Mathematics REAL VARIABLES (Rational numbers ) course

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, (−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.

Examples 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 λ, m, and n are positive rational numbers, and m > n, then λ(m2 − n 2 ), 2λmn, and λ(m2 + 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.

4. The positive rational numbers may be arranged in the form of a simple series as follows:

1 /1 , 2/ 1 , 1 /2 , 3 /1 , 2/ 2 , 1/ 3 , 4/ 1 , 3 /2 , 2 /3 , 1/ 4 , . . . .

Show that p/q is the [ 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 1/ 1 , 2/ 2 , 3/ 3 , . . . . 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.