The sets of numbers
Natural numbers
$\mathbb{N}=\{1,2,3,\ldots\}$. They are used for counting.
Integers
$\mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$. They are used for counting and taking away.
$\mathbb{Z}=\mathbb{N}^{-}\cup\{0\}\cup\mathbb{N}^{+}$
Rational numbers
$\mathbb{Q}=\left\{\dfrac{a}{b}\ \middle|\ a,b\in\mathbb{Z},\ b\neq 0\right\}$. They are used for sharing out.
$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}$. The rationals are the naturals, the integers and the decimal numbers:
- exact: $2{,}81$
- pure repeating: $5{,}\overline{3}$
- mixed repeating: $5{,}2\overline{15}$
Irrational numbers
$\mathbb{I}=\{\text{cannot be written as a fraction}\}$.
Examples: $\pi,\ e,\ \sqrt{2},\ \varphi,\ 1{,}234567\ldots,\ 1{,}010010001\ldots$
$\mathbb{Q}$ and $\mathbb{I}$ are disjoint sets.
Real numbers
$\mathbb{R}=\mathbb{Q}\cup\mathbb{I}$
Where does each number live?
Pick a number and the diagram lights up all the sets it belongs to. A few extra numbers are included to test the tricky cases (such as $\sqrt{4}$).
Represent $\sqrt{5}$
An irrational such as $\sqrt{5}$ has an exact place on the line: just draw a right triangle using Pythagoras. In class: legs $2$ and $1$, hypotenuse $\sqrt{5}$, carried to the line with a compass. Here you can try it with other roots.