# 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 **R**^{n} (where **R** is the set of real numbers)
together with the **distance function**, which is obtained
by defining the distance between two points (*x*_{1}, ..., *x*_{n}) and (*y*_{1}, ...,*y*_{n})
to be the square root of (sigma) (*x _{i}* -

*y*)

_{i}^{2}, where the sum is over

*i*= 1, ...,

*n*. This distance function is based on Pythagoras' theorem and is called the

**Euclidean metric**.