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. In the multicomplex number systems one also requires that (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 ( when m ≠ n for Clifford). Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: despite and, and despite and. 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.