It is convenient, in many branches of mathematical analysis, to make a good deal of use of geometrical illustrations.
The use of geometrical illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more, and are employed merely for the sake of clearness of exposition. This being so, it is not necessary that we should attempt any logical analysis of the ordinary notions of elementary geometry, we may be content to suppose, however, far it may be from the truth, that we know what they mean.
Assuming, then, that we know what is meant by a straight line, a segment of a line, and the length of a segment, let us take a straight line A, produced indefinitely in both directions, and a segment of any length. We call the origin, or the point 0, and the point 1, and we regard these points as representing the numbers 0 and 1.
In order to obtain a point which shall represent a rational number , we choose the point A, such that
, being a stretch of the line extending in the same direction along the line as , a direction which we shall suppose to be from left to right when, the line is drawn horizontally across the paper. In order to obtain a point to represent a negative rational number , it is natural to regard length as a magnitude capable of sign, positive if the length is measured in one direction (that of , and negative if measured in the other, so that , and to take as the point representing r the point
We thus obtain a point on the line corresponding to every rational value of r, positive or negative, and such that ;
and if, as is natural, we take as our unit of length, and write , then we have
We shall call the points , the rational points of the line.