Lecture · 14 min · basic

Fractions as operators

A ratio is a stretch of the unit

Euler, Elements of Algebra · §§70–90

The slash in $\frac{3}{4}$ is not furniture. It is an operation: take the unit, cut it into four equal parts, take three of them. Once a fraction is a stretch, multiplication of fractions is composition of stretches — not a ritual of cross-canceling.

On the board today

—Read $\frac{a}{b}$ as “ $a$ copies of a $b$ -th of the unit.”
—Multiply fractions by composing stretches.
—Add fractions only after they share a unit of subdivision.

Product of stretches

\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}

Do $c / d$ first, then $a / b$ . The unit is cut into $b d$ parts.

Two fractions can be added only when they name the same kind of piece. $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$ — those are halves and thirds, different coins. Rewrite both in sixths: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ . The common denominator is a common unit of length.

Worked example · A stretch then a stretch

Compute $\frac{2}{3} \times \frac{5}{4}$ by thinking in pieces of a rectangle.

1.Cut the unit into $3 \times 4 = 12$ equal cells.
2.Taking $2/3$ then $5/4$ keeps $2 \cdot 5 = 10$ of those $12$ cells.
$\frac{10}{12} = \frac{5}{6}$

$\frac{2}{3} \times \frac{5}{4} = \frac{5}{6}$ .

Pause the lecture

Why is $\frac{1}{2} + \frac{1}{3} \neq = \frac{2}{5}$ ? Give a one-sentence geometric reason.

Common mistake. “Cancel the $2$ s” in $\frac{2 + x}{2}$ is illegal — $2$ is not a factor of the whole numerator. Cancel only common factors of the entire top and entire bottom.

Practice this lecture

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