200 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSequilateral and oblong. The former class consists of the squares of rational numbers,
for example ø, and the latter are all other positive rational numbers, such as |.
One cannot help wondering why Theodorus stopped at 17 after proving that
the numbers 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, and 15 have no square roots. The
implication is that Theodorus "got stuck" trying to prove this fact for a square of
area 17. What might have caused him to get stuck? Most assuredly the square root
of 17 is irrational, and the proof commonly given nowadays to show the irrationality
of \/3, for example, based on the unique prime factorization of integers, works just
as well for 17 as for any other number. If Theodorus had our proof, he wouldn't
have been stuck doing 17, and he wouldn't have bothered to do so many special
cases, since the proofs are all the same. Therefore, we must assume that he was
using some other method.
An ingenious conjecture as to Theodorus' method was provided by Knorr
(1945-1997) (1975). Knorr suggests that the proof was based on the elementary
distinction between even and odd. To see how such a proof works, suppose that 7
is an equilateral number in the sense mentioned by Theatetus. Then there must
exist two integers such that the square of the first is seven times the square of the
second. We can assume that both integers are odd, since if both are even, we can
divide them both by 2, and it is impossible for one of them to be odd and the other
even. For the fact that the square of one of them equals seven times the square of
the other would imply that an odd integer equals an even integer if this were the
case. But it is well known that the square of an odd integer is always 1 larger than
a multiple of 8. The supposition that the one square is seven times the other then
implies that an integer 1 larger than a multiple of 8 equals an integer 7 larger than
a multiple of 8, which is clearly impossible.
This same argument shows that none of the odd numbers 3, 5, 7, 11, 13, and
15 can be the square of a rational number. With a slight modification it can also
be made to show that none of the numbers 2, 6, 8, 10, 12, and 14 is the square of a
rational number, although no argument is needed in the case of 8 and 12, since it is
already known that \/2 and \/3 are irrational. Notice that the argument fails, as it
must, for 9: A number 9 larger than a multiple of 8 is also 1 larger than a multiple
of 8. However, it also breaks down for 17 and for the same reason: A number 17
larger than a multiple of 8 is also 1 larger than a multiple of 8. Thus, even though
it is true that 17 is not the square of a rational number, the argument just given,
based on what we would call arithmetic modulo 8, cannot be used to prove this
fact. In this way the conjectured method of proof would explain why Theodorus
got stuck at 17.
The Greeks thus found not only that there was no integer whose square is, say,
11 (which is a simple matter of ruling out the few possible candidates), but also
that there was not even any rational number having this property; that is, 11 is
not the square of anything they recognized as a number.
The geometric origin of irrationals: incommensurable magnitudes. A second, "geo-
metric" theory of the origin of irrational numbers comes from geometry and seems
less plausible. If we apply the Euclidean algorithm to the side and diagonal of
the regular pentagon in Fig. 1, we find that the pair AD and CD get replaced by
lines equal to CD and CF, which are the diagonal and side of a smaller penta-
gon. Thus, no matter how many times we apply the procedure of the Euclidean
algorithm, the result will always be a pair consisting of the side and diagonal of a