Terminating vs Repeating Decimal Calculator

Determine whether a fraction produces a terminating or repeating decimal by analyzing its denominator prime factors.

Enter Fraction

Result

Decimal Value
--
Fraction--
Simplified fraction--
GCD--
Simplified denominator--
Prime factorization of denom.--
Contains only 2s and 5s?--
Other prime factors--

Step-by-Step Solution

Terminating if denominator (in lowest terms) = 2^a * 5^b

Terminating vs Repeating Decimals

A fraction p/q (in lowest terms) produces a terminating decimal if and only if the prime factorization of the denominator q contains no prime factors other than 2 and 5. This is because our decimal system is base 10, and 10 = 2 x 5. If the denominator has any other prime factor, the decimal representation will repeat.

Key Concepts

Terminating Decimal

A decimal that ends after a finite number of digits. Example: 1/8 = 0.125

denom = 2^a x 5^b only

Repeating Decimal

A decimal with a block of digits that repeats forever. Example: 1/3 = 0.333...

denom has primes other than 2, 5

Simplify First

Always reduce the fraction to lowest terms before checking. Example: 6/12 = 1/2 (terminating).

GCD(p, q) to simplify

Mixed Decimals

Some fractions have non-repeating digits followed by a repeating block. Example: 1/6 = 0.1666...

denom = 2^a x 5^b x other

Examples

  • 1/2 = 0.5 (terminating) - denominator is 2 = 2^1
  • 1/4 = 0.25 (terminating) - denominator is 4 = 2^2
  • 1/5 = 0.2 (terminating) - denominator is 5 = 5^1
  • 1/8 = 0.125 (terminating) - denominator is 8 = 2^3
  • 1/3 = 0.333... (repeating) - denominator is 3 (prime factor 3)
  • 1/7 = 0.142857... (repeating) - denominator is 7 (prime factor 7)
  • 1/6 = 0.1666... (repeating) - denominator is 6 = 2 x 3 (has factor 3)

Why Does This Rule Work?

A fraction p/q terminates in decimal if we can write it as n/10^k for some integers n and k. Since 10^k = 2^k x 5^k, this is possible only if q divides some power of 10. That requires q to have no prime factors other than 2 and 5. Any other prime factor in q means no power of 10 is divisible by q, so the decimal must repeat.