Multicomplex number

In mathematics, the multicomplex number systems 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 number. Then $\text{C}_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \text{C}_n \rbrace$. In the multicomplex number systems one also requires that $i_n i_m = i_m i_n$ (commutativity). Then C₁ is the complex number system, C₂ is the bicomplex number system, C₃ is the tricomplex number system of Corrado Segre, and C_n is the multicomplex number system of order n.

Each C_n forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system C₂.

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).

With respect to subalgebra C_k, k = 0, 1, ..., n − 1, the multicomplex system C_n is of dimension 2^{n − k} over 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).