# Complex space

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