Decimal to Fraction Converter
Enter any decimal and instantly see it as a simplified fraction (in lowest terms) plus a mixed number. You can also flag the last N fractional digits as a repeating block, so recurring decimals like 0.142857… = 1/7 come back as exact rationals — not just floating-point approximations.
Please enter a valid decimal (e.g. 0.75 or -2.5).
Simplified fraction
3/4
Mixed number
3/4
Decimal check
0.75
Exact — terminates after the given digits.
Examples
Repeating decimal — only the first 12 digits are shown as a sanity check. For instance 1/7 = 0.142857142857… never terminates.
Formula
Terminating: n / 10^k. Repeating: (NR − N) / (10^(|N|+|R|) − 10^|N|), then divide both sides by gcd(num, den) to reach lowest terms.
- · A terminating decimal becomes n / 10^k, then simplified — e.g. 0.75 = 75/100 = 3/4.
- · Repeating decimals use the classic 9999… denominator trick: 0.6̄ = 6/9 = 2/3; 0.16̄ = (16 − 1) / 90 = 1/6.
- · The whole pipeline runs on BigInt so precision is exact — there is no float drift even for very long inputs.
- · Every result is reduced via gcd (Euclidean algorithm); the denominator is always positive and the sign sits on the numerator.
- · Mixed numbers follow the schoolbook convention: −5/2 = −2 1/2 (whole and fractional parts share the sign).
- · Inputs accept a leading minus, thousands separators, leading dot (".5") and trailing dot ("5.").
Frequently asked
How do I know if a decimal is "repeating" — and how many digits do I need to type?
Quick rule: if the denominator (after simplification) has only 2 and 5 as prime factors (4, 8, 25, 100, …) the decimal terminates; any other prime factor (3, 7, 11, 13, …) forces an infinite cycle. So 1/3 = 0.333… and 1/7 = 0.142857142857…. You only need to type one complete period and then set "repeating digits" to the length of that period — six for 0.142857 (= 1/7), one for 0.16 (= 1/6, the "6" repeats).
Is the decimal-to-fraction answer unique? Do 0.3 and 0.333… give the same fraction?
No — those are two different inputs. 0.3 is a terminating decimal exactly equal to 3/10; 0.3 is also an approximation of 0.333… but they are different real numbers. Type 0.3 with repeat = 0 and you get 3/10; type 0.3 with repeat = 1 (i.e. 0.333…) and you get 1/3. The simplified fraction itself is unique (every rational has one lowest-terms form), but the same printed decimal digits can stand for two different rationals.
Why does 0.9 repeating equal 1 exactly? Does typing 0.9 with repeat = 1 really give 1?
Yes. 0.999… = 9/9 = 1 is a classic identity, provable from infinite geometric series, the fraction-conversion trick, or the ε–δ definition of a limit. It is not an approximation — two different decimal expansions both denote the same real number, just like 3/4 = 6/8. The calculator uses the repeating formula (9 − 0) / (10 − 1) = 9/9 = 1, so typing 0.9 with repeat = 1 really does return 1/1 = 1.
Can I use this for the reverse direction (fraction → decimal)?
The tool itself is one-way (decimal → fraction), but the "decimal check" field converts the simplified fraction back to a decimal so you effectively see both directions side by side. Terminating fractions like 3/4 = 0.75 print exactly; repeating ones like 1/7 = 0.142857142857… print to 12-digit precision with an ellipsis.
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