Multicomplex number

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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 mn for Clifford).

With respect to subalgebra Ck, k = 0, 1, ..., n − 1, the multicomplex system Cn is of dimension 2nk over Ck.