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 . 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.
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.
- 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).