Multicomplex number

In mathematics, the multicomplex number systems Cn 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 number. Then \text{C}_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \text{C}_n \rbrace. 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 Cn is the multicomplex number system of order n.

Each Cn 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 \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 Ck, k = 0, 1, ..., n − 1, the multicomplex system Cn is of dimension 2n − k over Ck.

References

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