Analysis — Chapter I — Part II — Real Variables

 

The representation of rational numbers by points on a line.

It is convenient, in many branches of mathematical analysis, to make a good deal of use of geometrical illustrations.

The use of geometrical illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more, and are employed merely for the sake of clearness of exposition. This being so, it is not necessary that we should attempt any logical analysis of the ordinary notions of elementary geometry, we may be content to suppose, however, far it may be from the truth, that we know what they mean.

Assuming, then, that we know what is meant by a straight line, a segment of a line, and the length of a segment, let us take a straight line A, produced indefinitely in both directions, and a segment A_{0}A_{1} of any length. We call A_{0} the origin, or the point 0, and A_{1} the point 1, and we regard these points as representing the numbers 0 and 1.

In order to obtain a point which shall represent a rational number r=p/q, we choose the point A, such that

A_{0}A_{r}/A_{0}A_{1}=r,

A_{o}A_{r}, being a stretch of the line extending in the same direction along the line as A_{0}A_{1}, a direction which we shall suppose to be from left to right when, the line is drawn horizontally across the paper. In order to obtain a point to represent a negative rational number r=-s, it is natural to regard length as a magnitude capable of sign, positive if the length is measured in one direction (that of A_{0}A_{1}, and negative if measured in the other, so that AB=-BA, and to take as the point representing r the point

A_{-s} such that

A_{0}A_{-s}=-A_{-s}A_{0}=-A_{0}A_{s}.

We thus obtain a point A_{r} on the line corresponding to every rational value of r, positive or negative, and such that A_{0}A_{r}=r.A_{0}A_{1};

and if, as is natural, we take A_{0}A_{1} as our unit of length, and write A_{0}A_{1}, then we have

A_{0}A_{r}=r.

We shall call the points A_{r}, the rational points of the line.

More later…

Nalin Pithwa

 

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