Socrates: So the number on the bottom is two times that whole number, whatever it is?
Boy: Yes, Socrates.
Socrates: (standing) And if it is two times a whole number, then it must be an even number, must it not?
Boy: Yes.
Socrates: Then is cannot be a member of the group which has an odd number on the bottom, can it?
Boy: No, Socrates.
Socrates: So can it be a member of the ratios created by an even number divided by an odd number and then used as a root to create a square?
Boy: No, Socrates. And that must mean it can't be a member of the last group, doesn't it?
Socrates: Yes, my boy, although I don't see how we can continue calling you boy, since you have now won your freedom, and are far richer than I will ever be.
Boy: Are you sure we have proved this properly? Let me go over it again, so I can see it in my head.