Meno, second part | Page 7

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?
Meno: All the ones who stay awake 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 4 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