Terminating and Recurring Decimals

Key points

• Remainder becomes 0 → the decimal terminates.
• Remainders are always less than the divisor, so if 0 never comes, one must come back → the digits repeat.
• 1/7 = 0.142857 repeating (remainders 3, 2, 6, 4, 5, 1, then 3 again).
• In lowest form, a denominator with only 2s and 5s terminates; any other prime factor repeats.
• Rational ⇔ terminating or recurring; neither ⇒ irrational.

Why decimals stop or repeat

At every step of long division a remainder is left. If it becomes 0, the division stops. Dividing by q, the remainder can only be 0 to q − 1, so if 0 never comes, some remainder must come back, and from then on the digits repeat.

The shortcut

p/q in lowest form terminates ⇔ q = 2ᵐ × 5ⁿ

Recurring decimal to p/q

Let x = 0.2727…; then 100x = 27.2727…, so 99x = 27 and x = 27/99 = 3/11.

Try it yourself

Goal: Divide any fraction and see whether its decimal terminates or repeats, and why.

• Set p and q with the + and − buttons, or pick an example.
• Press Divide and follow the long division step by step.
• Watch the remainders light up on the wheel.
• See whether the remainder reaches 0 (terminates) or comes back (repeats).

Open the simulation full screen

Quick quiz

5 questions. Choose an answer for each, then submit to see your score and the explanations.