Hypercomplex number
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Hypercomplex Numbers
In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, split-quaternions, tessarines, coquaternions, octonions, split-octonions, biquaternions 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, tessarines and coquaternions; 8 for the octonions and biquaternions; and 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.
Examples of Extension Programs
The quaternions, octonions and sedenion can be generated by the Cayley-Dickson construction. The Clifford algebras are another family of hypercomplex numbers.
A modified Cayley-Dickson construction construction process leads to split-quaternions and split-octonions.
Split-quaternions and split-octonions are computationally equivalent to hyperbolic quaternions and hyperbolic octonions from the hypernumbers program. Whereas traditional Cayley-Dickson construction yields 8 dimensional octonions as widest normed algebra, conic sedenion arithmetic offers a 16 dimensional number system with multiplicative modulus. The hypernumbers program offers hyperbolic, circular, and conic quaternions, octonions, and sedenions, which all have a multiplicative modulus (except for hyperbolic sedenions).
See also
- - wikibook link
- Hypernumbers
- Split-octonions
External links
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