Common Number Sets in Mathematics

Symbol	Name	Description	Origin
$\mathbb{N}$	Natural numbers	$1, 2, 3, \dots$ (sometimes includes $0$ )	English Natural
$\mathbb{Z}$	Integers	$\dots, -2, -1, 0, 1, 2, \dots$	German Zahlen
$\mathbb{Q}$	Rational numbers	Numbers that can be written as $\tfrac{p}{q}$ , where $p,q\in\mathbb{Z}$ and $q\neq0$	English Quotient
$\mathbb{R}$	Real numbers	Rational and irrational numbers ( $\pi$ , $\sqrt{2}$ , $\dots$ )	English Real
$\mathbb{C}$	Complex numbers	Numbers of the form $a+bi$ , where $a,b\in\mathbb{R}$ and $i^2=-1$	English Complex

The symbols $\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}$ are written in blackboard bold, also called double-struck notation.

This style developed from the practice of drawing an extra stroke on letters written on a blackboard. Since ordinary bold type was difficult to reproduce by hand, mathematicians used the extra stroke to imitate bold letters and distinguish important sets of numbers.

These symbols are mainly used in set notation. For example:

$1 \in \mathbb{N}$ : $1$ is a natural number.
$\pi \in \mathbb{R}$ : $\pi$ is a real number.

They also follow a strict inclusion relationship:

\mathbb{N} \subsetneq \mathbb{Z} \subsetneq \mathbb{Q} \subsetneq \mathbb{R} \subsetneq \mathbb{C}