s, Socrates.

Socrates: Let's try odd over even next, shall we?

Boy: Fine.

Socrates: What happens when you multiply an even number by
an even number, what kind of number do you get, even or odd?

Boy: Even, of course. An even multiple of any whole number
gives another even number.

Socrates: Wonderful, you have answered two questions, but we
need only one at the moment. We shall save the other. So, with
odd over even, if we multiply any of these times themselves, we
well get odd times odd over even times even, and therefore odd
over even, since odd times odd is odd and even of even is even.

Boy: Yes. A ratio of odd over even, when multiplied times
itself, yields odd over even.

Socrates: And can our square root of two be in that group?

Boy: I don't know, Socrates. Have I failed?

Socrates: Oh, you know, you just don't know that you know.

Try this: after we multiply our number times itself,
which the learned call "squaring" the number which is the root,
we need to get a ratio in which the first or top number is twice
as large as the second or bottom number. Is this much correct?

Boy: A ratio which when "squared" as you called it, yields
an area of two, must then yield one part which is two times the
other part. That is the definition of a ratio of two to one.

Socrates: So you agree that this is correct?

Boy: Certainly.

Socrates: Now if a number is to be twice as great as another,
it must be two times that number?

Boy: Certainly.

Socrates: And if a number is two times any whole number,
it must then be an even number, must it not?

Boy: Yes, Socrates.

Socrates: So, in our ratio we want to square to get two,
the top number cannot be odd, can it?

Boy: No, Socrates. Therefore, the group of odd over even
rational numbers cannot have the square root of two in it!
Nor can the group ratios of odd numbers over odd numbers.

Socrates: Wonderful. We have just eliminated three of the
four groups of rational numbers, first we eliminated the
group of even over even numbers, then the ones with odd numbers
divided by other numbers. However, these were the easier part,
and we are now most of the way up the mountain, so we must rest
and prepare to try even harder to conquer the rest, where the
altitude is highest, and the terrain is rockiest. So let us sit
and rest a minute, and look over what we have done, if you will.

Boy: Certainly, Socrates, though I am much inv