Multicomplex number explained

In mathematics, the multicomplex number systems

\Complex _{n} are defined inductively as follows: Let C₀ be the real number system. For every let i_n be a square root of −1, that is, an imaginary unit. Then \Complex _{n+1} = \lbrace z = x + y i _{n+1} : x,y \in \Complex _{n} \rbrace . In the multicomplex number systems one also requires that i _{n} i _{m} = i _{m} i _{n} (commutativity). Then \Complex _{1} is the complex number system, \Complex _{2} is the bicomplex number system, \Complex _{3} is the tricomplex number system of Corrado Segre, and \Complex _{n} is the multicomplex number system of order n.

Each

\Complex _{n} forms a Banach algebra. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system \Complex _{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 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 ^{2} _{n} - i ^{2} _{m} = 0 despite i _{n} - i _{m} ≠ 0 and i _{n} + i _{m} ≠ 0 , and (i _{n} i _{m} - 1 )(i _{n} i _{m} + 1 ) = i ^{2} _{n} i ^{2} _{m} - 1 = 0 despite i _{n} i _{m} ≠ 1 and i _{n} i _{m} ≠ -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.\Complex _{k} , k = 0, 1, ...,, the multicomplex system \Complex _{n} is of dimension over \Complex _{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).