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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.

References

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

Calabiâ€“Yau space
Complex polytope
Riemann surface
Several complex variables
Affine space