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Real coordinate space
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{{short description|Space formed by the ''n''-tuples of real numbers}} {{no footnotes|date=February 2024}} [[File:Cartesian-coordinate-system v2.svg|thumb|right|[[Cartesian coordinates]] identify points of the [[Euclidean plane]] with pairs of real numbers]] In [[mathematics]], the '''real coordinate space''' or '''real coordinate ''n''-space''', of [[dimension]] {{mvar|n}}, denoted {{math|'''R'''<sup>{{mvar|n}}</sup>}} or {{nowrap|<math>\R^n</math>}}, is the set of all ordered [[tuple|{{mvar|n}}-tuples]] of [[real number]]s, that is the set of all sequences of {{mvar|n}} real numbers, also known as ''[[coordinate vector]]s''. Special cases are called the ''[[real line]]'' {{math|'''R'''<sup>1</sup>}}, the ''real coordinate plane'' {{math|'''R'''<sup>2</sup>}}, and the ''real coordinate three-dimensional space'' {{math|'''R'''<sup>3</sup>}}. With component-wise addition and scalar multiplication, it is a [[real vector space]]. The [[coordinate (vector space)|coordinates]] over any [[basis (vector space)|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 {{mvar|n}}, {{math|'''E'''<sup>n</sup>}} ([[Euclidean line]], {{math|'''E'''}}; [[Euclidean plane]], {{math|'''E'''<sup>2</sup>}}; [[Euclidean three-dimensional space]], {{math|'''E'''<sup>3</sup>}}) form a ''real coordinate space'' of dimension {{mvar|n}}. These [[one to one correspondence]]s 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 [[RenΓ© Descartes]] in the 17th century. It is widely used, as it allows locating points in Euclidean spaces, and computing with them.
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