PureMath.pdf

The section of 4 gives an example of the last

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The section of § 4 gives an example of the last possibility. An example of the first is obtained by taking P to be ‘ x 2 5 1’ and Q to be ‘ x 2 > 1’; here l = 1. If P is ‘ x 2 < 1’ and Q is ‘ x 2 = 1’, we have an example of the second possibility, with r = 1. It should be observed that we do not obtain a section at all by taking P to be ‘ x 2 < 1’ and Q to be ‘ x 2 > 1’; for the special number 1 escapes classification (cf. Ex. iii . 5). 7. Irrational numbers ( continued ). In the first two cases we say that the section corresponds to a positive rational number a , which is l in the one case and r in the other. Conversely, it is clear that to any such number a corresponds a section which we shall denote by α . * For we might take P and Q to be the properties expressed by x 5 a, x > a respectively, or by x < a and x = a . In the first case a would be the greatest member of L , and in the second case the least member of R . * It will be convenient to denote a section, corresponding to a rational number de- noted by an English letter, by the corresponding Greek letter.

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[I : 8] REAL VARIABLES 14 There are in fact just two sections corresponding to any positive rational number. In order to avoid ambiguity we select one of them; let us select that in which the number itself belongs to the upper class. In other words, let us agree that we will consider only sections in which the lower class L has no greatest number. There being this correspondence between the positive rational numbers and the sections defined by means of them, it would be perfectly legitimate, for mathematical purposes, to replace the numbers by the sections, and to regard the symbols which occur in our formulae as standing for the sections instead of for the numbers. Thus, for example, α > α 0 would mean the same as a > a 0 , if α and α 0 are the sections which correspond to a and a 0 . But when we have in this way substituted sections of rational numbers for the rational numbers themselves, we are almost forced to a generali- sation of our number system. For there are sections (such as that of § 4 ) which do not correspond to any rational number. The aggregate of sec- tions is a larger aggregate than that of the positive rational numbers; it includes sections corresponding to all these numbers, and more besides. It is this fact which we make the basis of our generalisation of the idea of number. We accordingly frame the following definitions, which will how- ever be modified in the next section, and must therefore be regarded as temporary and provisional. A section of the positive rational numbers, in which both classes exist and the lower class has no greatest member, is called a positive real number . A positive real number which does not correspond to a positive rational number is called a positive irrational number.
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