Information Technology Reference
In-Depth Information
The symbol ? stands for the sign to assign to the product between
−
(
n
+
k
)
and
−
(
m
+
h
)
. But, in order to get a result which does not depend on
n
,
m
, the sign ? has
got to be
+
, thereby yielding
(
−
k
)(
−
h
)=
kh
.
5.3
Rational and Real Numbers: Approximation and Infinite
Rational numbers are fractions of integers. When we add a “part of the unit” to
a natural number, we get a rational number. For example, 3 plus 1
/
7 is equal to
21
7. All positive fractions can be viewed as the sum of a natural
number, possibly zero, plus a fraction of the unit. Within fractions, with positive or
negative sign, all the four arithmetical operations are always defined (apart from
division by zero, always undefined). Fractions of the unit can be represented in
decimal notation by sums of negative powers of ten. For example, 1
/
7
+
1
/
7, that is 22
/
/
8 is equal to
10
−
1
10
−
2
10
−
3
, that in the usual decimal notation is written as 1
1
×
+
2
×
+
5
×
/
8
=
0
.
125. However, in some cases these sums are infinite. For example, 1
/
6
=
1
×
10
−
1
10
−
2
10
−
3
+
6
×
+
6
×
+
···
, that is, in the usual decimal notation it is written
6, meaning that digit 6 is infinitely repeated. Therefore fractions provide
either finite sequences of decimal digits, separated by a dot (or a comma or some
symbol different from digits) to distinguish the integer part from the fractional part,
or periodical sequences ending with an infinitely repeated subsequence of digits.
It is possible to show that finite and periodical sequences are the only two kinds
of decimal representations associated to fractions. In fact, the decimal notation of
a fraction is obtained by applying the usual division algorithm learned at primary
school (with many variants, but with the same essential procedure). According to
this algorithm, at each step, a difference is calculated between two decimal num-
bers, and then a digit of the greater number is appended to this difference, by pro-
viding a number of at most
n
as 1
/
6
=
0
.
+
1 digits, if
n
is the number of digits of the divisor.
The comparison of this number with the divisor provides a new digit of the result.
But, the possible decimal sequences of
n
1 digits are 10
n
+
1
, therefore surely be-
+
fore than 10
n
+
1
1 times, the same case must occur twice. The digits of the result
between these two occurrences are exactly the digits of the periodical part in the
decimal representation of fractions (finite decimal representations are a special case
of periodical representations with periodical part consisting of zeros).
The above argument raises a natural question. What kind of numbers are infinite
non-periodical decimal sequences of type 0
+
? It is easy to realize that
the four arithmetical operations can be naturally defined for them. Moreover, these
sequences have a direct reading as points of the unitary segment. If a segment is
partitioned into 10 contiguous intervals (for example, including the extreme left
point, but excluding the right extreme point), then each digit can be associated to
each interval. For example, a sequence of type 0
.−−−···
23 individuates the points internal
to the third part (associated to digit 2) of the unitary segment, and then internal to
the fourth part of it (associated to digit 3). In conclusion, infinite non-periodical
sequences correspond to infinite processes generating subintervals of the unitary
segment. These numbers are the
real numbers
. Here are two examples of irrational
.
