Terminating and recurring decimals

Learning outcome

Distinguish terminating decimals from recurring decimals, recognise repetition after an initial nonrepeating part, and classify fractions only after cancellation.

Understanding termination and repetition

A terminating decimal stops after finitely many decimal places. A recurring decimal continues forever with a fixed digit or block repeating, possibly after some initial nonrepeating digits. The dots in our recurring examples mean that the stated pattern continues forever.

For the classification test, write a fraction a/b with integer numerator and denominator, and b nonzero. Use a positive denominator. A reduced fraction has no common positive divisor of numerator and denominator other than 1; reach it by cancelling their common factors.

A fraction has a terminating decimal expansion exactly when its reduced denominator has no prime factors other than 2 and 5. Either prime may be absent. Denominator 1 also qualifies. The reason is that every power of ten contains only these primes, so precisely these denominators can be scaled into a power of ten.

If another prime remains, division does not terminate but eventually repeats. For a fixed denominator, only finitely many remainders are possible. A zero remainder ends division; a repeated nonzero remainder restarts the same sequence of digits. Use a terminating representation when one exists.

Worked examples

Example 1: Classify 18/45.

Divide numerator and denominator by 9: 18/45 = 2/5 = 0.4. The reduced denominator contains only the prime 5, so the decimal terminates. Testing the original denominator would give the wrong conclusion.

Example 2: Examine 7/12.

The reduced denominator is 12 = 2 × 2 × 3. In division, 70 gives digit 5 and remainder 10; 100 gives digit 8 and remainder 4; 40 gives digit 3 and remainder 4 again. Thus, 7/12 = 0.583333..., with only 3 recurring after the initial 58.

Example 3: Examine 5/11.

Division gives 50 ÷ 11: digit 4, remainder 6; then 60 ÷ 11: digit 5, remainder 5. The cycle restarts, so 5/11 = 0.454545..., repeating 45. To two decimal places, 5/11 ≈ 0.45. The symbol ≈ marks an approximation, not exact equality.

Common mistakes

Do not test an unreduced denominator. Repetition need not begin immediately after the decimal point. A displayed finite prefix is not the whole recurring decimal. Not every nonterminating decimal is recurring; fractions always terminate or eventually repeat.

Practice questions

1. Classify 21/84 and find its decimal.
2. Classify 13/40 and find its decimal.
3. Identify the nonrepeating beginning and recurring digit of 11/18.
4. Is 2/3 = 0.667 correct? Rewrite accurately to three decimal places.

Worked answers

1. Cancel 21: 21/84 = 1/4 = 0.25. Since 4 = 2 × 2, it terminates.
2. Since 40 = 2 × 2 × 2 × 5, it terminates. Multiply both numbers by 25: 13/40 = 325/1000 = 0.325.
3. Division gives digit 6 with remainder 2; thereafter 20 ÷ 18 gives digit 1 with remainder 2 repeatedly. Thus, 11/18 = 0.611111...; 6 is nonrepeating and 1 recurs.
4. No. Exactly, 2/3 = 0.666666..., repeating 6. The fourth decimal digit is 6, so round upward: 2/3 ≈ 0.667 to three decimal places.