Numbers can be classified into sets of numbers according to their properties. The table below lists the names, properties of and symbols used for the main number types.
Note: Many numbers are included in more than one set.
Name |
Symbol |
Properties |
Set/Examples |
|---|---|---|---|
Integers |
$\mathbb{Z}$ |
All positive and negative whole numbers. |
$\{...-1, -2 , 0, 1, 2, ...\}$ |
Natural |
$\mathbb{N}$ |
Numbers used for counting (all positive integers). |
$0, 1, 2, ...$ |
Real |
$\mathbb{R}$ |
Includes all numbers on the number line. |
$\frac{1}{5}, \sqrt{\frac{1}{5}}, 0, -2$ |
Rational |
$\mathbb{Q}$ |
All real numbers which can be expressed as a fraction, $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. All integers are rational numbers as 1 is a non-zero integer. |
$\frac{1}{5}, \frac{5}{1} (=5), \frac{2}{3}, \frac{3}{2}, \frac{0}{3} (=0)$ |
Irrational |
$\mathbb{I}$ |
All real numbers which can't be expressed as a fraction whose numerator and denominator are integers (i.e. all real numbers which aren't rational). |
$\pi, \sqrt{2}, \sqrt{3}$ |
Imaginary |
NA |
Numbers which are the product of a real number and the imaginary unit $i$ (where $i=\sqrt{-1}$). |
$3i=\sqrt{-9}, -5i=-\sqrt{-25}, 3\sqrt{2}i=\sqrt{-18}$ |
Complex |
$\mathbb{C}$ |
All numbers which can be expressed in the form $a+bi$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$. Each complex number is a combination of a real number ($a$) and an imaginary number ($bi$). |
$1+2i, 1, i, -3i, 0, -5+i$. |