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In mathematics, n-dimensional complex space is a multi-dimensional generalisation of the complex numbers, which have both real and imaginary parts or dimensions. The n-dimensional complex space can be seen as n cartesian products of the complex numbers with itself:

\( \C^n = \underbrace{\C \times \C \times \cdots \times \C}_{n\text{-times}} \)

The n-dimensional complex space consists of ordered n-tuples of complex numbers, called coordinates:

\( \C^n = \{ (z_1,\ldots,z_n) : z_i \in \C \text{ for all } 1 \le i \le n\} \)

The real and imaginary parts of a complex number may be treated as separate dimensions. With this interpretation, the space \( \C^n \) of n complex numbers can be seen as having \( 2 \times n \) dimensions represented by \( 2 \times n \) -tuples of real numbers. The two different interpretations can cause confusion about the dimension of a complex space.

The study of complex spaces, or complex manifolds, is called complex geometry.

One dimension

The complex line \( \C^1 \) 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.
Two dimensions

The term "complex plane" can be confusing. It is sometimes used to denote\( \C^2 \) , and sometimes to denote the \( \C^1 \) space represented in the Argand diagram (with the Riemann sphere referred to as the "extended complex plane"). In the present context of \( \C^n \) , it is understood to denote \( \C^2. \)

An intuitive understanding of the complex projective plane is given by Edwards (2003), which he attributes to Von Staudt.


Djoric, M. & Okumura, M.; CR Submanifolds of Complex Projective Space, Springer 2010
Edwards, L.; Projective geometry (2nd Ed), Floris, 2003.
Lindenbaum, S.D.; Mathematical methods in physics, World Scientific, 1996

See also

Calabi–Yau space
Complex polytope
Riemann surface
Several complex variables
Affine space

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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