How it works
A fraction converts to a decimal by long division. When the division never hits a remainder of zero, the digits eventually start repeating forever, shown here in parentheses, e.g. 0.(3) for one third.
decimal = numerator ÷ denominator
The fraction bar is a division sign. That is the whole idea: 3/4 is not a picture of three quarters of something, it is the instruction "divide 3 by 4", and carrying out that division gives 0.75. Every fraction-to-decimal conversion is that one operation.
Long division is how you do it by hand. Because the numerator is smaller than the denominator, you start by writing it as a decimal, so 3 becomes 3.000, then bring down one zero at a time, recording each digit of the answer as you go.
The interesting part is what happens to the remainder. If it reaches zero, the division stops and the decimal terminates. If it never reaches zero, the remainders must eventually repeat, because there are only so many possible remainders below the denominator. Once a remainder repeats, so does every digit that follows it, forever. That is why some fractions give a clean short answer and others run on.
Worked examples
One that repeats, one that stops, and one that repeats in a longer cycle. Watching the remainder in each is the quickest way to see why they behave differently.
- Divide 1 by 3: 10 ÷ 3 = 3 remainder 1.
- The remainder is 1 again, exactly where we started, so the digit 3 repeats forever.
1/3 = 0.3333…, written 0.(3).
- 30 ÷ 8 = 3 remainder 6 → first digit 3.
- 60 ÷ 8 = 7 remainder 4 → next digit 7.
- 40 ÷ 8 = 5 remainder 0 → next digit 5, and the remainder hits zero.
3/8 = 0.375 exactly. The division stops.
- The remainders run 3, 2, 6, 4, 5, then back to 1.
- Six steps before the starting remainder returns, so six digits repeat.
1/7 = 0.(142857), or 0.142857142857… forever.
Which fractions terminate, and which repeat forever
You can tell before dividing, and the rule is short. Put the fraction in lowest terms first, then look only at the denominator.
If the denominator's only prime factors are 2 and 5, the decimal terminates. Any other prime factor and it repeats.
The reason is that our decimal system is built on ten, and 10 = 2 × 5. A denominator made only of 2s and 5s can always be scaled up to a power of ten, so 3/8 becomes 375/1000, and anything over a power of ten is just a decimal written differently. A denominator containing a 3, or a 7, or an 11 can never be scaled to a power of ten, so the division never closes.
| Denominator | Prime factors | Result | Example |
|---|---|---|---|
| 4 | 2 × 2 | Terminates | 1/4 = 0.25 |
| 5 | 5 | Terminates | 2/5 = 0.4 |
| 8 | 2 × 2 × 2 | Terminates | 3/8 = 0.375 |
| 20 | 2 × 2 × 5 | Terminates | 7/20 = 0.35 |
| 3 | 3 | Repeats | 1/3 = 0.(3) |
| 6 | 2 × 3 | Repeats | 1/6 = 0.1(6) |
| 7 | 7 | Repeats | 1/7 = 0.(142857) |
| 12 | 2 × 2 × 3 | Repeats | 1/12 = 0.08(3) |
Lowest terms matters. 6/12 looks like it should repeat, but it simplifies to 1/2 and terminates at 0.5.
Common fractions as decimals
The ones worth recognising on sight, rather than converting every time.
| Fraction | Decimal | Fraction | Decimal |
|---|---|---|---|
| 1/2 | 0.5 | 1/3 | 0.(3) |
| 1/4 | 0.25 | 2/3 | 0.(6) |
| 3/4 | 0.75 | 1/6 | 0.1(6) |
| 1/5 | 0.2 | 5/6 | 0.8(3) |
| 1/8 | 0.125 | 1/7 | 0.(142857) |
| 5/8 | 0.625 | 1/9 | 0.(1) |
| 1/16 | 0.0625 | 1/11 | 0.(09) |
Common questions
What does the parentheses notation mean?
The digits inside the parentheses repeat forever. 0.1(6) means 0.16666…, and 3.(142857) means 3.142857142857142857… continuing indefinitely.
Does "Decimal → Fraction" handle repeating decimals?
No. It converts terminating decimals only (ones that end, like 0.75 or 0.125). Converting a repeating decimal like 0.(3) back to a fraction needs a different algebraic technique, which this calculator doesn't cover.
Why is the fraction always simplified?
0.75 could be written as 75/100, but 3/4 is the same value in lowest terms, and simplifying (dividing both parts by their greatest common divisor) gives the cleanest, most standard form.
How can I tell if a fraction will repeat before I divide it?
Simplify it, then look at the denominator. If its only prime factors are 2 and 5, the decimal terminates. Any other prime factor, a 3 or a 7 or an 11, and it repeats forever.
What is 1/3 as a decimal?
0.3333…, with the 3 repeating forever, written 0.(3). It is not 0.33, and rounding it to 0.33 will make sums that should reach a whole number fall slightly short.
Why does 1/7 repeat six digits when 1/3 repeats only one?
The repeating block is as long as it takes for a remainder to come back around. With 3 the remainder returns immediately, so one digit repeats. With 7 the remainders run 3, 2, 6, 4, 5 before returning to 1: six steps, so six digits repeat.