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directions by Eudoxus of Cnidus, Euclid of Alexandria, and Archimedes of Syra-
cuse).
The following argument is a modern elaboration of a proof of the incommensura-
bility between the side and the diagonal of a square, developed by the Pythagorean
school (6th century B.C.).
Fig. 5.2
The incommensurability between the side and the diagonal of a square
A proof of geometrical incommensurability
. Consider a unitary square (with
side of length one). According to Pythagoras' theorem the length of its diagonal is
√
2, therefore if the ratio between the diagonal and the side were a rational number,
then two natural numbers
p
q
should exist such that
√
2
q
. We show that this
hypothesis implies a contradiction. In fact, without loss of generality we can assume
,
=
p
/
that no common factor exists between
p
and
q
.If
√
2
p
2
q
2
,that
=
p
/
q
,then2
=
/
is,
p
2
2
q
2
. The last equation implies that
p
2
=
has number 2 as a factor, but this
2
r
for some
r
,thatis,4
r
2
2
q
2
,
requires that also
p
has 2 as a factor. Therefore
p
=
=
thus 2
r
2
q
2
. But, by the same argument used earlier, this fact implies that
q
has 2
as a factor. In conclusion, both
p
and
q
have 2 as a factor, and this contradicts our
initial assumption that no common factor exists between
p
and
q
. Whence, the initial
=
hypothesis
√
2
q
implies a contradiction, thus proving the
irrationality
of
√
2,
=
p
/
because
√
2
=
/
,
q
. In conclusion, this proof
shows the incommensurability between the side and the diagonal of any square. In
an analogous way it can be shown that
√
p
is irrational for any prime number
p
.
Real numbers completely express the continuous nature of the straight line and
of segments. If we consider a segment as a set of points arranged in a linear struc-
ture, then we realize that the elements of this set include points that we cannot
singularly individuate. This is a subtle argument implied by the analysis developed
by Georg Cantor in the 19th century. We do not enter in the details of this discus-
sion, we only want to remark that a continuous structure implies infinite divisibility
and undistinguishability of its components. This aspect is a crucial discrimination
p
q
for any pair of natural numbers
p