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




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