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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
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