A repeating decimal’s fraction is the fraction in lowest terms that, when you divide its numerator by its denominator, reproduces that decimal. This page goes from the endless to the finite: from 0.1666… to a fraction. Here 0.75 is written with a decimal point, and that is what the calculator reads; many countries write 0,75 with a comma instead.
Worked example: 0.1(6)
What fraction does the decimal 0.1666… give, where the 6 repeats forever?
- Call the number x. x = 0.1666…
- Shift the point twice. 10x = 1.666…, with the nonrepeating digit (the 1) already in the whole part, and 100x = 16.666…, with one full repeating block.
- Subtract. 100x − 10x = 16.666… − 1.666…, and the tail 666… disappears: 90x = 15.
- Solve and reduce. x = 15/90, and dividing by 15 leaves 1/6.
Result: 0.16̅ = 1/6.
Check: 10 ÷ 6 = 1 with remainder 4; 40 ÷ 6 = 6 with remainder 4. The remainder 4 comes back, so the division gives 0.1666…
Repeating block, nonrepeating digits, and the three kinds of decimal
The repeating block is the group of digits that repeats forever, and the nonrepeating digits are the decimal digits that come before the repetition. In 0.16̅, the 1 is the nonrepeating digit and the 6 is the repeating block. If the block starts right after the point, the decimal is purely repeating; if there are nonrepeating digits, it is mixed. The whole part, when there is one, does not change this classification.
| Kind | Decimal | Calculation | Fraction |
|---|---|---|---|
| Terminating | 0.25 | 25/100 | 1/4 |
| Pure repeating | 0.7̅ | 7/9 | 7/9 |
| Mixed repeating | 0.083̅ | (83 − 8)/900 | 1/12 |
| Mixed with a whole part | 2.04̅5̅ | (2045 − 20)/990 | 45/22 |
| Long block | 0.1̅4̅2̅8̅5̅7̅ | 142857/999999 | 1/7 |
In the denominator, each repeating digit contributes a 9, and each nonrepeating digit adds a 0 after the nines. In 2.04̅5̅ there is one nonrepeating digit and two repeating ones, so the denominator is 990 and the numerator is 2045 − 20. A terminating decimal, like 0.25, is the case with no repeating block: write it over a power of 10 and reduce. To enter the repeating block in the calculator, use parentheses, as in 2.0(45).
Why the subtraction removes the tail
Multiplying by a power of 10 only moves the point and leaves the digits after it unchanged. If you choose two powers whose difference leaves the same repeating tail in both numbers, subtracting them cancels the endless part and leaves an equation with integers. That is the idea behind the example of 0.16̅.
The nines-and-zeros rule is the shortcut for the same calculation, and it works as a check on the result. The numerator is the number formed by all the digits through the end of the first repeating block, minus the number formed by the digits before the block, with no point. For 0.083̅: (83 − 8)/900 = 75/900 = 1/12. If the denominator does not have a 9 for each repeating digit, something went wrong.
Cases that deserve attention
With a whole part there is no need to split it off: it goes into both numbers being subtracted, as in the 2045 and the 20 of 2.04̅5̅. With a negative sign, the calculator finds the fraction of the unsigned value and applies the sign at the end, so −0.7̅ gives −7/9.
The case of 0.9̅ is surprising: the subtraction gives 9x = 9, so 0.9̅ = 1. Any decimal whose block is 9 equals a terminating one, which is why the calculator never returns a fraction that is merely “almost equal.” If your decimal has one digit more or one digit less, it is a different number. The repeating block is not guessed: 0.1666 could be terminating or continue with any digit, and only the parentheses say which.
Which page to use for your decimal
Each kind of number has its own tool, and it is best to choose the one that changes the least of what you have. The table covers the four cases.
| You have | Example | Use |
|---|---|---|
| Repeating block marked | 0.41(6) | This page |
| Terminating decimal | 0.375 | Decimal to fraction |
| Rounded decimal from a measurement | 0.3333 | Approximate a decimal as a fraction |
| A fraction, to see its decimal | 5/12 | Fraction to decimal |
The result already comes out in lowest terms. If you want to see how a fraction like 75/900 is reduced, the simplify fractions tool shows the GCF. When the result is an improper fraction, the conversion to a mixed number rewrites it, and for practice by hand there are the decimals and percents worksheets. OpenStax presents terminating and repeating decimals as rational numbers (accessed October 17, 2026).
Frequently asked questions
Why does 0.(9) equal 1?
If x = 0.999…, then 10x = 9.999…, and subtracting leaves 9x = 9, so x = 1. Nothing is rounded: no number fits between 0.999… and 1. The guide to terminating and repeating decimals develops this proof and two others.
What happens if I type 0.333 without parentheses?
It is read as a terminating decimal with three digits, which equals 333/1000, not 1/3. If the decimal does not repeat, converting it is a different task: use decimal to fraction. To ask for the endless repeat you have to mark it, as in 0.(3).
Which fractions give a repeating decimal?
Fractions in lowest terms whose denominator has a prime factor other than 2 or 5. That is why 7/30 gives a repeating block while 3/40 stops at 0.075. The guide to fractions with terminating decimals explains the rule with more cases.
Does a decimal like √2 = 1.41421… have a fraction?
No. An endless decimal in which no block repeats is irrational and cannot be written as a quotient of two integers. The same is true of π. A fraction exists only for terminating and repeating decimals.
How do I check the result?
Divide the numerator by the denominator and see whether the original decimal comes back. For 5/12 the division gives 0.41666…, which is 0.41(6). The fraction to decimal tool does that division and marks the repeating block it finds.