Information Technology Reference
In-Depth Information
5.2.1
Natural Numbers
The set
of natural numbers can be constructed by starting from 0,
by applying iteratively the successor function, which always yields a new number:
N=
{
0
,
1
,
2
,
3
,...}
0
→
s
(
0
)
→
s
(
s
(
0
))
→
s
(
s
(
s
(
0
)))
... .
If we represent numbers by sequences of only one symbol, say 1, then zero cor-
responds to symbol 1 and the successor operation becomes the juxtaposition of a
symbol 1 to its argument:
1
→
11
→
111
→
111
→ ... .
The
sum
of two numbers
n
m
can be obtained by iterating the successor operation
n
times, starting from
m
(
s
occurring
n
times):
,
n
+
m
=
s
(
...
s
(
m
)
...
)
.
The
difference
is the inverse operation of the sum, that is,
n
−
m
is the number
k
,if
it exists, such that
m
+
k
=
n
.
The
product
of
n
,
m
is defined by iterating the
m
-sum (sum of
m
)
n
times, starting
from 0:
n
∗
m
= (((
0
+
m
)+
m
)+
...
m
)))
.
n
m
=
The
division
is the inverse operation of the product, that is,
n
.
Sum and product are commutative operations, that is, their result does not depend
on the order of their arguments. Difference and division are not commutative. The
product is distributive with respect to the sum:
a
k
if
m
∗
k
=
∗
(
+
)=
∗
+
∗
c
.
The number
x
raised to
power
of
n
, is defined by iterating the
x
-product
n
times,
starting from 1: 1
b
c
a
b
a
x
; it is denoted by
x
n
. Number
x
is called the
base
,
×
x
×
x
...×
and
n
the
exponent
.
When a base
b
is fixed, then the operation yielding
b
x
in correspondence of
x
is
called
exponential
(of base
b
).
The
root
√
n
m
m
√
n
is the inverse of the
m
-power, that is,
=
n
, while the
logarithm
(of base
b
) is the inverse of the exponential, so that
m
lg
m
k
k
.
Inverse operations are not always defined on natural numbers. For example, dif-
ference and division are not defined when the first number is smaller than the second
one. The extension of naturals to the set
=
of integers guarantees that difference is
always defined. The extension of integers to the set
Z
of rationals (fractions) guaran-
tees that division is always defined for any pair of rationals (apart from any division
by zero, which is always undefined). Square root is not always defined on positive
rationals, as mathematicians of Pythagoras' school discovered (no rational num-
Q
ber can equal
√
2). The extension of rationals to the set
of reals guarantees that
square root is always defined on positive reals. However, square roots (in general,
even roots) of negative reals cannot be real numbers. The extension of reals to the
set
R
C
of complex numbers guarantees that roots are always defined.
