Euclidean space

Euclidean space is any n-dimensional mathematical space that is a generalization of the familiar two- and three-dimensional spaces described by the axioms of Euclidean geometry. The term "n-dimensional Euclidean space" (where n is any positive whole number) is usually abbreviated to "Euclidean n-space", or even just "n-space". Formally, Euclidean n-space is the set Rn (where R is the set of real numbers) together with the distance function, which is obtained by defining the distance between two points (x1, ..., xn) and (y1, ...,yn) to be the square root of (sigma) (xi - yi)2, where the sum is over i = 1, ..., n. This distance function is based on Pythagoras' theorem and is called the Euclidean metric.