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.
Classifying numbers can be useful when solving equations: it is much easier to solve an equation when we know which type of number (real, a whole number, imaginary...) the solution will be.
Another purpose of classifying numbers into sets is so that we can talk about general rules for numbers that apply to some categories, but not to others. For example, the numerator and denominator of an arithmetic fraction must both be integers.
Note: Many numbers are included in more than one set.
Name |
Symbol |
Properties |
Set/Examples |
|---|---|---|---|
Integers |
$\mathbb{Z}$ |
All positive and negative whole numbers. |
$\{...-1, -2 , 0, 1, 2, ...\}$ |
Real |
$\mathbb{R}$ |
Includes all numbers on the number line. |
$\frac{1}{5}, \sqrt{\frac{1}{5}}, 0, -2$ |
Natural |
$\mathbb{N}$ |
Numbers used for counting (all positive integers). |
$\{0, 1, 2, ...\}$ |
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 ($bi$). |
$1+2i, 1, i, -3i, 0, -5+i$. |