Multicomplex number

In mathematics, the multicomplex number systems ${\displaystyle \mathbb {C} _{n}}${\displaystyle \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 ${\displaystyle \mathbb {C} _{n+1}=\lbrace z=x+yi_{n+1}:x,y\in \mathbb {C} _{n}\rbrace }${\displaystyle \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 ${\displaystyle i_{n}i_{m}=i_{m}i_{n}}${\displaystyle i_{n}i_{m}=i_{m}i_{n}} (commutativity). Then ${\displaystyle \mathbb {C} _{1}}${\displaystyle \mathbb {C} _{1}} is the complex number system, ${\displaystyle \mathbb {C} _{2}}${\displaystyle \mathbb {C} _{2}} is the bicomplex number system, ${\displaystyle \mathbb {C} _{3}}${\displaystyle \mathbb {C} _{3}} is the tricomplex number system of Corrado Segre, and ${\displaystyle \mathbb {C} _{n}}${\displaystyle \mathbb {C} _{n}} is the multicomplex number system of order n.

Each ${\displaystyle \mathbb {C} _{n}}${\displaystyle \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 ${\displaystyle \mathbb {C} _{2}.}${\displaystyle \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 (${\displaystyle i_{n}i_{m}+i_{m}i_{n}=0}${\displaystyle 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: ${\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0}${\displaystyle (i_{n}-i_{m})(i_{n}+i_{m})=i_{n}^{2}-i_{m}^{2}=0} despite ${\displaystyle i_{n}-i_{m}\neq 0}${\displaystyle i_{n}-i_{m}\neq 0} and ${\displaystyle i_{n}+i_{m}\neq 0}${\displaystyle i_{n}+i_{m}\neq 0}, and ${\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0}${\displaystyle (i_{n}i_{m}-1)(i_{n}i_{m}+1)=i_{n}^{2}i_{m}^{2}-1=0} despite ${\displaystyle i_{n}i_{m}\neq 1}${\displaystyle i_{n}i_{m}\neq 1} and ${\displaystyle i_{n}i_{m}\neq -1}${\displaystyle i_{n}i_{m}\neq -1}. Any product ${\displaystyle i_{n}i_{m}}${\displaystyle i_{n}i_{m}} of two distinct multicomplex units behaves as the ${\displaystyle j}${\displaystyle 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 ${\displaystyle \mathbb {C} _{k}}${\displaystyle \mathbb {C} _{k}}, k = 0, 1, ..., n − 1, the multicomplex system ${\displaystyle \mathbb {C} _{n}}${\displaystyle \mathbb {C} _{n}} is of dimension 2^{n − k} over ${\displaystyle \mathbb {C} _{k}.}${\displaystyle \mathbb {C} _{k}.}

• 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).

• Hypercomplex numbers