Fraction to Decimal Calculator
Convert a fraction to a decimal and find out whether the decimal stops or repeats — with the repeating block clearly marked.
Calculator
- Long division
- Repeating-cycle detection
- Terminating vs recurring
- Exact decimal form
Result & calculation receipt
- Method
- Long division of numerator by denominator, detecting any repeating cycle
- Formula
decimal = n ÷ d- Substitution
- 1 ÷ 8 = 0.125
- Steps
- Divide 1 by 8.
- The division terminates: 0.125.
- Verification
- 1/8 × 8 = 1, matching the terminating decimal.
- Exact result
- 1/8 = 0.125
- Decimal result
- 0.125
- Engine
- math-solvers v1.4.0
How this calculator works
The calculator performs long division of the numerator by the denominator, tracking each remainder. If a remainder ever repeats, the digits between the two occurrences repeat forever, and that cycle is shown in parentheses. If a remainder reaches zero, the decimal terminates. Either way the exact decimal form is reported, not a rounded approximation.
Worked example
3/8 divides evenly to 0.375, a terminating decimal. 1/3 produces a remainder that never clears, so it is 0.(3) — the 3 repeats. 1/7 cycles through six digits before repeating: 0.(142857). The parentheses mark exactly which digits recur.
Key insight
A fraction in lowest terms terminates only when its denominator’s prime factors are just 2s and 5s; any other prime factor forces a repeating decimal. That is why 1/8 stops but 1/3 and 1/7 go on forever.
Frequently asked questions
How do I convert a fraction to a decimal?
Divide the numerator by the denominator. The calculator does this by long division and reports whether the decimal terminates or repeats.
How does it detect a repeating decimal?
It records each remainder during long division. The moment a remainder repeats, the digits produced since the last time it appeared will repeat forever, so that block is marked with parentheses.
Which fractions give terminating decimals?
A fraction in lowest terms terminates exactly when the only prime factors of its denominator are 2 and 5. Any other prime factor (3, 7, 11, …) produces a repeating decimal.
What does 0.(3) mean?
The parentheses mark the repeating block, so 0.(3) is 0.3333… forever, and 0.1(6) is 0.16666…. It is the same idea as a bar drawn over the repeating digits.
Is the decimal rounded?
No. The exact terminating or repeating form is shown. A separate decimal value is provided for calculations, but the cycle notation is exact.
Methodology & verification
The decimal is produced by long division, recording each remainder. A repeated remainder marks the start of the repeating cycle, which is shown in parentheses; a zero remainder means the decimal terminates. Inputs are validated as integers with a non-zero denominator. The exact terminating or repeating form is reported separately from a rounded decimal value, and the classification is justified by the remainder behaviour.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.