Multicomplex number

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

Each ${\displaystyle \mathbb {C} _{n}}$ forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system ${\displaystyle \mathbb {C} _{n}.}$

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 (${\displaystyle i_{n}i_{m}+i_{m}i_{n}=0}$ when m ≠ n for Clifford).

Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: ${\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0}$ despite ${\displaystyle i_{n}-i_{m}\neq 0}$ and ${\displaystyle i_{n}+i_{m}\neq 0}$, and ${\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0}$ despite ${\displaystyle i_{n}i_{m}\neq 1}$ and ${\displaystyle i_{n}i_{m}\neq -1}$. Any product ${\displaystyle i_{n}i_{m}}$ of two distinct multicomplex units behaves as the ${\displaystyle j}$ 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 ${\displaystyle \mathbb {C} _{k}}$, k = 0, 1, ..., n − 1, the multicomplex system ${\displaystyle \mathbb {C} _{n}}$ is of dimension 2^{n − k} over ${\displaystyle \mathbb {C} _{k}.}$