multicomplex number

In mathematics, the multicomplex number systems $\backslash Complex\_n$ are defined inductively as follows: Let C₀ be the real number system. For every {{nowrap|n > 0}} let i_n be a square root of −1, that is, an imaginary unit. Then $\backslash Complex\_\{n+1\}\; =\; \backslash lbrace\; z\; =\; x\; +\; y\; i\_\{n+1\}\; :\; x,y\; \backslash in\; \backslash Complex\_n\; \backslash rbrace$. In the multicomplex number systems one also requires that $i\_n\; i\_m\; =\; i\_m\; i\_n$ (commutativity). Then $\backslash Complex\_1$ is the complex number system, $\backslash Complex\_2$ is the bicomplex number system, $\backslash Complex\_3$ is the tricomplex number system of Corrado Segre, and $\backslash Complex\_n$ is the multicomplex number system of order n.

Each $\backslash Complex\_n$ forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system $\backslash Complex\_2\; .$

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 {{nowrap|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 $i\_n\; -\; i\_m\; \backslash neq\; 0$ and $i\_n\; +\; i\_m\; \backslash neq\; 0$, and $(i\_n\; i\_m\; -\; 1)(i\_n\; i\_m\; +\; 1)\; =\; i\_n^2\; i\_m^2\; -\; 1\; =\; 0$ despite $i\_n\; i\_m\; \backslash neq\; 1$ and $i\_n\; i\_m\; \backslash neq\; -1$. 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 $\backslash Complex\_k$, k = 0, 1, ..., {{nowrap|n − 1}}, the multicomplex system $\backslash Complex\_n$ is of dimension {{nowrap|2^{n − k}}} over $\backslash Complex\_k\; .$