Multicomplex number

In mathematics, the multicomplex number systems $$\Complex_n$$ are defined inductively as follows: Let C0 be the real number system. For every 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 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 $$i_n - i_m \neq 0$$ 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, ..., n − 1, the multicomplex system $$\Complex_n$$ is of dimension 2n − k over $$\Complex_k .$$