Boy: You want me to call the numbers made from ratios of whole numbers something called rational? A ratio makes a rational number?
Socrates:Yes boy, can you do that? Boy:Certainly, Socrates.
Socrates: Do you agree with the way I told him this, Meno? Does it violate our agreement?
Meno: You added -nal to the word ratio, just as we add -nal to the French work "jour" to create the word journal which mean something that contains words of the "jour" or of today. So we now have a word whichmeans a number made from a ratio. This is more than acceptable to me, Socrates. A sort of lesson in linguistics, perhaps, but certainly not in mathematics. No, I do not see that you have told him how to solve anything about the square root of two, but thank you for asking. I give you your journalistic license to do so.
Socrates: Good. Now boy, I need your attention. Please get up and stretch, if it will help you stay and thing for awhile.
Boy:(stretches only a little)I am fine, Socrates.
Socrates: Now think carefully, boy, what kind of ratios can we make from even numbers and odd numbers?
Boy: We could make even numbers divided by odd numbers, and odd numbers divided by even numbers.
Socrates:Yes, we could.Could we make any other kind?
Boy: Well . . . we could make even numbers divided by even numbers, or odd numbers divided by odd.
Socrates:Very good.Any other kind?
Boy: I'm not sure, I can't think of any, but I might have to think a while to be sure.
Socrates:(to Meno)Are you still satisfied.
Meno: Yes, Socrates. He knows even and odd numbers, and ratios; as do all the school children his age.
Socrates: Very well, boy. You have named four kinds of ratios: Even over odd, odd over even, even over even, odd over odd, and all the ratios make numbers we call rational numbers.
Boy:That's what it looks like, Socrates.
Socrates:Meno, have you anything to contribute here? Meno:No, Socrates, I am fine.
Socrates: Very well. Now, boy, we are off in search of more about the square root of two. We have divided the rational numbers into four groups, odd/even, even/odd, even/even, odd/odd?
Boy:Yes.
Socrates: And if we find another group we can include them. Now, we want to find which one of these groups, if any, contains the numberyou found the other day, the one which squared is two.
Would that be fun to try?
Boy:Yes, Socrates, and also educational.
Socrates: I think we can narrow these four groups down to three, and thus make the search easier. Would you like that?
Boy:Certainly, Socrates.
Socrates:Let's take even over even ratios.What are they? Boy:We know that both parts of the ratio have two in them.
Socrates: Excellent. See, Meno, how well he has learned his lessons in school. His teacher must be proud, for I have taught him nothing of this, have I?
Meno: No, I have not seen you teach it to him, therefore he must have been exposed to it elsewhere.
Socrates: (back to the boy) And what have you learned about ratios of even numbers, boy?
Boy: That both parts can be divided by two, to get the twos out, over and over, until one part becomes odd.
Socrates: Very good.Do all school children know that, Meno? Meno:All the ones who stay away in class.(he stretches)Socrates: So, boy, we can change the parts of the ratios, without changing the real meaning of the ratio itself?
Boy: Yes, Socrates. I will demonstrate, as we do in class. Suppose I use 16 and 8, as we did the other day. If I make a ratio of 16 divided by 8, I can divide both the 16 and the 8 by two and get 8 divided by 4. We can see that 8 divided by four is the same as 16 divided by 8, each one is twice the other, as it should be. We can then divide by two again and get 4 over 2, and again to get 2 over 1. We can't do it again, so we say that this fraction has been reduced as far as it will go, and everything that is true of the other ways of expressing it is true of this.
Socrates: Your demonstration is effective. Can you divide by other numbers than two?
Boy: Yes, Socrates. We can divide by any number which goes as wholes into the parts which make up the ratio.We could have started bydividing by 8 before, but I divided by three times, each time by two, to show you the process, though now I feel ashamed because I realize you are both masters of this, and that I spoke to you in too simple a manner.
Socrates: Better to speak too simply, than in a manner in which part or all of your audience gets lost, like the Sophists.
Boy:I agree, but please stop me if I get too simple.
Socrates: I am sure we can survive a simple explanation. (nudges Meno, who has been gazing elsewhere) But back to your simple proof: we know that a ratio of two even number can be divided until reduced until one or both its parts are odd?
Boy:Yes, Socrates.Then it is a proper ratio.
Socrates: So we can eliminate one of our four groups, the one where even was divided by even, and now we have odd/odd, odd/even and even/odd?
Boy:Yes, 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 question, 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 inwhich the first or top number is twice as large as the second or bottom number.Is this much correct?
