Multicomplex number
Encyclopedia
In mathematics
, the multicomplex number systems Cn are defined inductively as follows: Let C0 be the real number
system. For every n > 0 let in be a square root of minus one, that is, an imaginary number
. Then
In the multicomplex number systems one also requires when n ≠ m that 
(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 Cn is the multicomplex number system of order n.
Each Cn 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 minus one anti-commute (
for Clifford). Furthermore, the multicomplex systems differ from the n-complex numbers introduced by Silviu Olariu in 2002. In particular, Olariu's tricomplex number
s differ from Segre's tricomplex numbers C3 defined above.
With respect to subalgebra
Ck, k = 0, 1, ... n−1, the multicomplex system Cn is of dimension
2n−k over Ck.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the multicomplex number systems Cn are defined inductively as follows: Let C0 be the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
system. For every n > 0 let in be a square root of minus one, that is, an imaginary number
Imaginary number
An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...
. Then
In the multicomplex number systems one also requires when n ≠ m that (commutative property). Then C1 is the complex numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
system, C2 is the bicomplex number system, C3 is the tricomplex number system of Corrado Segre
Corrado Segre
Corrado Segre was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry....
, and Cn is the multicomplex number system of order n.
Each Cn forms a Banach algebra
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 which at the same time is also a Banach space...
. 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
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
), since Clifford's square roots of minus one anti-commute (
for Clifford). Furthermore, the multicomplex systems differ from the n-complex numbers introduced by Silviu Olariu in 2002. In particular, Olariu's tricomplex numberTricomplex number
In mathematics, the tricomplex numbers are a three-dimensional hypercomplex number system developed by Silviu Olariu in his book Complex Numbers in n Dimensions .-Definition:...
s differ from Segre's tricomplex numbers C3 defined above.
With respect to subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
Ck, k = 0, 1, ... n−1, the multicomplex system Cn is of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
2n−k over Ck.