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Hypercomplex number
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Hypercomplex number

In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, octonions and sedenions.

Whereas complex numbers can be viewed as points in a plane, hypercomplex numbers can be viewed as points in some higher-dimensional Euclidean space (4 dimensions for the quaternions, 8 for the octonions, 16 for the sedenions). More precisely, they form finite-dimensional algebras over the real numbers. But none of these extensions forms a field, essentially because the field of complex numbers is algebraically closed — see fundamental theorem of algebra.

The quaternions, octonions and sedenion can be generated by the Cayley-Dickson construction. The Clifford algebras are another family of hypercomplex numbers.

Topics in mathematics related to quantity Edit
Numbers | Natural numbers | Integers | Rational numbers | Real numbers | Complex numbers | Hypercomplex numbers | Quaternions | Octonions | Sedenions | Hyperreal numbers | Surreal numbers | Ordinal numbers | Cardinal numbers | p-adic numberss | Integer sequences | Mathematical constants | Infinity