Editing Octonion (section)

{{Short description|Hypercomplex number system}}
{{CS1 config|mode=cs2}}
{{Infobox number system
| official_name = Octonions
| symbol = <math>\mathbb O</math>
| type = [[Hypercomplex number|Hypercomplex]] [[algebra over a field|algebra]]
| units = e<sub>0</sub>, ..., e<sub>7</sub>
| identity = e<sub>0</sub>
| properties = {{ubl|[[Commutative property|Non-commutative]]|
[[Associative|Non-associative]]}}
}}
In [[mathematics]], the '''octonions''' are a [[normed division algebra]] over the [[real number]]s, a kind of [[Hypercomplex number|hypercomplex]] [[Number#Classification|number system]]. The octonions are usually represented by the capital letter O, using boldface {{math|'''O'''}} or [[blackboard bold]] <math>\mathbb O</math>. Octonions have eight [[dimension (vector space)|dimensions]]; twice the number of dimensions of the [[quaternion]]s, of which they are an extension. They are [[commutative property|noncommutative]] and [[associative property|nonassociative]], but satisfy a weaker form of associativity; namely, they are [[alternative algebra|alternative]]. They are also [[Power associativity|power associative]].

Octonions are not as well known as the quaternions and [[complex number]]s, which are much more widely studied and used. Octonions are related to exceptional structures{{what|date=November 2024}} in mathematics, among them the [[Simple Lie group#Exceptional cases|exceptional Lie group]]s. Octonions have applications in fields such as [[string theory]], [[special relativity]] and [[quantum logic]]. Applying the [[Cayley–Dickson construction]] to the octonions produces the [[sedenion]]s.