Multicomplex number
title: "Multicomplex number" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["hypercomplex-numbers"] topic_path: "general/hypercomplex-numbers" source: "https://en.wikipedia.org/wiki/Multicomplex_number" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
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 i**n be a square root of −1, that is, an imaginary unit. Then \Complex_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \Complex_n \rbrace. In the multicomplex number systems one also requires that i_n i_m = i_m i_n (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 (i_n i_m + i_m i_n = 0 when m ≠ n for Clifford).
Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: (i_n - i_m)(i_n + i_m) = i_n^2 - i_m^2 = 0 despite i_n - i_m \neq 0 and i_n + i_m \neq 0, and (i_n i_m - 1)(i_n i_m + 1) = i_n^2 i_m^2 - 1 = 0 despite i_n i_m \neq 1 and i_n i_m \neq -1. 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 .
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).
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