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 octinions, <math>\mathbb{O}</math>, are a eight-dimensional normed division algebra over the | The octinions, <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=7}^7a_i e_i|a_0, \dots, a_7 \in {\mathbb{R}}\right\rbrace</math><br/> | :<math>\mathbb{O}=\left\lbrace a_0 + \sum_{i=7}^7a_i e_i|a_0, \dots, a_7 \in {\mathbb{R}}\right\rbrace</math><br/> | ||
== Properties == | == Properties == | ||
== Applications == | == Applications == | ||
==See also== | |||
*[[Cayley-Dickinson construction]] | |||
==Related topics== | |||
*[[Geometric Algebra]] | |||
*[[Fano plane]] | |||
*[[Quaternions]] | |||
== References == | == References == | ||
==External links== | |||
*[http://mathworld.wolfram.com/Octonion.html Octonion] at MathWorld | |||
Revision as of 03:25, 19 December 2007
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 octinions, , are a eight-dimensional normed division algebra over the real numbers.
Properties
Applications
See also
Related topics
References
External links
- Octonion at MathWorld