The existence of root two

The existence of the square root of two.

How could there possibly be any doubt about the existence of the<br>square root of two? This is a very reasonable question, but one that<br>few people dare to ask when their lecturers solemnly stand in front of<br>them and prove that the square root of two exists, either by defining<br>x to be the supremum of all rationals r such that r22=2, or by applying the intermediate value<br>theorem (which itself confuses people by seeming too obvious to need a<br>proof).

Imagine that you did not know any advanced mathematics (if you<br>actually don't, then that is fine) and were confronted by somebody who<br>denied the existence of the square root of two. What would you say?<br>The conversation might go something like this.

What do you mean when you say that the square root<br>of two doesn't exist?

I mean the obvious thing: if you take any real number x and square<br>it, the answer is never 2.

But what about 1.41421356237309... ?

What about it? You haven't told me how the sequence continues.

Well, all I'm doing is taking the decimal expansion of the square<br>root of two.

That sounds pretty circular to me.

You're right. I'm sorry. But it isn't really as circular as it<br>sounds. What I mean is that I am calculating the decimal expansion<br>of the real number x with the property that x2=2.

That still sounds circular. Aren't you still assuming that<br>a number with this property exists?

No, because I can tell you how to calculate the sequence of<br>digits, and that will be my proof that the number exists.

Go on then.

Well, 12=12=4>2 so I know that<br>the the number must be one point something. Then by trial and<br>error I discover that 1.42=1.962=2.25>2 so the decimal expansion must start with<br>1.4. I then just continue this process: if I have calculated the<br>first 38 digits, say, then I try all the possibilities for the<br>39th, picking the largest one that results in a number whose square<br>is less than 2.

All right, I see what it is you are doing, and that it leads<br>to an unambiguously defined infinite decimal. But what makes you<br>call that a real number and what makes you so sure that it squares<br>to two?

I don't understand the first question. Surely a real number is<br>something with a possibly infinite decimal expansion.

That sounds fishy to me. I notice that you say something<br>with a decimal expansion, rather than just a decimal expansion.<br>So what is the actual thing, of which you calculate the expansion?

Well, it's just a number ... you know, something like 1,2,3,..<br>or 25/38 or pi or the square root of 3.

I notice you didn't have the courage to say the square root of<br>2! I'm beginning to think that you don't really have any idea what<br>a real number is. You've given me a few examples, but you haven't<br>said what they have in common.

I think you are being quite unnecessarily pedantic. Just think<br>of the number line. It's got all the numbers on it, in order (ignoring<br>the complex numbers for now). We know that some numbers are irrational,<br>but we can still describe them, by means of their decimal expansions.

Describe what?

Positions on the number line, lengths, whatever you want to call them.

That's no good at all. What is this number line that<br>you assume I am familiar with? What is a length? Note that for the<br>second question you can't fob me off with an answer about rulers<br>and so on, because they only work to a certain accuracy.

All right, I take the point, but I still don't think it is a<br>serious problem. If it makes you feel better, I shall simply<br>define a real number to be a decimal expansion.

So when you say "the real number x" what you really mean is<br>"the decimal expansion x"?

Well, it's not always what I think of when I talk about real numbers,<br>but if you insist on a precise definition, then I can fall back on this<br>one.

Does that mean that 0.999999.... and 1 are different numbers?

Oh yes, I forgot about that. Different decimal expansions correspond<br>to different real numbers except in cases like 2.439999999.... equalling<br>2.44. So I suppose my definition is that real numbers are finite or<br>infinite decimals except that a finite decimal can also be written as,<br>and is considered equal to, the "previous" finite decimal with an<br>infinite string of nines on the end. Happy now?

We've hardly started, because you haven't told me how to do<br>arithmetic with these real numbers of yours, and you certainly<br>haven't convinced me that there is a real number that squares to<br>give 2.

Are you going to ask me how to multiply two infinite decimals<br>together?

Yes.

Well, you just do it in the obvious way, by a sort of infinite<br>long multiplication.

It sounds to me as though long multiplication would be a pretty<br>accurate description of whatever process you have in mind, but I<br>notice that you are somewhat vague about it.

Do I really have to go into this? Surely you can see how it<br>would work.

No I can't.

Well, let's take the example of the square root of two. If you<br>take the numbers 1, 1.4, 1.41, 1.414 and so...