Terminating Decimal — Definition, Formula & Examples
A terminating decimal is a decimal number that has a finite number of digits after the decimal point — it eventually stops. For example, 0.75 and 3.125 are terminating decimals, while 0.333... is not.
A decimal representation of a rational number is terminating if and only if, when the corresponding fraction is written in lowest terms, the denominator has no prime factors other than 2 and 5.
How It Works
To determine whether a fraction produces a terminating decimal, reduce it to lowest terms and then examine the denominator. Factor the denominator completely into primes. If the only prime factors are 2 and 5 (or a combination of both), the decimal terminates. If any other prime factor appears, the decimal will repeat instead. For instance, has a denominator of , so it terminates. But has a denominator of , and the factor of 3 means it repeats.
Worked Example
Problem: Does the fraction 3/16 produce a terminating decimal? If so, find it.
Factor the denominator: The fraction is already in lowest terms. Factor 16 into primes.
Check for termination: The only prime factor of 16 is 2, so the decimal will terminate.
Divide to find the decimal: Divide 3 by 16.
Answer: Yes, 3/16 is a terminating decimal equal to 0.1875.
Why It Matters
Recognizing terminating decimals helps you convert between fractions and decimals quickly, which comes up constantly in pre-algebra and algebra. It also matters in science and engineering, where measurements need exact decimal values rather than repeating approximations.
Common Mistakes
Mistake: Checking the denominator for factors of 2 and 5 without first reducing the fraction to lowest terms.
Correction: Always simplify first. For example, 6/12 simplifies to 1/2. The denominator 12 contains a factor of 3, but 2 does not — so the decimal terminates (0.5).