Multicomplex number

In mathematics, the multicomplex number systems C_n are defined inductively as follows: Let C₀ be the real number system. For every n > 0 let i_n be a square root of −1, that is, an imaginary number. Then C n + 1 = { z = x + y i n + 1 : x , y ∈ C n } . In the multicomplex number systems one also requires that i n i m = i m i n (commutativity). Then C₁ is the complex number system, C₂ is the bicomplex number system, C₃ is the tricomplex number system of Corrado Segre, and C_n is the multicomplex number system of order n.

Each C_n forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system C₂.

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 m ≠ n 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 ≠ 0 and i n + i m ≠ 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 ≠ 1 and i n i m ≠ − 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 C_k, k = 0, 1, ..., n − 1, the multicomplex system C_n is of dimension 2^{n − k} over C_k.