Multicomplex number

In mathematics, the multicomplex number systems [math]\displaystyle{ \Complex_n }[/math] 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 [math]\displaystyle{ \Complex_{n+1} = \lbrace z = x + y i_{n+1} : x,y \in \Complex_n \rbrace }[/math]. In the multicomplex number systems one also requires that [math]\displaystyle{ i_n i_m = i_m i_n }[/math] (commutativity). Then [math]\displaystyle{ \Complex_1 }[/math] is the complex number system, [math]\displaystyle{ \Complex_2 }[/math] is the bicomplex number system, [math]\displaystyle{ \Complex_3 }[/math] is the tricomplex number system of Corrado Segre, and [math]\displaystyle{ \Complex_n }[/math] is the multicomplex number system of order n. Each [math]\displaystyle{ \Complex_n }[/math] forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system [math]\displaystyle{ \Complex_n . }[/math]

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 ([math]\displaystyle{ i_n i_m + i_m i_n = 0 }[/math] when m ≠ n for Clifford).

Because the multicomplex numbers have several square roots of –1 that commute, they also have zero divisors: [math]\displaystyle{ (i_n - i_m)(i_n + i_m) = i_n^2 - i_m^2 = 0 }[/math] despite [math]\displaystyle{ i_n - i_m \neq 0 }[/math] and [math]\displaystyle{ i_n + i_m \neq 0 }[/math], and [math]\displaystyle{ (i_n i_m - 1)(i_n i_m + 1) = i_n^2 i_m^2 - 1 = 0 }[/math] despite [math]\displaystyle{ i_n i_m \neq 1 }[/math] and [math]\displaystyle{ i_n i_m \neq -1 }[/math]. Any product [math]\displaystyle{ i_n i_m }[/math] of two distinct multicomplex units behaves as the [math]\displaystyle{ j }[/math] 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 [math]\displaystyle{ \Complex_k }[/math], k = 0, 1, ..., n − 1, the multicomplex system [math]\displaystyle{ \Complex_n }[/math] is of dimension 2^{n − k} over [math]\displaystyle{ \Complex_k . }[/math]

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).