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 $\mathbb {C} _{n+1}=\lbrace z=x+yi_{n+1}:x,y\in \mathbb {C} _{n}\rbrace$. In the multicomplex number systems one also requires that $i_{n}i_{m}=i_{m}i_{n}$ (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 $\mathbb {C} _{2}.$

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 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 $i_{n}-i_{m}\neq 0$ and $i_{n}+i_{m}\neq 0$, and $(i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0$ despite $i_{n}i_{m}\neq 1$ and $i_{n}i_{m}\neq -1$. 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 $\mathbb {C} _{k}$, 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).