# Multicomplex number

In mathematics, the **multicomplex number** systems C_{n} are defined inductively as follows: Let C_{0} be the real number system. For every *n* > 0 let *i*_{n} be a square root of −1, that is, an imaginary number. Then
. In the multicomplex number systems one also requires that
(commutativity). Then C_{1} is the complex number system, C_{2} is the bicomplex number system, C_{3} is the tricomplex number system of Corrado Segre, and C_{n} is the multicomplex number system of order *n*.

Each C_{n} forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system C_{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 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 C_{k}, *k* = 0, 1, ..., *n* − 1, the multicomplex system C_{n} is of dimension 2^{n − k} over C_{k}.

## 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).