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

What the notation means

In this guide 0.(9) means that the entire block 9 repeats forever. It is not the terminating decimal 0.9. The exact fraction is 1. 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.(9). The block has 1 digit, so multiply by 10. Then 10x = 9.(9). The repeating tails now line up exactly.

Subtract x from that equation. The repeating parts cancel, leaving 9x = 9. Divide both sides by 9: x = 9/9. This algebra explains why an all-nine denominator appears.

Reduce and verify the fraction

The greatest common divisor of 9 and 9 is 9. Divide numerator and denominator by that value to obtain 1. The fraction is exact even if a calculator displays only 1.00000000.

Multiply the reduced fraction by 9: (1) × 9 = 9. That returns the subtraction equation. For a repeating block of nines, the exact value is 1; the infinite repeating decimal does not leave a final positive gap.

Distinguish terminating and repeating values

The terminating decimal 0.9 equals 9/10. Its denominator before simplification is 10, not 9. These expressions are different because one stops and the other continues indefinitely.

To round the repeating value to four decimal places, use 1.0000. 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.(9) = 2 + 1 = 3. 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.