Information Technology Reference
In-Depth Information
Representations of base 60 and 20 are found in ancient civilizations, the Assyro-
Babilonyan and the Maya, respectively. The latter is a positional representation
proper, for it also had a symbol for zero. This is an essential aspect of positional
representations.
The word
zero
comes from an Arabic root for
zephyr
(a gentle wind, or some-
thing ineffable). Zero is the first great abstraction of modern algebra (the empty
set is its set-theoretic equivalent). It is essential in positional notation and is the
basis for computations with big numbers, a problem that Greek mathematics ad-
dressed with Archimedes'
Arenarius
, but without a complete and definite solution.
However, the informational power of zero and of positional notation relies on the
reduction of numbers to sequences of digits. It seems noteworthy that the etymology
of the word algorithm is rooted in the positional representation. It is used to mean
the computational procedures (on representation of numbers) after al-Khwarizmi's
seminal work, which was translated into Latin (several versions in the 12th and 13th
centuries) and published with title “Algorismi de numero Indorum” (in the 19th
century) [170, 180, 167]. This was the beginning of the algebraic methods for elab-
orating numerical information.
5.2.3
Integer Numbers
The set
to express negative quantities.
An integer number is a pair of a sign and of a natural number providing an absolute
value (usually, the sign is omitted when positive). The extension of the four fun-
damental operations to the integers is straightforward, except for product, which is
obtained by the product of the absolute values equipped with a sign according to
the rules given in Table 5.3, which, at a first glance, may appear somewhat arbitrary.
The rules of product sign are necessary for having an extension of the product op-
eration to integers that is coherent with the product defined on naturals. It is easy to
accept the first two rules of Table 5.3, what about the third rule?
Let us evaluate the product
Z
of integer numbers was obtained from
N
(
−
k
)(
−
h
)
.Wehave
(
−
k
)=[
n
−
(
n
+
k
)]
and
(
−
h
)=
[
m
−
(
m
+
h
)]
for any pair of natural numbers
n
,
m
, therefore:
(
−
k
)(
−
h
)=[
n
−
(
n
+
k
)][
m
−
(
m
+
h
)] =
nm
−
n
(
m
+
h
)
−
m
(
n
+
k
)
?
(
n
+
k
)(
m
+
h
)=
nm
−
nm
−
nh
−
mn
−
mk
?
(
nm
+
nh
+
km
+
hk
)
.
Ta b l e 5 . 3
The sign multiplication rules
+
∗
+=+
+
∗−
=
−
−∗−
=+
