Hypercomplex Number

There are at least two definitions of hypercomplex numbers. Clifford algebraists call their higher dimensional numbers hypercomplex, even though they do not share all the properties of complex numbers and no classical function theory can be constructed over them.

According to van der Waerden (1985), a hypercomplex number is a number having properties departing from those of the real and complex numbers. The most common examples are biquaternions, exterior algebras, group algebras, matrices, octonions, and quaternions.

One type of hypercomplex number due to Davenport (1996) and sometimes called "the" hypercomplex numbers are defined according to the multiplication table

			(1)
			(2)
			(3)

and therefore satisfy

(4)

Note that these are not quaternions, and that the multiplication of these hypercomplex numbers is commutative. Unlike real and complex numbers, not all nonzero hypercomplex numbers have a multiplicative inverse. An application of this sort of hypercomplex number can be found in the julia_fractal command in POVRay.