1. Rational Numbers. A fraction , where p and q are positive or negative integers, is called a rational number. We can assume (i) that p and q have no common factors, as if they have a common factor we can divide each of them by it, and (ii) that q is positive, since
To the rational numbers thus defined we may add the “rational number 0” obtained by taking .
We assume that you are familiar with the ordinary arithmetic 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 , , and are rational numbers, unless in the last case (when is meaningless, of course).
2. If , m, and n are positive rational numbers, and , then
, , and are positive rational numbers. Hence, show how to determine any number of right angled triangles the lengths of all of whose sides are rational.
Proof: Let the hypotenuse be and the two arms of the right angled triangle be
and . Then, the Pythagoras’s theorem holds. But, the sides and the hypotenuse are all 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.
(We will look into this matter a bit deeper, a little later).
4. The positive rational numbers may be arranged in the form of a simple series as follows:
Show that is the th term of the series.
(In this series, every rational number is repeated indefinitely. Thus 1 occurs as We can of course avoid this by omitting every number which has already occurred in a simple form, but then the problem of determining the precise position of becomes more complicated.) Check this for yourself! If you do not get the answer, just write back in the comment section and I will help clarify the matter.