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 3 — Real Variables — Irrational numbers

Part 3. Irrational numbers.

If the reader will mark off on the line all the points corresponding to the rational numbers whose denominators are 1,2,3, …in succession, he will readily  convince himself that he can cover the line with rational points, as closely as he likes. We can state this more precisely as follows: If we take any segment BC on A, we can find as many rational points on it as we please on BC.

Suppose, for example, that BC falls within the segment A_{1}A_{2}. it is evident that if we choose a positive integer k such that

k.BC>1 Equation I

(The assumption that this is possible is equivalent to the assumption of what is known as the Axiom of Archimedes.)

and divide A_{1}A_{2} into k equal parts, then at least one of the points of division (say P) must fall inside BC, without coinciding with either B or C. For if this were not so, BC would be entirely included in one of the k parts into which A_{1}A_{2} has been divided, which contradicts the supposition I. But P obviously corresponds to a rational number whose denominator is k. Thus at least one rational point P lies between B and C. But, then we can find another such point Q between B and P, another between B and Q, and so on indefinitely; that is, as we asserted above, we can find as many as we please. We may express this by saying that BC includes infinitely many 

rational points. (We will investigate the meaning of infinite more closely later).

From these considerations, the reader might be tempted to infer that an adequate view of the nature of the line could be obtained by imagining it to be formed simply by the rational points which lie on it. And, it is certainly the case that if we imagine the line to be made up of  solely of the rational points, and all other points (if there are any such) to be eliminated, the figure would possess most of the properties which common sense attributes to the straight line, and would, to put the matter roughly, look and behave very much like a line.

A little further consideration, however, shows that this view would involve us in serious difficulties.

Let us look at the matter for a moment with the eye of common sense, and consider some of the properties which we may reasonably expect a straight line to possess if it is to satisfy the idea which we have formed of it in elementary geometry.

The straight line must be composed of points, and any segment of it by all the points which lie between its end points.  With any such segment must be associated a certain entity called its length, which must be a quantity capable of numerical measurement in terms of any standard or unit length, and these lengths must be capable of combination with another, according to the ordinary rules of algebra, by means of addition or multiplication. Again, it must be possible to construct a line whose length is the sum or product of any two given lengths. If the length PQ along a given line is a, and the length QR, along the same straight line, is b, the length PR must be a+b.

Moreover, if the lengths OP and OQ, along one straight line, are 1 and a, and the length OR along another straight line is b, and if we determine the length OS by Euclid’s construction for a fourth proportional to the lines OP, OQ, OR, this length must be ab, the algebraic fourth proportional to 1, a and b. And, it is hardly necessary to remark that the sums and products thus defined must obey the ordinary laws of algebra; viz.,

a+b=b+a

a+(b+c)=(a+b)+c

ab=ba

a(bc)=(ab)c

a(b+c)=ab+ac

The lengths of our lines must also obey a number of obvious laws concerning inequalities as well as equalities: thus, if A, B, C are three points lying along A from left to right, we must have AB<AC, and so on. Moreover, it might be possible, on our fundamental line \Lambda to find a point P such that A_{0}P is equal to any segment whatever taken along \Lambda or along any other straight line. All these properties of a line, and more, are involved in the presuppositions of our elementary geometry.

Now, it is very easy to see that the idea of a straight line as composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements. There are various elementary geometrical constructions, for example, which purport to construct a length x such that x^{2}=2. For instance, we may construct an isosceles right angled triangle ABC such that AB=AC=1.. Then, if BC=x, x^{2}=2. Or we may determine the length x by means of Euclid’s construction for a mean proportional to a and 2, as indicated in the figure. Our requirements therefore involve the existence of a length measured by a number x, and a point P on \Lambda such that A_{0}P=x, x^{2}=2.

But, it is easy to see that there is no rational number such that its square is 2. In fact, we may go further and say that there is no rational number whose square is m/n, where m/n is say positive fraction in its lowest terms, unless m and n are both perfect squares.

For suppose, if possible, that \frac {p^{2}}{q^{2}}=m/n.

p having no common factor with q, and m no common factor with n. Thus, np^{2}=mq^{2}. Every factor of q^{2} must divide np^{2}, and as p and q have no common factor, every factor of q^{2} must divide n. Hence,

n={\lambda}q^{2}, where \lambda is an integer. But, this involves m={\lambda}p^{2}: and as m and n have common factor, \lambda must be unity. Thus, m=p^{2} and n=q^{2}, as was to be proved. In particular, it follows by taking n=1, that an integer cannot be the square of a rational number, unless that rational number is itself integral.

it appears that our requirements involve the existence of a number x and a point P, not one of the rational points already constructed, such that A_{0}P=x and x^{2}=2; and, (as the reader will remember from elementary algebra) we write x = \sqrt {2}.

Alternate proof. 

The following alternate proof that no rational number can have its square equal to 2 is interesting.

Suppose, if possible, that p/q is a positive fraction, in its lowest terms such that (p/q)^{2}=2. It is easy to see that this involves (2q-p)^{2}=2(p-q)^{2}, and so \frac {2q-p}{p-q} is also another fraction having the same property. But, clearly,

q<p<2q and so p-q<q. Hence, there is another fraction equal to p/q and having a smaller denomination, which contradicts the assumption that p/q is in its lowest terms.

In the next blog, we shall look at examples,

More later,

Nalin Pithwa