Multicomplex number

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In mathematics, the multicomplex number systems C n {\displaystyle \mathbb {C} _{n}} {\displaystyle \mathbb {C} _{n}} 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 unit. Then C n + 1 = { z = x + y i n + 1 : x , y ∈ C n } {\displaystyle \mathbb {C} _{n+1}=\lbrace z=x+yi_{n+1}:x,y\in \mathbb {C} _{n}\rbrace } {\displaystyle \mathbb {C} _{n+1}=\lbrace z=x+yi_{n+1}:x,y\in \mathbb {C} _{n}\rbrace }. In the multicomplex number systems one also requires that i n i m = i m i n {\displaystyle i_{n}i_{m}=i_{m}i_{n}} {\displaystyle i_{n}i_{m}=i_{m}i_{n}} (commutativity). Then C 1 {\displaystyle \mathbb {C} _{1}} {\displaystyle \mathbb {C} _{1}} is the complex number system, C 2 {\displaystyle \mathbb {C} _{2}} {\displaystyle \mathbb {C} _{2}} is the bicomplex number system, C 3 {\displaystyle \mathbb {C} _{3}} {\displaystyle \mathbb {C} _{3}} is the tricomplex number system of Corrado Segre, and C n {\displaystyle \mathbb {C} _{n}} {\displaystyle \mathbb {C} _{n}} is the multicomplex number system of order n.

Each C n {\displaystyle \mathbb {C} _{n}} {\displaystyle \mathbb {C} _{n}} forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system C 2 . {\displaystyle \mathbb {C} _{2}.} {\displaystyle \mathbb {C} _{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 {\displaystyle i_{n}i_{m}+i_{m}i_{n}=0} {\displaystyle 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 {\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0} {\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0} despite i n − i m ≠ 0 {\displaystyle i_{n}-i_{m}\neq 0} {\displaystyle i_{n}-i_{m}\neq 0} and i n + i m ≠ 0 {\displaystyle i_{n}+i_{m}\neq 0} {\displaystyle 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 {\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0} {\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0} despite i n i m ≠ 1 {\displaystyle i_{n}i_{m}\neq 1} {\displaystyle i_{n}i_{m}\neq 1} and i n i m ≠ − 1 {\displaystyle i_{n}i_{m}\neq -1} {\displaystyle i_{n}i_{m}\neq -1}. Any product i n i m {\displaystyle i_{n}i_{m}} {\displaystyle i_{n}i_{m}} of two distinct multicomplex units behaves as the j {\displaystyle j} {\displaystyle 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 C k {\displaystyle \mathbb {C} _{k}} {\displaystyle \mathbb {C} _{k}}, k = 0, 1, ..., n − 1, the multicomplex system C n {\displaystyle \mathbb {C} _{n}} {\displaystyle \mathbb {C} _{n}} is of dimension 2n − k over C k . {\displaystyle \mathbb {C} _{k}.} {\displaystyle \mathbb {C} _{k}.}

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