## 1.Rational Numbers

A fraction \(r = p/q\), where \(p\) and \(q\) are positive or negative integers, is called a *rational number*. We can suppose (i) that \(p\) and \(q\) have no common factor, as if they have a common factor we can divide each of them by it, and (ii) that \(q\) is positive, since \[p/(-q) = (-p)/q,\quad (-p)/(-q) = p/q.\] To the rational numbers thus defined we may add the ‘rational number \(0\)’ obtained by taking \(p = 0\).

We assume that the reader is familiar with the ordinary arithmetical rules for the manipulation of rational numbers. The examples which follow demand no knowledge beyond this.

## 2. The representation of rational numbers by points on a line.

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 \(\Lambda\), produced indefinitely in both directions, and a segment \(A_{0}A_{1}\) of any length. We call \(A_{0}\) the *origin*, or *the point \(0\)*, and \(A_{1}\) *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 positive rational number \(r = p/q\), we choose the point \(A_{r}\) such that \[A_{0}A_{r}/A_{0}A_{1} = r,\] \(A_{0}A_{r}\) being a stretch of the line extending in the same direction along the line as \(A_{0}A_{1}\), a direction which we shall suppose to be from left to right when, as in Fig. 1, the line is drawn horizontally across the paper.

In order to obtain a point to represent a negative rational number \(r = -s\), it is natural to regard length as a magnitude capable of sign, positive if the length is measured in one direction (that of \(A_{0}A_{1}\)), and negative if measured in the other, so that \(AB = -BA\); and to take as the point representing \(r\) the point \(A_{-s}\) such that \[A_{0}A_{-s} = -A_{-s}A_{0} = -A_{0}A_{s}.\]

We thus obtain a point \(A_{r}\) on the line corresponding to every rational value of \(r\), positive or negative, and such that \[A_{0}A_{r} = r \cdot A_{0}A_{1};\] and if, as is natural, we take \(A_{0}A_{1}\) as our unit of length, and write \(A_{0}A_{1} = 1\), then we have \[A_{0}A_{r} = r.\] We shall call the points \(A_{r}\) the *rational points* of the line.

Main Page | 3-7. Irrational Numbers $\rightarrow$ |