Analysis: Chapter 1: part 10: algebraic operations with real numbers

Algebraic operations with real numbers.

We now proceed to meaning of the elementary algebraic operations such as addition, as applied to real numbers in general.

(i),  Addition. In order to define the sum of two numbers \alpha and \beta, we consider the following two classes: (i) the class (c) formed by all sums c=a+b, (ii) the class (C) formed by all sums C=A+B. Clearly, c < C in all cases.

Again, there cannot be more than one rational number which does not belong either to (c) or to (C). For suppose there were two, say r and s, and let s be the greater. Then, both r and s must be greater than every c and less than every C; and so C-c cannot be less than s-r. But,

C-c=(A-a)+(B-b);

and we can choose a, b, A, B so that both A-a and B-b are as small as we like; and this plainly contradicts our hypothesis.

If every rational number belongs to (c) or to (C), the classes (c), (C) form a section of the rational numbers, that is to say, a number \gamma. If there is one which does not, we add it to (C). We have now a section or real number \gamma, which must clearly be rational, since it corresponds to the least member of (C). In any case we call \gamma the sum of \alpha and \betaand write 

\gamma=\alpha + \beta.

If both \alpha and \beta are rational, they are the least members of the upper classes (A) and (B). In this case it is clear that \alpha + \beta is the least member of (C), so that our definition agrees with our previous ideas of addition.

(ii) Subtraction.

We define \alpha - \beta by the equation \alpha-\beta=\alpha +(-\beta).

The idea of subtraction accordingly presents no fresh difficulties.

More later,

Nalin Pithwa

Analysis — Chapter 1: continued — Real Variables part 9

9. Relations of magnitude between real numbers. 

It is plain, that, now that we have extended our conception of number, we are bound to make corresponding extensions of our conceptions of equality, inequality, addition, multiplication, and so on. We have to show that these ideas can be applied to the  new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of Chapter 1, part 1 blog. To do all this systematically would occupy considerable space/time, and we shall be content to indicate summarily how a more systematic discussion would proceed.

We denote a real number by a Greek letter such as \alpha, \beta, \gamma\ldots; the rational numbers of its lower and upper classes by the corresponding English letters a, A; b, B; c, C; …We denote the classes themselves by (a), (A),…

If \alpha and \beta are two real numbers, there are three possibilities:

i) every \alpha is a b and every A a B; in this case, (a) is identical with (b) and (A) with (B);

ii) every a in a b, but not all A’s are B’s; in this case (a) is a proper part of (b)^{*}, and (B) a proper part of (A);

iii) every A is a B, but not all a’s are b’s.

(These three cases may be indicated graphically on a number line).

In case (i) we write \alpha=\beta, in case (ii) \alpha=\beta, and in case (iii) \alpha>\beta. It is clear that, when \alpha and \beta are both rational, these definitions agree with the ideas of equality and inequality between rational numbers which we began by taking for granted; and that any positive number is greater than any negative number.

It will be convenient to define at this stage the negative -\alpha of a positive number \alpha. If

(\alpha), (A) are the classes, which consitute \alpha, we can define another section of the rational numbers by putting all numbers -A in the lower class and all numbers -\alpha in  the upper. The real number thus defined, which is clearly negative, we denote by -\alpha. Similarly, we can define

-\alpha when \alpha is negative or zero; if \alpha is negative, -\alpha is positive, It is plain also  that -(-\alpha)=\alpha. Of the two numbers \alpha and -\alpha one is always positive (unless \alpha=0). The one which is positive we denote by |\alpha| and call the modulus of \alpha.

More later,

Nalin Pithwa

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