Multicomplex number

In mathematics, the multicomplex number systems ${\mathbb{ℂ}}_n$ 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 ${\mathbb{ℂ}}_{n+1}=\{z=x+y{i}_{n+1}:x,y\in {\mathbb{ℂ}}_n\}$. In the multicomplex number systems one also requires that $i_n{i}_m=i_m{i}_n$ (commutativity). Then ${\mathbb{ℂ}}_1$ is the complex number system, ${\mathbb{ℂ}}_2$ is the bicomplex number system, ${\mathbb{ℂ}}_3$ is the tricomplex number system of Corrado Segre, and ${\mathbb{ℂ}}_n$ is the multicomplex number system of order n. Each ${\mathbb{ℂ}}_n$ forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system ${\mathbb{ℂ}}_n.$

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 ($i_n{i}_m+i_m{i}_n=0$ when m ≠ n for Clifford).

Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: $(i_n-i_m)(i_n+i_m)=i_n^2-i_m^2=0$ despite $i_n-i_m\ne 0$ and $i_n+i_m\ne 0$, and $(i_n{i}_m-1)(i_n{i}_m+1)=i_n^2{i}_m^2-1=0$ despite $i_n{i}_m\ne 1$ and $i_n{i}_m\ne -1$. 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 ${\mathbb{ℂ}}_k$, k = 0, 1, ..., n − 1, the multicomplex system ${\mathbb{ℂ}}_n$ is of dimension 2^{n − k} over ${\mathbb{ℂ}}_k.$

References

• G. Baley Price (1991) An Introduction to Multicomplex Spaces and Functions, Marcel Dekker.
• Corrado Segre (1892) "The real representation of complex elements and hyperalgebraic entities" (Italian), Mathematische Annalen 40:413–67 (see especially pages 455–67).