One of the most fundamental properties of the set of rational numbers is their order. We shall find in the sequel that the idea of order is one of the most essential to the understanding of sets of points, and that we habitually use the order of some or all of the rational numbers as a standard of comparison.

The order of the rational numbers as a whole is such that we cannot say which is the next rational number in order of magnitude after any given one a, or before a given one c; indeed, if a and c be any two rational numbers, we can always insert a rational number b between them.

It is of assistance to the imagination that we can set up a (1, 1)-correspondence between the rational numbers and certain points of the straight line, in such a way that the order is maintained, that is to say if Ap, Aq, Ar are three of the points, corresponding to the rational numbers p, q, r, Aq lies between Ap and Ar if, and only if, q lies between p and r, and vice versa.

We shall now discuss shortly, how and under what assumptions with respect to the nature of the straight line, this correspondence can be extended to the irrational numbers.

In setting up the (1, 1)-correspondence referred to, measurement may be entirely avoided; in this way various difficulties which have nothing to do with the subject in hand do not come into the discussion.