Complex space
Encyclopedia
In mathematics
, n-dimensional complex space is a multi-dimensional generalisation of the complex number
s, which have both real
and imaginary
parts or dimensions. The n-dimensional complex space can be seen as n cartesian product
s of the complex numbers with itself:
The n-dimensional complex space consists of ordered n-tuples of complex numbers, called coordinates:
The real and imaginary parts of a complex number may be treated as separate dimensions. With this interpretation, the space
of n complex numbers can be seen as having 
dimensions. This can cause confusion.
The study of complex spaces, or complex manifold
s, is called complex geometry
.
has one real and one imaginary dimension. It is analogous in some ways to two-dimensional real space, and may be represented as an Argand diagram in the real plane.
In projective geometry
, the complex projective line includes a point at infinity in the Argand diagram and is an example of a Riemann sphere
.
, and sometimes to denote the 
space represented in the Argand diagram (with the Riemann sphere referred to as the "extended complex plane"). In the present context of 
, it is understood to denote 
.
An intuitive understanding of the complex projective plane is given by Edwards (2003), which he attributes to Von Staudt.
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...
, n-dimensional complex space is a multi-dimensional generalisation of the complex number
Complex 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...
s, which have both real
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...
and imaginary
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...
parts or dimensions. The n-dimensional complex space can be seen as n cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
s of the complex numbers with itself:
The n-dimensional complex space consists of ordered n-tuples of complex numbers, called coordinates:
The real and imaginary parts of a complex number may be treated as separate dimensions. With this interpretation, the space
of n complex numbers can be seen as having dimensions. This can cause confusion.The study of complex spaces, or complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s, is called complex geometry
Complex geometry
In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
.
One dimension
The complex lineComplex line
In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C , it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a...
has one real and one imaginary dimension. It is analogous in some ways to two-dimensional real space, and may be represented as an Argand diagram in the real plane.In projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, the complex projective line includes a point at infinity in the Argand diagram and is an example of a Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
.
Two dimensions
The term "complex plane" can be confusing. It is sometimes used to denote, and sometimes to denote the space represented in the Argand diagram (with the Riemann sphere referred to as the "extended complex plane"). In the present context of , it is understood to denote .An intuitive understanding of the complex projective plane is given by Edwards (2003), which he attributes to Von Staudt.
See also
- Calabi-Yau space
- Complex polytopeComplex polytopeA complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one....
- Riemann surfaceRiemann surfaceIn mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
- Several complex variablesSeveral complex variablesThe theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...