In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, 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, 4 for the tessarines, 4 for the coquaternions, 8 for the octonions, 8 for the biquaternions, 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.
See also