# Real coordinate space

In mathematics, the **real coordinate space** of dimension n, denoted **R**^{n} (/ɑːrˈɛn/ *ar-EN*) or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.

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The coordinates over any basis of the elements of a real vector space form a *real coordinate space* of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a *real coordinate space* of dimension n.

These one to one correspondences between vectors, points and coordinate vectors explain the names of *coordinate space* and *coordinate vector*. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.