PureMath.pdf

# 8 real numbers we have confined ourselves so far to

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8. Real numbers. We have confined ourselves so far to certain sec- tions of the positive rational numbers, which we have agreed provisionally to call ‘positive real numbers.’ Before we frame our final definitions, we must alter our point of view a little. We shall consider sections, or divisions into two classes, not merely of the positive rational numbers, but of all ra- tional numbers, including zero. We may then repeat all that we have said about sections of the positive rational numbers in §§ 6 , 7 , merely omitting

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[I : 8] REAL VARIABLES 15 the word positive occasionally. Definitions. A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a real number , or simply a number . A real number which does not correspond to a rational number is called an irrational number. If the real number does correspond to a rational number, we shall use the term ‘rational’ as applying to the real number also. The term ‘rational number’ will, as a result of our definitions, be ambiguous; it may mean the rational number of § 1 , or the corresponding real number. If we say that 1 2 > 1 3 , we may be asserting either of two different propositions, one a proposition of elementary arithmetic, the other a proposition concerning sections of the rational numbers. Ambiguities of this kind are common in mathematics, and are perfectly harmless, since the relations between different propositions are exactly the same whichever interpretation is attached to the propositions themselves. From 1 2 > 1 3 and 1 3 > 1 4 we can infer 1 2 > 1 4 ; the inference is in no way affected by any doubt as to whether 1 2 , 1 3 , and 1 4 are arithmetical fractions or real numbers. Sometimes, of course, the context in which ( e.g. ) ‘ 1 2 ’ occurs is sufficient to fix its interpretation. When we say (see § 9 ) that 1 2 < q 1 3 , we must mean by ‘ 1 2 ’ the real number 1 2 . The reader should observe, moreover, that no particular logical importance is to be attached to the precise form of definition of a ‘real number’ that we have adopted. We defined a ‘real number’ as being a section, i.e. a pair of classes. We might equally well have defined it as being the lower, or the upper, class; indeed it would be easy to define an infinity of classes of entities each of which would possess the properties of the class of real numbers. What is essential in math- ematics is that its symbols should be capable of some interpretation; generally they are capable of many , and then, so far as mathematics is concerned, it does not matter which we adopt. Mr Bertrand Russell has said that ‘mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true’, a remark which is expressed in the form of a paradox but which in reality embodies a number of important truths.
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