2. Representation on the real line and ordering — Fractions

Representing fractions on the real line

→ The denominator says how many equal parts we cut each gap between two integers into.

→ The numerator says how many of those parts we count from $0$.

Example. Let us represent $\dfrac{7}{3}$: we cut each unit into $3$ pieces and count $7$ of them from $0$.

Each gap between two integers is cut into 3 parts, because the denominator is $3$. Counting $7$ thirds from $0$ lands between $2$ and $3$.

$<$ means less than.

$>$ means greater than.

Ordering means putting the fractions from largest to smallest, or from smallest to largest.

→ Work out the least common multiple of every denominator.

→ Scale the fractions up so that they all share that denominator.

→ Order them by their numerators.

\dfrac{2}{3}, \quad \dfrac{1}{2}, \quad \dfrac{3}{5}, \quad \dfrac{8}{15}

Least common multiple of the denominators. Since $3$ and $5$ already divide $15$, only $2$ and $15$ matter:

\operatorname{lcm}(3, 2, 5, 15) = \operatorname{lcm}(2, 15) = 30

We scale every fraction up to denominator $30$:

\dfrac{20}{30}, \quad \dfrac{15}{30}, \quad \dfrac{18}{30}, \quad \dfrac{16}{30}

Now only the numerators matter: $15 < 16 < 18 < 20$.

\dfrac{1}{2} < \dfrac{8}{15} < \dfrac{3}{5} < \dfrac{2}{3}