
- Multiplication
- Fano plane mnemonic
- Conjugate, norm, and inverse
- Exponentiation and polar form
- Properties
- Commutator and cross product
- Automorphisms
- Isotopies
- Matrix representation
- Applications
- Integral octonions
- See also
- Notes
- References
- External links
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 or blackboard bold \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 a weaker form of associativity; namely, they are alternative. They are also power associative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. 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. Applying the Cayley–Dickson construction to the octonions produces the sedenions.