Coordinate system

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in a tuple (a coordinate tuple), or by a label, such as in "the x-coordinate". Some coordinate tuples such as Cartesian coordinates are also coordinate vectors, satisfying the requirements of vector spaces (superposition and scalar multiplication), whereas tuples of curvilinear coordinate are not strictly vectors.

The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.