What this calculator does
A decimal is what you get from dividing a numerator by a denominator, carried out digit by digit. Some fractions divide evenly and stop, like 1/4 becoming 0.25. Others never stop: 1/3 becomes 0.333... forever, and this calculator marks that repeating block in brackets, 0.(3), rather than leaving it trailing off unclearly.
Whether a fraction terminates or recurs depends entirely on its denominator once fully simplified. A denominator built only from powers of 2 and 5 always terminates, because those are the prime factors of 10. Any other prime factor in the denominator, such as 3 or 7, produces a recurring decimal.
The formula
The calculator performs long division of the numerator by the denominator one digit at a time, keeping track of every remainder it has already seen. Division stops naturally if the remainder reaches zero, which means the decimal terminates. If a remainder repeats, the digits produced since that remainder first appeared form the recurring block, shown in brackets after the non-repeating part.
| Term | Meaning |
|---|---|
| Numerator | The number being divided, on top of the fraction. |
| Denominator | The number being divided by, on the bottom of the fraction. |
| Terminating decimal | A decimal that ends, because the division eventually leaves no remainder. |
| Recurring decimal | A decimal with a block of digits that repeats forever, shown here in brackets. |
The inputs explained
| Field | What to enter |
|---|---|
| Numerator | The top number of the fraction. Can be negative. |
| Denominator | The bottom number of the fraction. Cannot be zero. |
When to use it
Converting a measurement fraction for a calculation
Recipes, timber sizes and machine tolerances are often given as fractions, but most calculators and spreadsheets need a decimal to work with.
Checking whether a fraction terminates before rounding it
Knowing a decimal recurs explains why rounding it always loses information, however many places you show, and why the exact repeating notation is the only fully accurate way to write it.
Spotting the pattern in cyclic fractions
Sevenths are a well-known example: every fraction with denominator 7 produces the same six digits in rotation, just starting at a different point.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What do the sevenths look like as decimals?
Every fraction out of seven produces the same six recurring digits, starting from a different point each time.
| Fraction | Decimal | Type |
|---|---|---|
| 1/7 | 0.(142857) | Recurring decimal, repeating block of 6 digits |
| 2/7 | 0.(285714) | Recurring decimal, repeating block of 6 digits |
| 3/7 | 0.(428571) | Recurring decimal, repeating block of 6 digits |
| 4/7 | 0.(571428) | Recurring decimal, repeating block of 6 digits |
| 5/7 | 0.(714285) | Recurring decimal, repeating block of 6 digits |
| 6/7 | 0.(857142) | Recurring decimal, repeating block of 6 digits |
How do common unit fractions convert to decimals?
A range of common denominators applied to a numerator of 1.
| Fraction | Decimal | Type |
|---|---|---|
| 1/2 | 0.5 | Terminating decimal |
| 1/3 | 0.(3) | Recurring decimal, repeating block of 1 digit |
| 1/4 | 0.25 | Terminating decimal |
| 1/5 | 0.2 | Terminating decimal |
| 1/6 | 0.1(6) | Recurring decimal, repeating block of 1 digit |
| 1/8 | 0.125 | Terminating decimal |
| 1/9 | 0.(1) | Recurring decimal, repeating block of 1 digit |
| 1/11 | 0.(09) | Recurring decimal, repeating block of 2 digits |
Questions
Why do some fractions convert to a decimal that never ends?
It depends on the denominator once the fraction is fully simplified. If its only prime factors are 2 and 5, the decimal terminates, because those are the factors of 10. Any other prime factor, like 3, 7 or 11, produces a decimal that eventually repeats.
What does the number in brackets mean?
It is the block of digits that repeats forever. 0.1(6) means 0.16666..., with the 6 repeating indefinitely, not 0.16 followed by a single extra 6.
How accurate is the rounded decimal value shown?
It is rounded to a small, sensible number of places for quick reading. For an exact value, use the full decimal with the repeating block marked, since any rounded figure necessarily drops information from a recurring decimal.
Can I enter a negative fraction?
Yes. The sign is taken from whichever of the numerator or denominator is negative, and the result is shown with a leading minus sign.
To go the other way, from a decimal to the closest fraction at a chosen precision, see the decimal to fraction calculator. For arithmetic on fractions directly, use the fraction calculator.