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That's a distinction between a vector space and an affine space, but I often find the term "Euclidean space" defined as a vector space, as hereThe reason the distinction is made is that the point 0 is "marked" in a vector space, but it is not in Euclidean space.

http://mathworld.wolfram.com/EuclideanSpace.html

So I'm interested in what the ideal way of defining "Euclidean space" and "R

^{n}" might be, and also in the (apparently varied) ways that they're defined in practice, so that I can understand mathematical texts. I think your point is that it's best to regard Euclidean space as an affine space.

The definitions that I've read of an affine space allFor our purposes, we don't need the vector space structure, so it's superfluous.

*include*the concept of a vector space. For example, I've seen affine space defined as the 3-tuple (S,V,f) where S is a nonempty set (whose elements are called the points), V is a vector space, and f is a function, +, relating points in S and vectors of V, such that

1. (p +

**u**) +

**v**= p + (

**u**+

**v**).

2. For all q in S, there exists a unique vector

**v**in V such that q = p +

**v**.

You mean bound vectors? That reminds me... Does this line from the articleIn addition, it's confusing, since we need to be able to identify "vectors" with differing base points, which we certainly can't do in the vector space context.

*Euclidean vector*seem the wrong way round to you: "This more general type of spatial vector is the subject of vector spaces (for bound vectors) and affine spaces (for free vectors)"?