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:
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 manifolds, is called complex geometry.
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The complex line 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.
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.