0.(81) as a Fraction: Convert a Repeating Decimal Exactly

What the notation means

In this guide 0.(81) means that the entire block 81 repeats forever. It is not the terminating decimal 0.81. The exact fraction is 9/11. Parentheses mark the repeating part only.

A finite screen display can show just a few repetitions. Work with the repeating block itself when you need an exact answer rather than treating the last displayed digit as the end of the number.

Shift by the full block length

Let x = 0.(81). The block has 2 digits, so multiply by 100. Then 100x = 81.(81). The repeating tails now line up exactly.

Subtract x from that equation. The repeating parts cancel, leaving 99x = 81. Divide both sides by 99: x = 81/99. This algebra explains why an all-nine denominator appears.

Reduce and verify the fraction

The greatest common divisor of 81 and 99 is 9. Divide numerator and denominator by that value to obtain 9/11. The fraction is exact even if a calculator displays only 0.81818182.

Multiply the reduced fraction by 99: (9/11) × 99 = 81. That returns the subtraction equation. Keeping the original block length also protects leading zeros in problems such as 0.(03), where the denominator is 99 rather than 9.

Distinguish terminating and repeating values

The terminating decimal 0.81 equals 81/100. Its denominator before simplification is 100, not 99. These expressions are different because one stops and the other continues indefinitely.

To round the repeating value to four decimal places, use 0.8182. Label that result as rounded. Replacing an exact value with a rounded value halfway through a longer calculation can introduce error.

Extend the example

Adding 2 gives 2.(81) = 2 + 9/11 = 31/11. The repeating fractional part is unchanged. First convert the repeating part, then combine it with the whole number.

A decimal with a nonrepeating prefix needs two shifts so the repeating tails align. For example, a prefix before the repeating block changes the subtraction setup. Do not apply the all-nine denominator shortcut to every decimal that happens to repeat.

Check the mathematical definitions

For fractions, percentages and worked arithmetic, see OpenStax Prealgebra. For number definitions, see OpenStax on prime and composite numbers and Wolfram MathWorld on digital roots. Worked examples on this page are calculated for this guide.