The notion of vector space grew up from the discovery by Descartes that points in the Euclidian plane can be represented by ordered pairs of real numbers (and points in 3-dimensional space by ordered triples). We are faced with two completely different descriptions: a point in the plane, or a pair of real numbers. Moreover, operations on vectors look quite different according to which description is used. By example, we add vectors by the parallelogram law used in mechanics, and we add the real numbers componentwise; but the result is the same.