1. Definitions and types of fractions — Fractions

Definition

Fractional numbers are a set of numbers that arise from the need to share out.

They have the form $\dfrac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

The denominator cannot be $0$, because you cannot share something into zero parts.

\dfrac{\textcolor{#06b6d4}{a}}{\textcolor{#ec4899}{b}}

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

$\dfrac{2}{5}$ The unit is cut into $5$ pieces and we shade $2$.

$\dfrac{4}{10}$ The same unit is cut into $10$ pieces and we shade $4$.

Notice that the shaded portion is the same in both drawings.

By comparing the numerator $N$ with the denominator $D$:

→ If $N < D$ they are proper fractions. They are worth less than one unit.

→ If $N = D$ the fraction is worth exactly one unit.

→ If $N > D$ they are improper fractions. They are worth more than one unit.

$\dfrac{2}{5}$ Proper: $2 < 5$.

$\dfrac{3}{3}$ $3 = 3$: it fills one whole unit.

$\dfrac{5}{3}$ Improper: $5 > 3$, two units are needed.

In every drawing the unit is the same size. What changes is how many pieces we cut it into and how many we shade.

If the fraction is negative, the comparison is made in absolute value: $-\dfrac{5}{3}$ is also improper, because $5 > 3$.