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A coordinate system is a system for assigning an tuple of scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or members of any of many other rings or ring-like algebraic structures.

Although any specific coordinate system is useful for numerical calculations in a given space, the space itself is independent of any particular choice of coordinates. Some choices of coordinate systems lead to paradoxes, for example, close to a black hole, but can be understood by changing the choice of coordinate system.

A coordinate transformation is a conversion from one system to another, to describe the same space.

An example of a coordinate system is to describe a point P in the Euclidean space Rn by an n-tuple P=(r1,...,rn) of real numbers r1,...,rn.

These numbers r1,...,rn are called the coordinates of the point P.

If a subset S of an Euclidean space is mapped continuously onto another topological space, this defines coordinates in the image of S.

Some coordinate systems are the following:  