Denominator

What it names

$$\frac{3}{\textbf{4}} \quad\longleftarrow\quad \text{the denominator}$$

It is a size, not a count. Four means quarters; eight means eighths. The numerator does the counting.

Bigger denominator, smaller parts

This is the fact that trips people up.

$$\frac{1}{8} < \frac{1}{3}$$

More cuts make thinner pieces, so one of them is less. Sharing a pizza between eight people gives each less than sharing it between three. See comparing fractions.

Why it never changes when you add

$$\frac{2}{5} + \frac{1}{5} = \frac{3}{5}$$

Collecting more fifths does not make the fifths smaller. The denominator is the unit, and units survive addition — the same way two metres plus one metre is three metres, not three square metres.

A denominator of zero

$\tfrac{3}{0}$ has no meaning. Cutting a whole into zero parts is not something you can do, so a denominator is never $0$.

Common denominators

Adding, subtracting and comparing all need the denominators to match first. Finding the smallest one that works is the job of the least common denominator.