The classical sets of numbers

The following notations are standard notations.

Notation 3   The ``natural numbers'', whose set is denotet by $\mathbb{N}$, are

$$\{ 0, 1,2,3,\dots \}.$$

The integers are

$$\{ \dots, -2, -1, 0, 1, 2, 3, \dots \}.$$

The set of all the integers is denoted by $\mathbb{Z}$; the set of all the non-negative integers is equal to $\mathbb{N}$ and is sometimes denoted also by $\mathbb{Z}^+$ and the set of non positive integers is denoted by $\mathbb{Z}^-$. There are also the rational numbers, whose set is denoted by $\mathbb{Q}$ and the set of real numbers, denoted by $\mathbb{R}$.

Actually, these sets are generally assumed to be ``well known'', but most students never saw their construction. We will devote the following sections to these constructions.