Terminating vs. Repeating Decimals

The two kinds of decimals

A terminating decimal stops. When you do the division, you eventually hit a remainder of $0$ and there is nothing left to bring down: $\dfrac{1}{4} = 0.25$, $\dfrac{7}{8} = 0.875$.

A repeating decimal never stops, but it cycles through the same digits forever: $\dfrac{1}{3} = 0.333\ldots$ and $\dfrac{5}{6} = 0.8333\ldots$. Instead of writing dots, put a bar over the part that repeats: $0.\overline{3}$ and $0.8\overline{3}$.

The denominator rule

Put the fraction in simplest form, then factor the denominator into primes. If the only prime factors are $2$s and $5$s, the decimal terminates. If any other prime shows up — a $3$, a $7$, an $11$ — the decimal repeats forever.

Why $2$ and $5$? Because our decimal system is built on $10$, and $10 = 2 \cdot 5$. A denominator made of $2$s and $5$s can be scaled up to a power of ten, and any fraction with a power-of-ten denominator is just a decimal that stops. Any other prime can never be scaled into a power of ten, so the division never ends.

Simplifying first matters. The fraction $\dfrac{9}{60}$ looks like it should repeat because $60$ has a factor of $3$ — but $\dfrac{9}{60} = \dfrac{3}{20}$, and $20 = 2 \cdot 2 \cdot 5$, so it terminates: $0.15$. Only the simplified denominator tells the truth.

Put the bar in the right place

The bar covers only the digits that actually repeat. $\dfrac{5}{12} = 0.41666\ldots$ — the $4$ and $1$ happen once, and only the $6$ cycles, so you write $0.41\overline{6}$. Writing $0.\overline{416}$ would mean $0.416416416\ldots$, which is a different number.

When you divide by hand, the moment a remainder you've already seen shows up again, the digits are about to repeat — that's your repeating block.

Worked examples

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Common questions