Analysis — Chapter 1 — Real Variables — Part 4 Irrational numbers continued

Part 4. Irrational numbers (continued).

The result of our geometrical interpretation of the rational numbers is therefore to suggest the desirability of enlarging our conception of “number” by the introduction of further numbers of a new kind.

The same conclusion might have been reached without the use of geometrical language. One of the central problems of algebra is that of the solution of equations, such as

x^{2}=1, x^{2}=2.

The first equation has the two rational roots 1 and -1. But, if our conception of number is to be limited to the rational numbers, we can only say that the second equation has no roots; and the same is the case with such equations as x^{3}=2, x^{4}=7. These facts are plainly sufficient to make some generalization of our idea of number desirable, if it should prove to be possible.

Let us consider more closely the equation x^{2}=2.

We have already seen that there is no rational number x which satisfies this equation. The square of any rational number is either less than or greater than 2. We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than 2, and the other those whose squares are greater than 2. We shall confine our attention to the positive rational numbers, and we shall call these two classes the class L, or the lower class, or the left-hand class, and the class R, or the upper class, or the right hand class. It is obvious that every member of R is greater than all the members of class R. Moreover, it is easy to convince ourselves that we can find a member of the class L whose square, though less than 2, differs from 2 by as little as possible, and a member of R whose square, though greater than 2, also differs from 2 by as little as we please. In fact, it we carry out the ordinary arithmetical process for the extraction of the square root of 2, we obtain a series of rational numbers, viz.,

1,1.4, 1.41, 1.414, 1.4142, \ldots

whose squares

1, 1.96, 1.9881, 1.999396, 1.99996164, \ldots

are all less than 2, but approach nearer and nearer to it, and by taking a sufficient number of the figures given by the process we can obtain as close an approximation as we want. And if we increase the last figure, in each of the approximations given above, by unity, we obtain a series of rational numbers

2, 1.5, 1.42, 1.415,1.413, \ldots

whose squares

4, 2.25, 2.0164, 2.002225, 2.00024449, \ldots

are all greater than 2, but approximate to 2 as closely as we please.

It follows also that there can be no largest member of L or smallest member of R. For if x is any member of L, then

x^{2} < 2. Suppose that x^{2}=2-\delta. Then we can find a member x, of L such that {x_{1}}^{2} differs from 2 by less than \delta, and {x_{1}}^{2}>x^{2} or x_{1}>x. Thus there are larger members of L than x; and, as x is any member of L, it follows that no member of L can be larger than all the rest. Hence, L has no largest member, and similarly, it has no smallest.

Note: A rigorous analysis of the above can be easily carried out. If you need help, please let me know and I will post it in the next blog.

More later,

Nalin Pithwa

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