structural:scholar_edition
Motivation of the modern definition

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below).

In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts — the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

The set R n {\displaystyle \mathbb {R} ^{n}} of n -tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and R n {\displaystyle \mathbb {R} ^{n}} viewed as a Euclidean space.

It follows that everything that can be said about a Euclidean space can also be said about R n. {\displaystyle \mathbb {R} ^{n}.} Therefore, many authors, especially at elementary level, call R n {\displaystyle \mathbb {R} ^{n}} the standard Euclidean space of dimension n, or simply the Euclidean space of dimension n.

A reason for introducing such an abstract definition of Euclidean spaces, and for working with E n {\displaystyle \mathbb {E} ^{n}} instead of R n {\displaystyle \mathbb {R} ^{n}} is that it is often preferable to work in a coordinate-free and origin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world.