 Chapter 1

Sets and Numbers

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:

• The fractions with integer terms   , etc.)
• The finite decimals ( , , etc.)
• The periodic decimals (  , etc.)
There is a set even wider than the rational numbers, that is the set of real numbers which is denoted with R.
This set contains not only the rational numbers but other numbers as well, who cannot be written as fractions since they are represented by non periodic infinite decimals. The irrationality of was known to the ancient Greeks.

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 .      