Socrates: Friend Meno, how hard do you think it will be for
this boy to prove the irrationality of the square root of two?
Meno: You know that I think it is impossible, Socrates.
Socrates: Well, how long did it take the Pythagoreans?
Meno: I should think it took them years.
Socrates: And how many of them were there?
Meno: Quite a few, though not all worked equally, and some
hardly at all, for they were most interested in triangles of the
right and virtuous variety, and not in squares and their roots.
Socrates: Can you give me an estimate?
Meno: No, I can't say that I can. I am sorry, Socrates.
Socrates: No problem, would you accept five thinkers as an estimate.
Meno: I think that should be fair.
Socrates: And shall we assume they worked for two years, that is the
smallest number which retains the plural, and our assumption was that
they worked for years.
Meno: Two years is indeed acceptable to me, Socrates.
Socrates: Very well then, Meno, it would appear that the
Pythagoreans spent 10 total years of thinking time to solve the
riddles of the square root of two.
Meno: I agree.
Socrates: And would you like to hire the Pythagoreans
to run your household, Meno?
Meno: Surely I would, Socrates, if they were only for hire,
but, as you well know, they are a secret lot, and hire to no one.
Socrates: Well, if I could get you one, perhaps one of the
best of them, in fact the leader of the group that solved the
square root of two, would you not hire him, and at high wages?
Meno: Certainly, Socrates. I'd be a fool not to.
Socrates: And you would put him in charge of your house.
Meno: And all my lands, too, Socrates.
Socrates: Possibly. Then I would like to propose, that if
this boy should solve the proof of the square root of two being
irrational, in the next few hours of our discussion, that he be
given wages equal to those due to your most highly placed servant
for ten years of service, as he shall perform ten years service
for you in the next few hours, should he succeed.
Meno: That sounds quite fair, Socrates, I like your logic.
Socrates: (the boy tugs his tunic, to complain that he wants
his freedom, not a mere bucket of gold) Hush, boy, did you not
promise to hold your tongue?
Boy: Yes, Socrates, but....
Socrates: (turning to Meno) And, of course, with the
monetary rewards for such a position, go all the rest of it.
Meno: Of course, Socrates. I never thought to cheat you.
Socrates: I know that, friend Meno, but I merely ask for the
boy's sake, who is not used to hearing about high finance and
the powers and rank which accompany such things.
Meno: Of course, Socrates. Shall I tell the boy what he
shall receive?
Socrates: You are very kind to do so, my friend Meno.
Meno: (turns to the boy) You are aware that a servant may not own
the amount of gold I would have to give you, should you win the day?
Boy: Yes, sir.
Meno: Therefore, I would have to give to you the freedom to
own the money, before I could give you the money, would I not?
Boy: Yes, sir.
Meno: And in giving you freedom, I would be remiss if I did
not give you a job and a coming out party of equal position
with your wealth, would I not?
Boy: I can't really say, sir, though I suppose so.
Meno: You suppose correctly. I will feed you for a week of
partying, and dress you in the finest garments, while you are
introduced to the finest ladies and gentlemen of Athens, from
whom you are free to select for your interests as friends,
business partners, social acquaintances, connections, and perhaps
even a wife, should you find someone you like for that. Do you
now understand that there is nothing I would leave out that you
would have to ask for, or that if you did have to ask, I would
give it immediately, and ask your forgiveness for my error?
Boy: It is hard to understand, but I take your word.
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 of a 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 as square 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 numbers?
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 word "jour" to create the word journal which means
something that contains words of the "jour" or of today. So we
now have a word which means 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 think 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 number you 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?