multicomplex number

In mathematics, the multicomplex number systems \Complex_n are defined inductively as follows: Let C0 be the real number system. For every {{nowrap|n > 0}} let in be a square root of −1, that is, an imaginary unit. Then \Complex_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \Complex_n \rbrace. In the multicomplex number systems one also requires that i_n i_m = i_m i_n (commutativity). Then \Complex_1 is the complex number system, \Complex_2 is the bicomplex number system, \Complex_3 is the tricomplex number system of Corrado Segre, and \Complex_n is the multicomplex number system of order n.

Each \Complex_n forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system \Complex_2 .

The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute (i_n i_m + i_m i_n = 0 when {{nowrap|mn}} for Clifford).

Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: (i_n - i_m)(i_n + i_m) = i_n^2 - i_m^2 = 0 despite i_n - i_m \neq 0 and i_n + i_m \neq 0, and (i_n i_m - 1)(i_n i_m + 1) = i_n^2 i_m^2 - 1 = 0 despite i_n i_m \neq 1 and i_n i_m \neq -1. Any product i_n i_m of two distinct multicomplex units behaves as the j of the split-complex numbers, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane.

With respect to subalgebra \Complex_k, k = 0, 1, ..., {{nowrap|n − 1}}, the multicomplex system \Complex_n is of dimension {{nowrap|2nk}} over \Complex_k .

References

{{Number systems}}

Category:Hypercomplex numbers