Analysis — Chapter 1 — Real Variables — part 8

8. Real numbers. 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 part 6 and 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 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 line.

The term “rational number” will, as a result of our definitions, be ambiguous, it may mean the rational number of part 1, or the, corresponding real number. If we say that 1/2 > 1/3, we may  be asserting either of the 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 arithmetic fractions or real numbers. Sometimes, of course, the context in which (example) ‘1/2‘ occurs is sufficient to fix its interpretation. When we say (next blog part 9) that 1/2 < \sqrt{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, that is, a pair of classes. We might equally well have defined it to being the lower, or the upper class; indeed it would be easy to define an infinity of classes of entities of 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 what we say about it is true”, a remark which is expressed in the form of paradox but which in reality embodies a number of important truths. It would take too long to analyze the meaning of Mr Russell’s epigram in detail, but one at any rate of the implications is this, that the symbols of mathematics are capable of varying interpretations, and that we are in general at liberty to adopt whatever 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 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. 

Note: The difference between our presentation of a positive real number here and that or part 7 of the blogs 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 part 6 of the blogs 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.

More later,

Nalin Pithwa

 

 

Analysis — Chapter 1 Real Variables — part 7 — continued

Part 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

\alpha^{*}. For we might take P and Q to be the properties expressed by

x \leq a, x > a

respectively, or by x<a and x \leq a. In the first case, a would be the greatest number of L, and in the second case the least member of R. These 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,

\alpha > \alpha^{'} would mean the same as a > a^{'}. If \alpha and \alpha^{'} are

the sections which correspond to a and a^{'}.

But, when we have in this way substituted sections of rational numbers for the rational numbers themselves, we are almost forced to a generalization of our number system. For there are sections (such as that of blog on Chapter 1 — part 4) which do not correspond to any rational number. The aggregate of sections 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 generalization of the idea of a number. We accordingly frame the following definitions, which will however be modified in the next blog, 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.

More later,

Nalin Pithwa

Analysis — Chapter I Part I — Real Variables

Real Variables.

1Rational Numbers. A fraction r=p/q, where p and q are positive or negative integers, is called a rational number. We can assume (i) that p and q have no common factors, as if they have a common factor we can divide each of them by it, and (ii) that q is positive, since

p/(-q)=(-p)/q and (-p)/(-q)=p/q.

To the rational numbers thus defined we may add the “rational number 0” obtained by taking p=0.

We assume that you are familiar with the ordinary arithmetic rules for the manipulation of rational numbers. The examples which follow demand no knowledge beyond this.

Example I. 1. If r and s are rational numbers, then r+s, r-s, rs and r/s are rational numbers, unless in the last case s=0 (when r/s is meaningless, of course).

2. If \lambda, m, and n are positive rational numbers, and m>n, then

\lambda (m^{2}-n^{2}), 2\lambda mn, and \lambda (m^{2}+n^{2}) are positive rational numbers. Hence, show how to determine any number of right angled triangles the lengths of all of whose sides are rational.

Proof: Let the hypotenuse be \lambda (m^{2}+n^{2}) and the two arms of the right angled triangle be

2\lambda mn and \lambda (m^{2}-n^{2}). Then, the Pythagoras’s theorem holds. But, the sides and the hypotenuse are all rational.

3. Any terminated decimal represents a rational  number whose denominator contains no  factors other than 2 or 5. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.

(We will look into this matter a bit deeper, a little later).

4. The positive rational numbers may be arranged in the form of a simple series as follows:

1/1,2/1,1/2, 3/1,2/2, 1/3,4/1,3/2,2/3,1/4, \ldots.

Show that p/q is the \{ (1/2)(p+q-1)(p+q-2) +q \}th term of the series.

(In this series, every rational number is repeated indefinitely. Thus 1 occurs as 1/1, 2/2, 3/3, \ldots We can of course avoid this by omitting every number which has already occurred in a simple form, but then the problem of determining the precise position of p/q becomes more complicated.) Check this for yourself! If you do not get the answer, just write back in the comment section and I will help clarify the matter.

More later…

Nalin Pithwa