1-2. Rational Numbers

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.

Example I

1. If \(r\) and \(s\) are rational numbers, then \(r + s\), \(r – s\), \(rs\), and \(r/s\) are rational numbers, unless in the last case \(s = 0\) (when \(r/s\) is of course meaningless).

2. If \(\lambda\), \(m\), and \(n\) are positive rational numbers, and \(m > n\), then \(\lambda(m^{2} – n^{2})\), \(2\lambda mn\), and \(\lambda(m^{2} + n^{2})\) are positive rational numbers. Hence show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.

3. Any terminated decimal represents a rational number whose denominator contains no factors other than \(2\) or \(5\). Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.

[The general theory of decimals will be considered in Ch. IV.]

4. The positive rational numbers may be arranged in the form of a simple series as follows: \[\tfrac{1}{1},\quad \tfrac{2}{1},\quad \tfrac{1}{2},\quad \tfrac{3}{1},\quad \tfrac{2}{2},\quad \tfrac{1}{3},\quad \tfrac{4}{1},\quad \tfrac{3}{2},\quad \tfrac{2}{3},\quad \tfrac{1}{4},\ \dots.\]

Show that \(p/q\) is the \([\frac{1}{2}(p + q – 1)(p + q – 2) + q]\)th term of the series.

[In this series every rational number is repeated indefinitely. Thus \(1\) occurs as \(\frac{1}{1}\), \(\frac{2}{2}\), \(\frac{3}{3}, \dots\). We can of course avoid this by omitting every number which has already occurred in a simpler form, but then the problem of determining the precise position of \(p/q\) becomes more complicated.]

 

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.

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