Socrates: Now don't let this all go to your head, boy. This is something you could have figured out for yourself, if you had applied your mind to it as you did to squares the other day. Can you do as well, today?
Boy: I should think and hope so, friend Socrates, for I see you are indeed my friend, and I should hope I am more capable today, for having learned some the other day.
Socrates: We shall see, boy. Let us on to the test. Now you remember the squares we dealt with the other day.
Boy:Yes, Socrates.
Socrates: And the one particular square on the diagonal we made, whose area was two, do you remember that one?
Boy:Yes, Socrates.
Socrates: And you remember that the length of the side of a square, when multiplied by itself, yields the area of the square.
Boy:Everyone at school knows that, Socrates.
Socrates: Well, maybe. However, it is about that side, which when multiplied time itself yields an area of two, that I would like to speak further today. How is that with you?
Boy: That is fine, Socrates. I remember that line, and I sort of liked it the best, if you know what I mean.
Socrates: Good, then we should have a great time. Do you know how long that line is, boy?
Boy:Well, I know that you both thought it wise when I said it was ofa length which when made a square of, yielded a square with an area of two, so I suppose I should answer that way.
Socrates: And a good answer it is, too. We are going to make it an even better answer as we proceed.
Boy:Good.
Socrates: Do you remember when you tripped up and fell on your face the other day, when you thought that the square of area nine was actually a square of area eight?
Boy: Oh yes, Socrates! And I am sorely ashamed, because I still do not know enough to make sure I never make such an error again, and therefore I know my virtue and rightness are lacking.
Socrates: They are not lacking so much that they cannot be improved, are they boy?
Boy:I should hope and pray not.
Socrates: Well today, you are going to tell us some things about that number, which when multiplied by itself gives us two.
Boy: I will tell you everything I know, or think I know, Socrates, and hope that I am correct or can be corrected.
Socrates:To Meno, surely he is a fine boy, eh Meno?
Meno: Yes, I am proud to own him, but I don't see how he can be smart enough to do the work today that would take a Pythagorean monk ten years of cloistered life to accomplish.
Socrates: We shall see. Boy, you are doing fine. I think I could even make a scholar of you, though I fear you might turn to wine and women with your new found wealth, if you succeed, rather than continue to polish the wit which should get you that reward.
Boy: I don't think I would want to spend that much time with women or with wine, Socrates.
Socrates: You will find something, no doubt. So, back to the number which when square gives us two. What can we say about such a number? Is it odd or even? Well it would have to be a whole number to be one of those, would it not, and we saw the other day what happens to whole numbers when they are squared?They give us 1,4,9 and 16 assquare areas, did they not?
Boy: Yes, Socrates, though I remember thinking that there should have been a number which would give eight, Socrates?
Socrates: I think we shall find one, if we keep searching. Now, this number, do you remember if it had to be larger or smaller than one?
Boy: Larger, Socrates. For one squared gives only an area of one, and we need and area of two, which is larger.
Socrates:Good.And what of two?
Boy:Two gives a square of four, which is too large.
Socrates: Fine. So the square root of two is smaller than the side two which is the root of four, and larger than the side one which yields one?
Boy:Yes, Socrates.
Socrates: (Turning to Meno) So now he is as far as most of us get in determining the magnitude of the square root of two? And getting farther is largely a matter of guesswork, is it not?
Meno:Yes, Socrates, but I don't see how he will do it.
Socrates: Neither does he. But I do. Watch! (turning to the boy) Now I am going to tell you something you don't know, so Meno will listen very closely to make sure he agrees that I can tell you. You know multiplication, boy?
Boy:I thought I had demonstrated that, Socrates? Socrates:So you have, my boy, has he not Meno? Meno:Yes, Socrates, I recall he did the other day. Socrates:And you know the way to undo multiplication?
Boy: It is called division, but I do not know it as well as multiplication, since we have not studied it as long.
Socrates: Well, I will not ask you to do much division, but rather I will ask you only whether certain answers may be called odd or even, and the like. Does that suit you?
Boy:It suits me well, Socrates.
Socrates:Then you know what odd and even are, boy? Boy:Yes, shall I tell you?
Socrates:Please do.I would love to hear what they teach.
Boy: (the boy recites) A number can only be odd or even if it is a whole number, that is has no parts but only wholes of what it measures. Even numbers are special in that they have only whole twos in them, with no ones left over, while odd numbers always have a one left over when all the twos are taken out.
Socrates: An interesting, and somewhat effective definition. Do you agree, Meno.
Meno:Yes, Socrates.Please continue.
Socrates: Now boy, what do you get when you divide these odd and even numbers by other odd and even numbers.
Boy: Sometimes you get whole numbers, especially when you divide an even number by an even number, but odd numbers sometimes give whole numbers, both odd and even, and sometimes they give numbers which are not whole numbers, but have parts.
Socrates: Very good, and have your teachers ever called these numbers ratios?
Boy: Sometimes, Socrates, but usually only with simple numbers which make one-half, one-third, two-thirds and the like.
Socrates: Yes, that is usually what people mean by ratios. The learned people call numbers made from the ratios, rational. Does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers, whether they be odd or even whole number?
