Theorem

When a rational number's decimal expansion terminates

Arithmetic • Number System • Rational Numbers

Statement

A rational number

\frac{p}{q}

in lowest terms has a terminating decimal expansion if and only if

q = 2^{m} 5^{n}

for some integers

m, n \geq 0

; otherwise the expansion is non-terminating and repeating.

How to use it

1Reduce the fraction first, then factorise only the denominator — the numerator is irrelevant.
2Scan the denominator's prime factors: a single prime other than $2$ or $5$ is enough to force a repeating expansion.
3Every rational is either terminating or repeating, so 'not terminating' and 'repeating' are the same verdict here.

Does

\frac{21}{56}

have a terminating decimal expansion?

Step-by-step

1
$\frac{21}{56} = \frac{3}{8}$
Why: Cancel the common factor $7$ before judging the denominator.
2
$8 = 2^{3}$
Why: Only the prime $2$ appears, so the expansion terminates.
3
$\frac{3}{8} = 0.375$

Answer

Yes, it terminates:

0.375

Watch out

The test is invalid before reduction:

\frac{21}{56}

has the prime

7

in its printed denominator yet still terminates, because that

7

cancels.

Builds on

Also known as

terminating decimal, recurring decimal, decimal expansion of a rational, denominator 2^m 5^n