In mathematics, the multicomplex number systems are defined inductively as follows: Let C0 be the real number system. For every 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 is the complex number system, is the bicomplex number system, is the tricomplex number system of Corrado Segre, and is the multicomplex number system of order n. Each forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system 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 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 of two distinct multicomplex units behaves as the 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 , k = 0, 1, ..., , the multicomplex system is of dimension over References G. Baley Price (1991) An Introduction to Multicomplex Spaces and Functions, Marcel Dekker. Corrado Segre (1892) "The real representation of complex elements and hyperalgebraic entities" (Italian), Mathematische Annalen 40:413–67 (see especially pages 455–67). Hypercomplex numbers