Number sets — Rational numbers

Natural numbers

Used for counting.

\mathbb{N} = \{1,\ 2,\ 3,\ 4,\ \ldots\}

Integers

Used for counting forwards and backwards.

\mathbb{Z} = \{\ldots,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ \ldots\}

Rational numbers

Used, among other things, for sharing out. They are all the numbers that can be written as a fraction.

\mathbb{Q} = \left\{ \dfrac{a}{b} \;\middle|\; a, b \in \mathbb{Z},\ b \neq 0 \right\}

Read it as: “the set of fractions $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not $0$”.

Examples

$\dfrac{3}{4}$✓ is a fraction

$\dfrac{5{,}2}{3}$✗ is not a fraction: $5{,}2$ is not an integer.

$\dfrac{0}{3} = 0$✓ is a fraction

$\dfrac{-5}{0}$✗ is not a fraction: the denominator cannot be $0$: you cannot share something into zero parts.

Parts of a fraction

\dfrac{\textcolor{#0891b2}{a}}{\textcolor{#db2777}{b}}

→ Numerator ($a$): the parts we take. → Denominator ($b$): the parts each unit is divided into.

Fractions on the number line

Method

→ The denominator tells you how many equal parts each unit is divided into.

→ The numerator tells you how many of those parts to count from $0$: to the right if it is positive, to the left if it is negative.

Example. To place $\dfrac{7}{3}$, divide each unit into $3$ parts and count $7$ of them from $0$: the point lands between $2$ and $3$.

Try it: change the numerator and the denominator and see where the fraction lands.

Numerator7 Denominator3

Set diagram

Every natural number is an integer, and every integer is a rational number: $\;\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$.

$\mathbb{Q}$

$\mathbb{Z}$

$\mathbb{N}$$2$$5$

$-3$$0$

$\dfrac{7}{3}$ $\dfrac{3}{4}$ $5{,}2$terminating $5{,}\overline{2}$purely recurring $5{,}0\overline{2}$mixed recurring

Rational decimals

→ Terminating decimal: it has a finite number of decimal digits. $5{,}2$

→ Purely recurring decimal: a digit or group of digits (the period) repeats for ever straight after the decimal comma. $5{,}\overline{2} = 5{,}222\ldots$

→ Mixed recurring decimal: before the period there are digits that do not repeat. $5{,}0\overline{2} = 5{,}0222\ldots$

All three can be written as a fraction, which is why they are rational.