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 $\mathrm {C} _{n+1}=\lbrace z=x+yi_{n+1}:x,y\in \mathrm {C} _{n}\rbrace .$ In the multicomplex number systems one also requires when n ≠ m that $i_{n}i_{m}=i_{m}i_{n}$ (commutative property). 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 ($i_{n}i_{m}+i_{m}i_{n}=0$ 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).