**Real Variables.**

**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

and .

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.**

More later…

Nalin Pithwa

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