Page:A Course of Modern Analysis - 3rd edition - 1920.pdf/18

 4 THE PROCESSES OF ANALYSIS [CHAP. I

A rational number x may be represented to the eye in the following manner:

Tf, on a straight line, we take an origin O and a fixed segment OP, (2, being on the right of 0), we can measure from 0 a length OP, such that the ratio OP,/OP, is equal w a; the point P, is taken on the right or left of G according as the number « is positive or negative. We may regard either the point P, or the displacement OP, (which will be written OP) as representing the number 2.

All the rational numbers can thus be represented by points on the line, but the converse is not true. Vor if we measure off on the line a length OQ equal to the diagonal of a square of which OP, is one side, it can be proved that Q docs not correspond to any rational number.

Points on the line which do not represent rational numbers may be said to represent irrational numbers; thus the point @ is said to represent the irrational number /2=1.414213..... But while such an explanation of the existence of irrational mumbers satisfied the mathematicians of the eighteenth century and may still be sufficient for those whose interest Hes in the applications of mathematics rather than in the logical upbuilding of the theory, yet from the logical standpoint it is improper to introduce geometrical intuitions to supply deficiencies in arithmetical arguments; and it was shown by Dedekind in 1855 that the theory of irrational numbers can be established on a purely arithmetical basis without any appeal to geometry.

12. Dedehind’s* theory of irrational numbers.

The geometrical property of points on a line which suggested the starting point of the arithmetical theory of irrationals was that, if all points of a line are separated into two classes such that every point of the first class is on the right of every point of the second class, there exists one and only once point at which the line is thus severed.

Following up this idea, Dedekind considered rules by which a separation+ or section of all rational numbers into two classes can be made, these classes Qvhich will be called the L-e and the H-class, or the left class and the right class) being such that they possess the following properties:

(i) At least one member of each class exists.

(ii) Every member of the £-class is less than every member of the R-class.

It is obvious that such a section is made by any rational number a; and x is either the greatest number of the L-class or the least number of the


 * ‘The theory, though elaborated m 1858, was not published before the appearance of Dedekind’s tract, Stetigheit und terationate 7 Other theories are due to Weicrstiass [see von Dantscher, Die Weirrstrass'sche Theorie der irrationden Zahlen (Leipzig, 1908}) avd Cantor, Math. dan. ¥. (1872), pp. 125-180.

} This procedure formed the basis of the treatment of irrational numbers by the Greek mathematicians in the sixth and fifth centuries B.C. The advance made by Dedekind consisted in observing that a purely arithmetical theory could be built up on it.