Information Technology Reference
In-Depth Information
n
i
=
1
a
i
standing for
a
1
+
a
2
,
+
...
+
a
n
. The extreme values can be indicated in many ways,
for example:
∑
a
i
i
=
1
,
n
or similar, clearly understandable, notations. The following equation expresses the
appearance
of the sum variable (its identity is not relevant).
n
i
=
1
a
i
=
n
j
=
1
a
j
other properties of
∑
follow from the properties of sum operation:
n
i
=
1
a
i
=
k
i
=
1
a
i
+
n
∑
a
i
i
=
k
+
1
n
i
=
1
(
a
i
+
b
i
)=
n
i
=
1
a
i
+
n
i
=
1
b
i
n
i
=
1
ba
i
=
b
n
i
=
1
a
i
j
=
1
b
j
n
i
=
1
a
i
i
=
1
a
i
m
j
=
1
b
j
m
n
n
i
=
1
m
j
=
1
a
i
b
j
=
m
j
=
1
n
i
=
1
a
i
b
j
.
=
=
Given a natural number
b
1, called
base
, for any natural number
n
, there exists a
natural
k
such that the following univocal representation of base
b
holds:
>
=
∑
i
=
0
,
k
c
i
b
i
n
where
c
i
<
(usually writ-
ten in the reverse order) univocally identifies, with respect to base
b
, the natural
number
n
.When
b
different symbols, called
digits
, are chosen to represent the num-
bers smaller than
b
, then any sequence of digits, with
c
k
=
b
for
i
≤
k
and
c
k
>
0. Therefore, sequence
(
c
0
,
c
1
,...,
c
k
)
c
0
, is a representation of
a natural number.
All classical algorithms to compute the four arithmetical operations can be gen-
eralized to any base, when the tables for products and carries are given for any pair
of values smaller than
b
(Pythagorean tables).
Other positional representations different from the decimal one are: the binary
representation (introduced by Leibniz), the octal and hexadecimal representations
(basis 8 and 16 respectively), which, besides the binary representation, are mostly
used in computer representations of numbers.
