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| In [[mathematics]], the '''multicomplex number''' systems C<sub>''n''</sub> are defined inductively as follows: Let C<sub>0</sub> be the [[real number]] system. For every {{nowrap|''n'' > 0}} let ''i''<sub>''n''</sub> be a square root of −1, that is, an [[imaginary number]]. Then <math>\text{C}_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \text{C}_n \rbrace</math>. In the multicomplex number systems one also requires that <math>i_n i_m = i_m i_n</math> ([[commutativity]]). Then C<sub>1</sub> is the [[complex number]] system, C<sub>2</sub> is the [[bicomplex number]] system, C<sub>3</sub> is the ''tricomplex number'' system of [[Corrado Segre]], and C<sub>''n''</sub> is the multicomplex number system of order ''n''.
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| Each C<sub>''n''</sub> forms a [[Banach algebra]]. [[Griffith Baley Price|G. Bayley Price]] has written about the function theory of multicomplex systems, providing details for the bicomplex system C<sub>2</sub>.
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| 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 (<math>i_n i_m + i_m i_n = 0</math> when {{nowrap|''m'' ≠ ''n''}} for Clifford).
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| With respect to [[subalgebra]] C<sub>''k''</sub>, ''k'' = 0, 1, ..., {{nowrap|''n'' − 1}}, the multicomplex system C<sub>''n''</sub> is of [[dimension]] {{nowrap|2<sup>''n'' − ''k''</sup>}} over C<sub>''k''</sub>.
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| ==References==
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| * [[Griffith Baley Price|G. Baley Price]] (1991) ''An Introduction to Multicomplex Spaces and Functions'', [[Marcel Dekker]].
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| * [[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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| [[Category:Hypercomplex numbers]]
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Latest revision as of 19:27, 30 November 2014
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