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

 

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