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 for the numbers of one class and 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
and , or and . 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 “” and Q the property ““. 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 , then 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.)