What this tool does
Turning a fraction into a decimal simply means dividing the numerator by the denominator. Sometimes the division ends cleanly and you get a terminating decimal, like 3/4 = 0.75. Other times it runs forever in a repeating pattern, like 1/3 = 0.333…. Going the other way — decimal to fraction — means writing the number over a power of ten and reducing it.
This calculator handles both directions. In Fraction → decimal mode you type a numerator and a denominator and get the decimal back; when it repeats, the tool finds the recurring block and shows it in parentheses, so 1/3 becomes 0.(3) and 1/6 becomes 0.1(6). In Decimal → fraction mode you type a finite decimal and get the fraction already reduced with the greatest common divisor (GCD). Everything runs in your browser — nothing is uploaded.
How to use it
- Pick a mode with the two tabs at the top.
- In Fraction → decimal, enter the numerator and denominator. The result updates instantly and tells you whether the decimal terminates or repeats.
- In Decimal → fraction, enter a value like
0.75. You get the reduced fraction, plus the un-reduced form it came from underneath. - Hit Copy to grab the result.
If the denominator is zero you will see a notice: dividing by zero is undefined.
The method
Fraction to decimal. This is long division. You divide, take the remainder, multiply it by ten, and divide again. If a remainder ever hits 0, the decimal terminates. If a remainder repeats, the digits from that point on cycle forever — that cycle is the repetend. The length of the repeating block is never larger than the denominator minus one.
Decimal to fraction. A decimal with k digits after the point is the number without the dot over 10 to the power k. For example 0.75 = 75/100. Then you reduce by dividing top and bottom by their GCD, found with the Euclidean algorithm: GCD of 75 and 100 is 25, so 75/100 = 3/4.
A quick test for whether a fraction terminates: reduce it and look at the denominator. If its only prime factors are 2 and 5, the decimal terminates; if any other prime shows up (3, 7, 11…), it repeats.
| Fraction | Decimal | Type |
|---|---|---|
| 1/2 | 0.5 | terminating |
| 1/4 | 0.25 | terminating |
| 3/4 | 0.75 | terminating |
| 1/5 | 0.2 | terminating |
| 1/8 | 0.125 | terminating |
| 5/8 | 0.625 | terminating |
| 1/3 | 0.(3) | repeating |
| 2/3 | 0.(6) | repeating |
| 1/6 | 0.1(6) | repeating |
| 1/9 | 0.(1) | repeating |
Worked example
Convert 3/4 to a decimal with long division:
- 3 divided by 4 is 0, remainder 3. Whole part: 0.
- Bring down a zero: 30 divided by 4 is 7 (7×4 = 28), remainder 2. First decimal: 7.
- Bring down another zero: 20 divided by 4 is 5, remainder 0. Second decimal: 5.
The remainder reached 0, so the decimal terminates: 3/4 = 0.75.
Now the reverse, 0.75 to a fraction: there are two decimal places, so 0.75 = 75/100. The GCD of 75 and 100 is 25. Divide both: 75 ÷ 25 = 3 and 100 ÷ 25 = 4. Result: 0.75 = 3/4. As a repeating check, 1/3 never closes the division (remainder 1 keeps coming back), which is why it is 0.(3).
Frequently asked questions
What do the parentheses in 0.(3) mean?
They mark the repetend — the block of digits that repeats without end. 0.(3) is 0.3333… and 0.1(6) is 0.16666…, where only the 6 repeats. It is the exact, compact way to write a repeating decimal, more precise than rounding to a handful of places.
How do I know if a fraction terminates or repeats?
Reduce the fraction and factor the denominator. If it contains only 2 and 5 as prime factors, the decimal terminates. If any other prime appears, it repeats. That is why 1/8 (denominator 8 = 2×2×2) terminates while 1/6 (6 = 2×3) repeats.
Can I convert a repeating decimal to a fraction here?
The Decimal → fraction mode is built for finite decimals. A repeating value like 0.333… needs its period identified first by hand; its exact fraction is 1/3. For those, work the Fraction → decimal mode in reverse: try simple fractions until you recognize the pattern.
Does the tool always reduce the fraction?
Yes. Decimal → fraction returns the fraction in lowest terms using the GCD, and it also shows the un-reduced version so you can see where it came from. When the result is a whole number, it says so instead of leaving a denominator of 1.