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 6: Irrational numbers continued

6. Irrational numbers (continued).

In Part 4, we discussed a special mode of division of the positive rational numbers x into two classes, such that x^{2}<2 for the numbers of one class and x^{2}>2 for those of the others. Such a mode of division is called a section of the numbers in question. It is plain that we could equally well construct a section in which the numbers of the two classes were characterized by the inequalities

x^{3}<2 and x^{3}>2, or x^{4}>7 and x^{4}<7. Let us now attempt to state the principles of the construction of such “sections” of the positive rational numbers in quite general terms.

Suppose that P and Q stand for two properties which are mutually exclusive and one of which must be possessed by every positive rational number. Further, suppose that every such number which possesses P is less than any such number which possesses Q. Thus, P might be the property “x^{2}<2” and Q the property “x^{2}>2“. Then, we call the numbers which possess P the lower or left-hand class L and those which possess Q the upper or right hand class R. In general, both classes will exist; but, it may happen in special cases that one is non-existent and every number belongs to the other. This would obviously happen, for example, if P (or Q) were the property of being rational, or of being positive. For the present, however, we shall confine ourselves to cases  in which both the classes do exist; and then it follows, as in Part 4, that we can find a member of L and a member of R, whose difference is as small as we please.

In the particular case, which we considered in Part 4, L had no greatest member and R no least. This question of the existence of greatest or least members of the classes is of the utmost importance. We observe first that it is impossible in any case that L should have a greatest member and R least. For, if l were the greatest member of L, and r the least of R, so that l<r, then (1/2)(l+r) would be a positive rational number lying between l and r, and so could neither belong to L nor to R, and this contradicts our assumption that every such number belongs to one class  or to the other.  This being so, there are but three possibilities, which are mutually exclusive. Either

(i) L has a greatest member l, or (ii) R has a least member, r, or (iii) L has no greatest member and R no least.

(In Part 4, there is an example of the last possibility.)

More later,

Nalin Pithwa