Repeating Decimal — Definition, Formula & Examples
A repeating decimal is a decimal number in which a digit or group of digits after the decimal point repeats forever in a pattern. For example, and are both repeating decimals.
A repeating decimal (also called a recurring decimal) is the decimal representation of a rational number in which, after some initial sequence of digits, a finite block of one or more digits — called the repetend — cycles indefinitely. Every fraction where has prime factors other than 2 and 5 produces a repeating decimal.
Key Formula
Where:
- = The repeating decimal value
- = The repeating block of digits (the repetend)
- = The number of digits in the repeating block
How It Works
When you divide the numerator of a fraction by its denominator using long division, the remainder at each step determines the next digit. If the denominator has prime factors other than 2 and 5, the remainders eventually repeat, which forces the digits to repeat in a cycle. You write a repeating decimal by placing a bar (called a vinculum) over the repeating block: means , and means . To convert a repeating decimal back to a fraction, you use algebra to eliminate the repeating part.
Worked Example
Problem: Convert 0.272727... to a fraction.
Step 1: Let x equal the repeating decimal.
Step 2: The repeating block "27" has 2 digits, so multiply both sides by 100 to shift the decimal point past one full cycle.
Step 3: Subtract the original equation from this new equation. The repeating parts cancel out.
Step 4: Solve for x and simplify the fraction.
Answer:
Another Example
Problem: Convert the fraction 5/6 to a decimal.
Step 1: Divide 5 by 6 using long division. 6 goes into 50 eight times with remainder 2.
Step 2: Bring down a zero. 6 goes into 20 three times with remainder 2 again.
Step 3: The remainder 2 has appeared before, so the digit 3 will repeat forever.
Answer:
Why It Matters
Repeating decimals show up constantly in pre-algebra and algebra courses whenever you convert fractions to decimals or work with rational numbers. Understanding them helps you see that every fraction is a rational number — and that irrational numbers like never settle into a repeating pattern. In real life, fields like accounting and engineering require knowing when a decimal is exact versus when it repeats so you can round appropriately.
Common Mistakes
Mistake: Placing the repeat bar over digits that do not repeat.
Correction: Identify the exact block that cycles. In 0.1666..., only the 6 repeats, so write , not (which would mean 0.161616...).
Mistake: Rounding a repeating decimal and treating it as exact.
Correction: The value equals exactly . Writing 0.33 and using it in calculations introduces a small error. Use the fraction form when you need an exact answer.