Convert any fraction to a decimal instantly — see terminating vs repeating digits, the repetend highlighted, and the exact value. Free and precise.
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Every fraction a/b is a division problem waiting to happen: 3/4 = 3 ÷ 4 = 0.75. Converting a fraction to a decimal performs that division, revealing whether the decimal terminates (like 0.75) or repeats (like 0.333…). Our free Fraction to Decimal Converter does the long division for you, detects repeating cycles, and marks the repetend clearly.
The terminating-vs-repeating distinction is pure number theory: a fraction in lowest terms terminates exactly when its denominator’s prime factors are only 2s and 5s (the primes dividing 10). Anything else — a factor of 3, 7, 11 — forces a repeating cycle. The converter shows you which case you have and why.
Decimals are the language of measurement, money, and science, while fractions are the language of exact ratios. Converting fluently between them — and knowing when the decimal is exact versus rounded — is one of the most practical skills in arithmetic.
Long division of a by b either ends (remainder hits 0) or cycles (a remainder repeats, and the digit sequence loops forever). There is no third option: with only b possible remainders, either 0 appears or some remainder repeats within b steps. This is the pigeonhole principle guaranteeing the behavior.
The deep rule: in lowest terms, a/b terminates if and only if b = 2^i·5^j — its prime factors are only 2 and 5, the primes dividing 10. Denominators like 8 (2³), 25 (5²), 40 (2³·5) terminate; denominators with 3, 7, 11, or 13 repeat. 1/3 = 0.333… because of the factor 3; 1/7 = 0.142857 repeating because of 7.
The repetend (repeating block) has period at most b−1. For 1/7 the period is 6 (142857); for 1/17 it is 16. Full-reptend primes like 7 produce maximal cycles with beautiful properties — 142857’s rotations (428571, 285714, …) are its multiples, the famous cyclic number.
Notation: repeating digits wear a vinculum (overline): 0.3̅ = 0.333…. The converter displays the repetend overlined and reports its period, so the infinite tail is captured exactly in finite space.
a/b = a ÷ b — long division; terminates ⟺ denominator (reduced) is 2^i·5^j.
Reduce first: 6/15 = 2/5 terminates (denominator 5), even though 15 has a factor of 3.
Long division: 1.000 ÷ 3 gives 0.333 with remainder 1 forever. Result: 0.3̅ (period 1). The denominator 3 forces repetition.
3.00 ÷ 4 = 0.75 exactly, remainder 0. Terminating: denominator 4 = 2² has only prime factor 2.
Reduced: 5/6 (GCD 1). Denominator 6 = 2×3 has a factor 3 → repeats. Division: 0.8333… = 0.83̅ — the 8 is non-repeating, then 3 repeats (period 1 after a 1-digit preperiod).
Denominator 8 = 2³ → terminates. 7.000 ÷ 8 = 0.875 exactly. Three decimal places, remainder 0.
A machinist reads 5/16 inch on a blueprint: 5 ÷ 16 = 0.3125 — terminating, exact. The digital caliper displays 0.3125 with no rounding ambiguity because sixteenths always terminate.
A poll reports 2/3 support: 0.666… — repeating. Quoted as 66.7%, it is rounded; the exact value needs the repetend or the fraction. The converter shows 0.6̅ so the rounding is a conscious choice.
A recipe scales 1/3 cup by 4: as a decimal 0.333… × 4 = 1.333… cups — messy. Keeping the fraction: 4/3 cups = 1⅓ cups exactly. The converter’s repeating flag warns when decimals will smear exactness.
Why do only 2s and 5s permit termination? A terminating decimal 0.d₁d₂…d_k equals the fraction D/10^k. Reducing D/10^k = D/(2^k·5^k), the denominator’s primes can only be 2 and 5 (cancellation removes factors, never adds). Conversely, if b = 2^i·5^j, multiply top and bottom to reach 10^max(i,j): a/b = (a·2^{j−i}·5^{i−j}…)/10^max — a terminating decimal. The correspondence is exact.
For repeating decimals, the period relates to the multiplicative order of 10 modulo b (after removing 2s and 5s): the smallest k with 10^k ≡ 1 (mod b). For b = 7, ord = 6 (period 6); for b = 13, ord = 6; for b = 17, ord = 16. The converter’s cycle detection is long division made visible; the order is the theory behind it.
Every repeating decimal converts back: 0.3̅ = 3/9 = 1/3; 0.142857̅ = 142857/999999 = 1/7. The nines trick (repetend over as many 9s) inverts the division. Fractions and repeating decimals are two names for the same rational numbers — the converter walks the bridge both ways.
The period length of 1/p for prime p (not 2 or 5) divides p−1 (Fermat). For p = 7, 6 divides 6 — full period. For p = 13, the period is 6, dividing 12. Primes where the period is maximal (p−1) are full reptend primes (7, 17, 19, 23, …) — conjectured infinite, unproven. Your calculator’s cycle detection is brushing against an open problem every time it divides by 17.
The long-division tableau is the algorithm behind every conversion: each digit comes from divide–multiply–subtract–bring-down, and the remainders either hit zero or cycle. Doing 1/7 by hand once — watching 142857 emerge — teaches more about decimals than a hundred calculator readouts.
Binary decimals follow the same rules with base 2: 1/10 (decimal) = 0.000110011… repeating in binary, because 10 = 2×5 has the prime factor 5, foreign to base 2. This is why 0.1 + 0.2 ≠ 0.3 in floating point — the converter’s 2s-and-5s theorem generalizes to “primes dividing the base.”
For exact decimal arithmetic (money, science), professionals avoid binary floats and use decimal types or fractions — because 1/3 must stay exact until the final rounding. The converter’s repetend flag is the manual version of that discipline: know what is exact before you round.
Terminating decimals are dense in the reals but the rationals they represent are sparse among decimals: pick a random decimal and it is almost surely irrational (non-repeating, non-terminating). The fractions — terminating plus repeating — are a measure-zero subset. The converter maps that tiny, precious subset exactly.
The 0.999… = 1 identity falls out of the machinery: 0.9̅ = 9/9 = 1, since the repetend 9 over one 9 gives 9/9. Every terminating decimal has a repeating twin (0.75 = 0.74999…), and they name the same number — the converter’s exactness makes this visible rather than paradoxical.
The classic: 0.142857̅, period 6 — maximal for denominator 7. Multiples rotate the digits: 2/7 = 0.285714̅, 3/7 = 0.428571̅.
Denominator 20 = 2²×5 → terminates. 9 ÷ 20 = 0.45 exactly. Twentieths always give two clean decimal places.
Reduced (GCD 1); 12 = 2²×3 → repeats with preperiod. 11 ÷ 12 = 0.91666… = 0.916̅: the 91 is fixed, then 6 repeats.
Convert 5/12 to a decimal by hand (long division), marking the repetend. Then verify with the calculator.
Which terminate: 3/16, 7/24, 9/40, 5/14? Apply the 2s-and-5s rule to the reduced denominators first, then check.
Convert 0.454545… back to a fraction using the nines trick. What fraction is it? Verify by converting forward.
A contractor bills 1/3 of a $10,000 project per milestone: $3,333.33 per the invoice (rounded). Three milestones total $9,999.99 — one cent short of $10,000. The repeating decimal 0.333… forced rounding, and the rounding leaked a cent. The fix: bill the final milestone as $3,333.34, or keep fractions (1/3 exactly) in the ledger.
The accounting software now flags repeating-decimal conversions: 1/3 → 0.3̅ triggers a “repeating: rounding will lose value” warning. The converter’s explicit repetend display is the human version of that flag — it makes the exact-vs-rounded distinction visible before money moves.
The Fraction to Decimal Converter divides numerator by denominator with explicit long-division cycle detection (tracking seen remainders up to 60 digits), renders repeating blocks with an overline (vinculum), reports termination vs repetition with the repetend period, and handles signs and zero denominators correctly.
Divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75. The calculator performs the long division and shows whether digits terminate or repeat.
0.333… repeating, written 0.3̅. The remainder never hits zero because the denominator 3 forces a cycle.
In lowest terms, those whose denominators have only prime factors 2 and 5 (like 8, 25, 40). Any other prime factor (3, 7, 11…) forces repetition.
It marks the repetend — the digit block that repeats forever. 0.142857̅ means 0.142857142857…. It captures the infinite tail exactly.
Reduced denominator 6 = 2×3: the factor 3 forces repetition. Division gives 0.8 then 3 forever: 0.83̅ — one non-repeating digit, then the cycle.
Put the repetend over as many 9s: 0.3̅ = 3/9 = 1/3; 0.142857̅ = 142857/999999 = 1/7.
Yes. 0.9̅ = 9/9 = 1. Every terminating decimal has a repeating twin (0.75 = 0.74999…) — they are the same number.
0.875 exactly. Denominator 8 = 2³, so it terminates after three places with remainder zero.
Cite this page as: Fraction to Decimal Converter. Sigma Calculator. Retrieved October 10, 2026, from https://sigmacalculator.net/math-calculators/fraction-to-decimal-converter/
Last updated: October 10, 2026