# Difference between revisions of "Cayley numbers"

From Encyclopedia of Mathematics

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− | + | ''octonions'' | |

− | Hypercomplex numbers (cf. [[ | + | |

+ | Hypercomplex numbers (cf. [[Hypercomplex number]]), namely, the elements of the 8-dimensional algebra over the field of real numbers (the Cayley algebra), also termed octonions. They were first considered by A. Cayley. The Cayley algebra may be derived via the Cayley–Dickson process from the algebra of quaternions (see [[Cayley–Dickson algebra]]; [[Quaternion]]). It is the only 8-dimensional real alternative algebra without zero divisors (see [[Frobenius theorem]]). The Cayley algebra is an algebra with unique division and with an identity; it is [[Alternative rings and algebras|alternative]], non-associative and non-commutative. | ||

====References==== | ====References==== | ||

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|valign="top"|{{Ref|Ku}}|| valign="top"| A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) | |valign="top"|{{Ref|Ku}}|| valign="top"| A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) | ||

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|valign="top"|{{Ref|CS}}|| valign="top"| John H. Conway; Derek A. Smith, "On Quaternions and Octonions" (A.K. Peters, 2003). ISBN 1568811349. | |valign="top"|{{Ref|CS}}|| valign="top"| John H. Conway; Derek A. Smith, "On Quaternions and Octonions" (A.K. Peters, 2003). ISBN 1568811349. | ||

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+ | [[Category:Nonassociative rings and algebras]] |

## Latest revision as of 18:45, 18 October 2014

*octonions*

Hypercomplex numbers (cf. Hypercomplex number), namely, the elements of the 8-dimensional algebra over the field of real numbers (the Cayley algebra), also termed octonions. They were first considered by A. Cayley. The Cayley algebra may be derived via the Cayley–Dickson process from the algebra of quaternions (see Cayley–Dickson algebra; Quaternion). It is the only 8-dimensional real alternative algebra without zero divisors (see Frobenius theorem). The Cayley algebra is an algebra with unique division and with an identity; it is alternative, non-associative and non-commutative.

#### References

[Ku] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |

[CS] | John H. Conway; Derek A. Smith, "On Quaternions and Octonions" (A.K. Peters, 2003). ISBN 1568811349. |

**How to Cite This Entry:**

Cayley numbers.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cayley_numbers&oldid=30356

This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article