# Euclidean Space

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Related to Euclidean 3-space: N-dimensional Euclidean space

## euclidean space

[yü′klid·ē·ən ′spās] (mathematics)

A space consisting of all ordered sets (

*x*_{1}, …,*x*_{n }) of*n*numbers with the distance between (*x*_{1}, …,*x*_{n }) and (*y*_{1}, …,*y*_{n }) being given by the number*n*is called the dimension of the space.McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Euclidean Space

in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an *n*-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point *M* has the coordinates (*x*_{1}*x*_{2}, …, *x*_{n} and point *M*^{*} has the coordinates (*x*_{1}^{*}, *x*_{2}^{*}, …, *x*_{n}^{*}), then the distance between these points is

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.