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Recurring Decimal — Definition, Formula & Examples

A recurring decimal is a decimal number that has a digit or group of digits that repeats forever. For example, 0.333...0.333... and 0.142857142857...0.142857142857... are recurring decimals.

A recurring (or repeating) decimal is a decimal representation of a rational number in which, after some point, a finite sequence of digits (called the repetend) repeats indefinitely. It is denoted by placing a bar (vinculum) over the repeating block, such as 0.3‾0.\overline{3} for 0.333...0.333...

How It Works

Recurring decimals appear whenever you divide a number and the division never terminates. To identify the repeating part, carry out long division until you see the same remainder appear again — the digits between those two points form the repeating block. You write the result using a bar over the repeating digits: 0.1666...=0.16‾0.1666... = 0.1\overline{6} and 0.272727...=0.27‾0.272727... = 0.\overline{27}. Every recurring decimal can be converted back into a fraction, which confirms it represents a rational number.

Worked Example

Problem: Convert the recurring decimal 0.36‾0.\overline{36} into a fraction.
Step 1: Let x equal the recurring decimal.
x=0.363636...x = 0.363636...
Step 2: Since two digits repeat, multiply both sides by 100 to shift the decimal point past one full repeating block.
100x=36.363636...100x = 36.363636...
Step 3: Subtract the original equation from the new one to eliminate the repeating part.
100x−x=36.3636...−0.3636...  ⟹  99x=36100x - x = 36.3636... - 0.3636... \implies 99x = 36
Step 4: Solve for x and simplify the fraction.
x=3699=411x = \frac{36}{99} = \frac{4}{11}
Answer: 0.36‾=4110.\overline{36} = \dfrac{4}{11}

Why It Matters

Recurring decimals show up whenever you convert common fractions like 13\frac{1}{3}, 27\frac{2}{7}, or 56\frac{5}{6} into decimal form. Understanding them helps you move fluently between fractions and decimals in algebra, measurement, and science calculations.

Common Mistakes

Mistake: Rounding a recurring decimal and treating it as exact (e.g., writing 13=0.33\frac{1}{3} = 0.33).
Correction: A rounded value is only an approximation. Use the bar notation 0.3‾0.\overline{3} or keep the fraction 13\frac{1}{3} when an exact value is needed.