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numbers obtained by defining infinite sequences of digits that are not periodical. The
first one is an infinite sequence which cannot be periodic because 1 occurs once, 2
twice,
...
, 0 ten times, 10 eleven times, and so on.
.
...
0
1223334444555556666667777777
Another infinite sequence is due to Champernowne, a mathematician of the begin-
ning of the 20th century; it has the digits of all natural numbers arranged after the
dot in their enumeration order (of course this prevents any possibility of a periodical
repetition of the same group of digits).
0
.
1234567891011121314151617
...
5.3.1
Incommensurability, Divisibility, and Distinguishability
The way of introducing irrational numbers by means of infinite decimal sequences is
related to the positional representation of numbers (see [179]). However, it does not
correspond to the historical development of the concept of number. In fact, real num-
bers were discovered by the Greeks, when no positional representation of numbers
was available. The Greek discovery of the necessity of real numbers was entirely
based on geometrical arguments that we want to present briefly.
Consider a square having side of length a .Let d denote the length of its diagonal,
as indicated in Fig. 5.10. If the side and the diagonal lengths are in a rational ratio,
then a
m ). Therefore, the discovery
that this ratio is not a rational number implies that, if c is the length of a portion of
the side such that a
/
d
=
n
/
m ,where n
,
m are natural numbers ( n
<
nc , even assuming the length of c as small as we want, no
natural number m can exist such that m times this portion could equal the diagonal,
that is, d
=
mc . For this reason, we say that the lengths of these two segments are
incommensurable . This argument seems completely counterintuitive, because it is
reasonable that at some, even very small level, a common part that is submultiple of
any two segments must exist. As we show, the mathematical power of this argument
is in its logical nature. Of course, our resolution ability has to stop at some level,
therefore the claim of incommensurability may appear meaningless. However, if we
assume a pure geometrical notion of segment where any two internal points define
a part of it, and any two points can be separated by a point between them, then the
argument holds perfectly. This argument has a pure logical nature; this is the reason
for the name ”logoi” given to numbers defined by such a kind of existential argument
(”logos” means reasoning). However, despite their purely logical existence, these
numbers are fundamental for the development of the whole mathematics.
From a historical viewpoint, we remark that a lot of Greek philosophy was famil-
iar with processes of time and space divisibility. The famous Zeno's paradoxes were
examples where specific processes of infinite divisibility are taken into account. The
deep understanding of “infinite” as a key feature of mathematical entities is surely
one of the most original and fruitful aspects of Greek thought (developed in several
=
 
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