irrational

The story of the discovery of irrational numbers is shrouded in mystery, and linked with a certain early Pythagorean named Hippasus. The popular version has rumours of cover-up, betrayal, perhaps even murder.

I have taken the proof of their existence, and, seeing that the Socratic dialogue is still a contemporary form, re-created a later conversation between Socrates and the slave boy we meet in Meno.







Socrates: So you’ve kept going with your mathematical studies?
Boy: Yes, you made me see that it was possible, Socrates. I have, as you said, approached the same questions in many different ways, and have indeed gone far beyond what we talked about last time.
Socrates: And what in particular have you been studying?
Boy: Too many things to say.
Socrates: Tell me at least one or two of the areas.
Boy: Where can I begin? Factors, square numbers, whole numbers, fractions…
Socrates: So, many of the kinds of numbers…
Boy: You could say all of the numbers, as all numbers are either whole numbers, or a fraction of some kind.
Socrates: Well that’s an interesting suggestion. Are you sure you’ve included all possible numbers?
Boy: How could there be any other?
Socrates: Well, you remember that diagonal we talked about last time?
Boy: When we were doubling the size of the square, yes.
Socrates: I think I can prove to you that the side of a square and its diagonal could not possibly both be whole numbers or fractions.
Boy: How could that be?
Socrates: Well let’s draw that same diagram again.










Socrates: Let’s propose that the side and diagonal of the small square are both whole numbers, the smallest possible whole numbers they could be. Do you understand what I am proposing?
Boy: Yes, I do.
Socrates: So they could not both be, let’s say, even, could they?
Boy: Obviously not. You would be able to halve both numbers and still have whole numbers, and you proposed that they were the smallest possible whole numbers.
Socrates: I see that you have indeed, as you say, been studying. Now, if the side of the small square is a whole number, you will agree that the square itself must be whole a whole number.
Boy: Yes, that’s obvious.
Socrates: Forgive me if I move in steps that are too obvious. I only want to be sure that we have not taken some wrong turning. Now we saw last time, did we not, that the square on the diagonal is twice the size of the small square?
Boy: Yes, how could I forget!
Socrates: Then that square on the diagonal, as it is double a whole number, must be an even number?
Boy: Agreed.
Socrates: And would you agree too that if the square is even, then the side of the square must be even too?
Boy: I’ll need to think about that a moment, let me see…











Boy: Yes, I can see now that all even squares will have an even side.
Socrates: So, we have established that the diagonal must be an even number.
But the way you’ve drawn those squares, shows something else, did you see? – They are always multiples of four too?
Boy: Yes…
Socrates: And the little square is half the size of the square on the diagonal, so it must, you’ll agree, be half a multiple of four?
Boy. Yes… which will be a multiple of two.
Socrates: I can see it’s not so easy for me to race ahead of you anymore. Yes, it will be an even number. And even squares have even sides as we said, so the side of the small square is even too.
Boy: Yes.
Socrates: But don’t you see, there is something absurd here? We started by saying the side and the diagonal would be the smallest possible numbers. But now we find that they are both even.
Boy: Yes, which means they could both be divided by two to get smaller whole numbers. How can that be?
Socrates: How indeed? Are you sure that all the steps we have taken are acceptable? I haven’t moved you on too quickly, making you agree with something that is uncertain by sheer force of personality, have I?
Boy: No, Socrates, each step was clear enough. And yet we arrive at an absurdity…?
Socrates: The only way out of it that I can see – tell me if you can see another – is that the proposition we started with, that there are two smallest whole numbers that could measure the side and diagonal of the square is an impossibility: there are no such numbers.
Boy: But that seems illogical. These lines must after all have a particular length?
Socrates: That I do not deny. But both of them cannot be whole numbers – or, as it is simple enough to demonstrate, fractions for that matter.
Boy: Is there not something unreasonable about that proposition, that there are measurements which cannot be expressed either as whole numbers or as fractions?
Socrates: We cannot exactly call it a proposition, since it is something more certain that we have arrived at by carefully following the trail left by the numbers themselves. But I know what you mean about it’s seeming unreasonableness, and with your permission I propose to call such numbers “irrational numbers”.