Analysis — Chapter I — Real Variables — Part 5 — Irrational numbers continued

We have thus divided the positive rational numbers into two classes, L and R, such that (i) every member of R is greater than every member of L, and (ii) we can find a member of L and a member of R, whose difference is as small as we please, (iii) L has no greatest and R has not least member. Our common-sense notion of the attributes of a straight line, the requirements of our elementary geometry and our elementary algebra, alike demand the existence of a number x greater than all the members of L and less than all the members of R, and of a corresponding point P on \Lambda such that P divides the points which correspond to members of L from those which correspond to members of R.

Let us suppose for a moment that there is such a number x and that it may be operated upon in accordance with laws of algebra, so that, for example, x^{2} has a definite meaning. Then x^{2} cannot either be less than or greater than 2. For suppose, for example, that x^{2} is less than 2. Then, it follows from what precedes that we can find a positive rational number \xi such that \xi^{2} lies between x^{2} and 2. That is to say, we can find a member of L greater than x; and this contradicts the supposition that x divides the members of L from those of R. Thus, x^{2} cannot be less than 2, and similarly, it cannot be greater than 2. We are therefore driven to the conclusion that x^{2}=2, and that x is the number which in algebra  we denote by \sqrt{2}. And, of course, this number \sqrt{2} is not rational, for no rational number has its square equal to 2. It is the simplest example of what is called an irrational number.

But the preceding argument may be applied to equations other than x^{2}=2, almost word for word; for example, to

x^{2}=N, where N is an integer which is not a perfect square, or to

$latex $x^{3}=3$ and x^{2}=7 and x^{4}=23,

or, as we shall see later on. to x^{3}=3x+8. We are thus led to believe for the existence of irrational numbers x and points P on \Lambda such that x satisfies equations such as these, even when these lengths cannot (as \sqrt{2} can) be constructed by means of elementary geometric methods.

The reader may now follow one or other of two alternative courses. He may, if he pleases, be content to assume that “irrational numbers” such as \sqrt{2} and \sqrt[5]{3} exist and are amenable to usual algebraic laws. If he does this, he will be able to avoid the more abstract discussions of the next few blogs.

If, on the other hand, he is not disposed to adopt so naive an attitude, he will be well advised to pay careful attention to the blogs which follow, in which these questions receive further consideration.

More later,

Nalin Pithwa

 

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