The simplest numbers are the natural
numbers:
0, 1, 2, 3, ...
The set of natural numbers is denoted
with N (N stands for natural)
If we append the opposite numbers of the
members of N, the set of natural numbers is extended in a wider set,
the integers:
..., -3, -2, -1, 0, 1, 2, 3, ...
The set of integers is denoted with Z
by the German word Zahl that means number.
A wider set than the set of integers is
the set of rational numbers which
is obtained by taking
quotients
of integers, where n is not equal to zero.
The set of rational numbers
is denoted by Q
(from the first letter of the word quotient).
Q consists of all
the fractions with integer terms. So, rational numbers are:
In 1760 the mathematicians Lambert proved that the number is irrational. Another proof for that is included in the book Calculus by M. Spivak.[see bibliography section for more details]
It is obvious that .