# Multicomplex number

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In [mathematics](/source/mathematics), the '''multicomplex number''' systems <math>\Complex_n</math> are defined inductively as follows: Let C<sub>0</sub> be the [real number](/source/real_number) system. For every {{nowrap|''n'' > 0}} let ''i''<sub>''n''</sub> be a square root of&nbsp;−1, that is, an [imaginary unit](/source/imaginary_unit). Then <math>\Complex_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \Complex_n \rbrace</math>. In the multicomplex number systems one also requires that <math>i_n i_m = i_m i_n</math> ([commutativity](/source/commutativity)). Then <math>\Complex_1</math> is the [complex number](/source/complex_number) system, <math>\Complex_2</math> is the [bicomplex number](/source/bicomplex_number) system, <math>\Complex_3</math> is the tricomplex number system of [Corrado Segre](/source/Corrado_Segre), and <math>\Complex_n</math> is the multicomplex number system of order ''n''.

Each <math>\Complex_n</math> forms a [Banach algebra](/source/Banach_algebra). [G. Bayley Price](/source/Griffith_Baley_Price) has written about the function theory of multicomplex systems, providing details for the bicomplex system <math>\Complex_2 .</math>

The multicomplex number systems are not to be confused with ''Clifford numbers'' (elements of a [Clifford algebra](/source/Clifford_algebra)), since Clifford's square roots of&nbsp;−1 anti-commute (<math>i_n i_m + i_m i_n = 0</math> when {{nowrap|''m'' ≠ ''n''}} for Clifford). 

Because the multicomplex numbers have several square roots of –1 that commute, they also have [zero divisor](/source/zero_divisor)s: <math>(i_n - i_m)(i_n + i_m) = i_n^2 - i_m^2 = 0</math> despite <math>i_n - i_m \neq 0</math> and <math>i_n + i_m \neq 0</math>, and <math>(i_n i_m - 1)(i_n i_m + 1) = i_n^2 i_m^2 - 1 = 0</math> despite <math> i_n i_m \neq 1</math> and <math>i_n i_m \neq -1</math>. Any product <math>i_n i_m</math> of two distinct multicomplex units behaves as the <math>j</math> of the [split-complex number](/source/split-complex_number)s, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane.

With respect to [subalgebra](/source/subalgebra) <math>\Complex_k</math>, ''k'' = 0, 1, ..., {{nowrap|''n'' − 1}}, the multicomplex system <math>\Complex_n</math> is of [dimension](/source/dimension) {{nowrap|2<sup>''n'' − ''k''</sup>}} over <math>\Complex_k .</math>

==References==
* [G. Baley Price](/source/Griffith_Baley_Price) (1991) ''An Introduction to Multicomplex Spaces and Functions'', [Marcel Dekker](/source/Marcel_Dekker).
* [Corrado Segre](/source/Corrado_Segre) (1892) "The real representation of complex elements and hyperalgebraic entities" (Italian), [Mathematische Annalen](/source/Mathematische_Annalen) 40:413&ndash;67 (see especially pages 455&ndash;67).

{{Number systems}}
Category:Hypercomplex numbers

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Adapted from the Wikipedia article [Multicomplex number](https://en.wikipedia.org/wiki/Multicomplex_number) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multicomplex_number?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.