Octonions: Difference between revisions
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'''Octonions''' are a [[Commutativity|non-commutative]] and [[Associative law|non-associative]] extension of the [[Real number|real numbers]]. They were were first discovered by John Graves, a friend of Sir William Rowan Hamilton who first described the related [[Quaternions|quaternions]]. | '''Octonions''' are a [[Commutativity|non-commutative]] and [[Associative law|non-associative]] extension of the [[Real number|real numbers]]. They were were first discovered by John Graves, a friend of Sir William Rowan Hamilton who first described the related [[Quaternions|quaternions]]. | ||
Although Hamilton offered to publicize Graves discovery, it took Arthur Cayley to rediscover and publish in 1845, for this reason octonions are also known as Cayley Numbers. | Although Hamilton offered to publicize Graves discovery, it took Arthur Cayley to rediscover and publish in 1845, for this reason octonions are also known as Cayley Numbers. | ||
== Definition & basic operations == | == Definition & basic operations == | ||
The | The octonions, <math>\mathbb{O}</math>, are a eight-dimensional normed division algebra over the real numbers.<br/><br/> | ||
:<math>\mathbb{O}=\left\lbrace a_0 + \sum_{i= | :<math>\mathbb{O}=\left\lbrace a_0 + \sum_{i=1}^7a_i e_i|a_1, \dots, a_7 \in {\mathbb{R}}\right\rbrace</math><br/> | ||
== Properties == | == Properties == | ||
== Applications == | == Applications == | ||
==See also== | |||
*[[Cayley-Dickson construction]] | |||
==Related topics== | |||
*[[Geometric Algebra]] | |||
*[[Fano plane]] | |||
*[[Quaternions]] | |||
== References == | == References == | ||
{{reflist}} | |||
Latest revision as of 06:31, 14 September 2013
Octonions are a non-commutative and non-associative extension of the real numbers. They were were first discovered by John Graves, a friend of Sir William Rowan Hamilton who first described the related quaternions. Although Hamilton offered to publicize Graves discovery, it took Arthur Cayley to rediscover and publish in 1845, for this reason octonions are also known as Cayley Numbers.
Definition & basic operations
The octonions, , are a eight-dimensional normed division algebra over the real numbers.
