# Analysis — Chapter I Part I — Real Variables

Real Variables.

1Rational Numbers. A fraction $r=p/q$, 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

$p/(-q)=(-p)/q$ and $(-p)/(-q)=p/q$.

To the rational numbers thus defined we may add the “rational number 0” obtained by taking $p=0$.

We assume that 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 $r+s$, $r-s$, $rs$ and $r/s$ are rational numbers, unless in the last case $s=0$ (when $r/s$ is meaningless, of course).

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.

Proof: Let the hypotenuse be $\lambda (m^{2}+n^{2})$ and the two arms of the right angled triangle be

$2\lambda mn$ and $\lambda (m^{2}-n^{2})$. 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:

$1/1,2/1,1/2, 3/1,2/2, 1/3,4/1,3/2,2/3,1/4, \ldots$.

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 $1/1, 2/2, 3/3, \ldots$ 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 $p/q$ 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

This site uses Akismet to reduce spam. Learn how your comment data is processed.