When the result is useful
Exact fraction form is useful when a recurring display such as 0.1666… must enter later algebra without premature rounding.
Math calculator
Repeating Decimal to Fraction Calculator
Convert a decimal with a stated repeating block into an exact reduced fraction. The calculation trail makes the reported exact fraction easier to reproduce.
Repeating Decimal to Fraction inputs
Values for this result
Example from start to finish
For 0.1(6), let x = 0.1666…. Then 100x − 10x = 16.666… − 1.666… = 15, so 90x = 15 and x = 1/6. This Repeating Decimal to Fraction example can be compared with decimal expansion.
Reading the calculation
A repeating decimal is rational because shifting it by a power of ten aligns a second copy of its repeating tail; subtraction then removes the infinite part.
A correct exact fraction is reliable for Repeating Decimal to Fraction only when its setup matches the relationship.
Boundaries and common traps
Enter digits only in the two decimal fields. A repeating block is required; leading zeros inside that block are meaningful. If the Repeating Decimal to Fraction assumptions do not fit, consider reduce the result.
Review the sign, scale, and unit of exact fraction after entering whole-number part, nonrepeating decimal digits and repeating block.
Working through Repeating Decimal to Fraction on paper
Build a numerator by subtracting the nonrepeating prefix number from the number formed by prefix plus one repeat. Use one 9 per repeating digit and one 0 per nonrepeating digit in the denominator, then reduce.
From calculation to usable answer
Source values may arrive in a different order from the form. Map them explicitly to whole-number part, nonrepeating decimal digits and repeating block, normalize units, and retain enough precision for the next step after exact fraction.
Adding another copy of the same repeating block to the written preview does not change the exact fraction. Watching this response separates a data-entry mistake from an unexpected but valid value.
If whole-number part, nonrepeating decimal digits and repeating block are exact counts, more result digits may be meaningful than when they are measured approximations. Let the least certain source guide the final presentation.
A terminating-decimal converter reads a finite value; this page treats the marked block as infinite. This page deliberately reports only the former interpretation.
Label the output as exact fraction in notes or tables. Store whole-number part, nonrepeating decimal digits and repeating block beside it when the result will be reused in a later stage.
Questions about Repeating Decimal to Fraction
Can the repeating block contain zero?
Yes, including leading zeros.
What about 0.999…?
It converts exactly to 1.
Is a terminating decimal repeating?
It can be written with repeating zeros, but the ordinary decimal-to-fraction converter is clearer.
Why are parentheses used in notation?
They mark exactly which digits repeat indefinitely.
Can there be nonrepeating digits first?
Yes; enter them in the separate prefix field.