Multicomplex number
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the multicomplex number systems \Complex_n are defined inductively as follows: Let C0 be the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
system. For every let ''i''''n'' be a square root of −1, that is, an
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Then \Complex_ = \lbrace z = x + y i_ : x,y \in \Complex_n \rbrace. In the multicomplex number systems one also requires that i_n i_m = i_m i_n (
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
). Then \Complex_1 is the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
system, \Complex_2 is the
bicomplex number In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^* = (w, -z), and the product of two bicomplex numbers as :(u,v)(w,z) = (u w - v z, u z ...
system, \Complex_3 is the tricomplex number system of
Corrado Segre Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. Early life Corrado's parents were Abramo Segre and Estella De Ben ...
, and \Complex_n is the multicomplex number system of order ''n''. Each \Complex_n forms a
Banach algebra In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach spa ...
. G. Bayley Price has written about the function theory of multicomplex systems, providing details for the bicomplex system \Complex_n . The multicomplex number systems are not to be confused with ''Clifford numbers'' (elements of a
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
), 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 divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s: (i_n - i_m)(i_n + i_m) = i_n^2 - i_m^2 = 0 despite i_n - i_m \neq 0 and i_n + i_m \neq 0, and (i_n i_m - 1)(i_n i_m + 1) = i_n^2 i_m^2 - 1 = 0 despite i_n i_m \neq 1 and i_n i_m \neq -1. Any product i_n i_m of two distinct multicomplex units behaves as the j of the
split-complex number In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane. With respect to
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
\Complex_k, ''k'' = 0, 1, ..., , the multicomplex system \Complex_n is of
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
over \Complex_k .


References

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G. Baley Price G. Baley Price (14 March 1905, Brookhaven, Mississippi – 7 November 2006, Lawrence, Kansas) was an American mathematician and historian of American mathematics. He was a president of the Mathematical Association of America. Career After graduat ...
(1991) ''An Introduction to Multicomplex Spaces and Functions'',
Marcel Dekker Marcel Dekker was a journal and encyclopedia publishing company with editorial boards found in New York City. Dekker encyclopedias are now published by CRC Press, part of the Taylor and Francis publishing group. History Initially a textbook pu ...
. *
Corrado Segre Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. Early life Corrado's parents were Abramo Segre and Estella De Ben ...
(1892) "The real representation of complex elements and hyperalgebraic entities" (Italian),
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
40:413–67 (see especially pages 455–67). {{Number systems Hypercomplex numbers