Information Technology Reference
In-Depth Information
Very often, different names for indicating variables relate to different levels of
variability, for example:
parameter, indeterminate, unknown
.
The essence of the notion of variable is that of an entity taking values within a
range
of variability, which can be specified by a set. The simplest kind of variables
are
boolean variables
, ranging over a set of two values (usually denoted by 0
,
1). An
important issue concerning variables, attains to the relationship between variables.
For example, let
X
and
Y
represent the measures of two quantities related to some
bacterial cells (say, the quantity of a nutrient in the environment and the quantity
of a protein inside the cell, respectively). Let us assume we discover that, in certain
conditions, the value of
Y
depends on the value of
X
. In this case we say that
Y
is a dependent variable with respect to
X
, and this dependence defines a function
f
:
A
→
B
, from the range
A
of
X
to the range
B
of
Y
:forany
a
∈
A
,
f
(
a
)=
b
when
b
is the value taken by
Y
in correspondence to the value
a
taken by
X
.
5.2
Numbers and Digits
Numbers appear as quantity ratios when measures are considered. Natural numbers
measure sizes of populations; rational numbers measure, with approximations, the
sizes of continuous quantities.
In counting populations, a relevant aspect is the distinguishability of elements. In
fact, the possibility of counting many objects does not imply the ability of recogniz-
ing them as single objects distinguishable from each other. This situation is typical
with molecules. One may be able to find 1000 molecules of a protein, but one can
seldom recognize each of them from another of the same group. If we measure pop-
ulations by using a standard population size of 1000 elements, say it a Kilo
K
(of
elements), as a unit, then we obtain fractions. For example, the size of a population
of 3200 elements is 3
K
+2/10
K
, that is, 3.2
K
.
Greeks had different notions of numbers. Natural numbers were “arithmoi”, frac-
tions were “megethe”, connected to the geometrical intuition of segments. The four
classical operations on numbers were defined in geometrical terms (by compass and
ruler constructions over segments). “Logoi” were related to the incommensurable
ratios (e. g., between the edge and the diagonal of a square) and to the notion of
infinite.
A definite, complete understanding of number systems was developed in the 19th
century, at the end of a long process of unification of the distinct Greek notions and
of the geometric Greek perspective with the algebraic notion of number [190]. This
process evolved along the new algorithmic perspective of the positional notation,
of Hindu-Arabic origins, that was introduced by Leonardo Fibonacci in his
Liber
Abaci
(1202), whereby numbers are represented by sequences of
digits
.
Sequences over an alphabet can always represent numbers. In fact, as we will
explain later on, if the pairwise distinct elements which may occur in a sequence are
fewer than
k
, then a sequence may be taken as a base-
k
representation of a number.
