Information Technology Reference
In-Depth Information
Iteration and inversion are the two basic mechanisms to define operations on
numbers, starting from the basic operation of successor which generates the infinite
sequence of natural numbers. As we have just seen, sums are iterations of successor,
products are iterations of sum, and powers and exponentials are iterations of prod-
uct. Differences are inverse operations of sums, divisions are inverse operations of
products, and root extractions and logarithms are inverse operations of powers. The
extensions of the number sets:
guarantee that inverse oper-
ations of difference, division, and root extraction are always defined.
It is interesting that the inverse operations can be also obtained by a chain of
iterations, using the operation
pred
, for predecessor, which is the inverse of succes-
sor. In fact, difference can be obtained by iterating predecessor, division by iterating
difference, and logarithm by iterating division.
Usual symbols for numbers came from Magreb. In fact, Leonardo Fibonacci
(1180-1250), who introduced them in Europe in his famous topic
Liber Abaci
,was
son of a merchant from Pisa and studied near Tunis. These digits are compact forms,
drawn by a continuous line, where the number of angles corresponds to the numer-
ical value of each digit (zero is a circle without angles), see Fig. 5.1. Hindu-Arabic
positional number representation had an enormous impact on all the western cul-
tures: it can be compared with the discovery of phonetic alphabets (the birth of ef-
ficient writing systems). Algorithms for numerical elaboration based on positional
representation were the beginning of a mathematical attitude from which modern
algebra stems, with systematic methods for solving equations.
N
⇒
Z
⇒
Q
⇒
R
⇒
C
Fig. 5.1
The angles of Arabic digits
The usual representation of numbers with decimal digits expresses a number, say
357, as a sum of powers of 10: 357
10
0
. It is possible to show
that this representation is univocal for any natural number, provided the leftmost
digit is nonzero. Moreover, the method may be applied with any base greater than
1. For example, the same number, with respect to base 8, becomes 545, because
357
10
2
10
1
=
∗
+
∗
+
∗
3
5
7
8
2
8
1
8
0
357. A more general and compact way for expressing
this kind of notation is given in the next subsection.
=
5
∗
+
4
∗
+
5
∗
=
5.2.2
Sums and Positional Representations
The sum operation can be extended to any sequence of numbers, by using the sum-
mation symbol
:
∑
