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Euclidean space

In [mathematics] , Euclidean space is the [Euclidean plane] and [three-dimensional space] of [Euclidean geometry] , as well as the generalizations of these notions to [higher dimension] s. The term “Euclidean” is used to distinguish these spaces from the [curved space] s of [non-Euclidean geometry] and [Einstein's] [general theory of relativity] .

In classical [Greek geometry] , the Euclidean plane and Euclidean three-dimensional space were defined using certain [postulates] , and the other properties of these spaces were deduced as [theorem] s. In modern mathematics, it is more common to define Euclidean space using [Cartesian coordinates] and the ideas of [analytic geometry] . This approach brings the tools of [algebra] and [calculus] to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three [dimension] s.

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the [real line] ; in dimension two it is the [Cartesian plane] ; and in higher dimensions it is the real [coordinate space] with three or more [real number] coordinates. Thus a [point] in Euclidean space is a [tuple] of real numbers, and distances are defined using the [Euclidean distance formula] . Mathematicians often denote the [''n''-dimensional] Euclidean space by \mathbb{R}^n, or sometimes \mathbb{E}^n if they wish to emphasize its Euclidean nature. Euclidian spaces have finite dimension.

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