- References
In mathematics, the multicomplex number systems Cn are defined inductively as follows: Let C0 be the real number system. For every {{nowrap|n > 0}} let in be a square root of −1, that is, an imaginary number. Then 
. In the multicomplex number systems one also requires that 
(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 (
when {{nowrap|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 Ck, k = 0, 1, ..., {{nowrap|n − 1}}, the multicomplex system Cn is of dimension {{nowrap|2n − k}} over Ck.
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).