We have confined ourselves so far to certain sections 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 rational numbers, including zero. We may then repeat all that we have said about sections of the positive rational numbers in §§ 6, 7, merely omitting 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 \(\frac{1}{2} > \frac{1}{3}\), 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 \(\frac{1}{2} > \frac{1}{3}\) and \(\frac{1}{3} > \frac{1}{4}\) we can infer \(\frac{1}{2} > \frac{1}{4}\); the inference is in no way affected by any doubt as to whether \(\frac{1}{2}\), \(\frac{1}{3}\), and \(\frac{1}{4}\) are arithmetical fractions or real numbers. Sometimes, of course, the context in which (e.g.) ‘\(\frac{1}{2}\)’ occurs is sufficient to fix its interpretation. When we say (see § 9) that \(\frac{1}{2} < \sqrt{\frac{1}{3}}\), we must mean by ‘\(\frac{1}{2}\)’ the real number \(\frac{1}{2}\).
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, 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 mathematics 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. It would take too long to analyse the meaning of Mr Russell’s epigram in detail, but one at any rate of its implications is this, that the symbols of mathematics are capable of varying interpretations, and that we are in general at liberty to adopt whichever we prefer.
There are now three cases to distinguish. It may happen that all negative rational numbers belong to the lower class and zero and all positive rational numbers to the upper. We describe this section as the real number zero. Or again it may happen that the lower class includes some positive numbers. Such a section we describe as a positive real number. Finally it may happen that some negative numbers belong to the upper class. Such a section we describe as a negative real number.1
The difference between our present definition of a positive real number \(a\) and that of § 7 amounts to the addition to the lower class of zero and all the negative rational numbers. An example of a negative real number is given by taking the property \(P\) of § 6 to be \(x + 1 < 0\) and \(Q\) to be \(x + 1 \geq 0\). This section plainly corresponds to the negative rational number \(-1\). If we took \(P\) to be \(x^{3} < -2\) and \(Q\) to be \(x^{3} > -2\), we should obtain a negative real number which is not rational.
- There are also sections in which every number belongs to the lower or to the upper class. The reader may be tempted to ask why we do not regard these sections also as defining numbers, which we might call the real numbers positive and negative infinity.There is no logical objection to such a procedure, but it proves to be inconvenient in practice. The most natural definitions of addition and multiplication do not work in a satisfactory way. Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word ‘infinity’; and experience seems to show that he is likely to be confused by any addition to their number.↩︎
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