Key takeaway: A repeating decimal is a decimal whose digits settle into a block that repeats forever, such as 0.454545…, and every one equals a fraction of two integers.
Grade 77.NS.A.2dNumbers and Operations
Definition of Repeating Decimal
A repeating decimal is a decimal number with a digit or group of digits that keeps recurring without end. The recurring group is called the repetend or period.
These decimals appear when you divide an integer by an integer and the remainders start to cycle. Since there are only so many possible remainders, the cycle must come back, so a repeating pattern is guaranteed whenever a fraction does not stop. That is why every repeating decimal is a rational number.
Formula and notation
A bar over the repeating block shows what repeats:
\[\frac{5}{11} = 0.454545\ldots = 0.\overline{45}\]
The bar covers only the digits that repeat, so \(\frac{7}{12} = 0.58\overline{3}\) has a bar over the 3 alone.
Key facts
- Division stops cycling as soon as a remainder appears a second time.
- The repeating block can be one digit long or many digits long.
- A fraction in lowest terms repeats when its denominator has a prime factor besides 2 and 5.
- A rounded copy such as 0.67 is close to the repeating value but not equal to it.
Repeating Decimal example
Use long division to write \(\frac{7}{12}\) as a decimal, and decide whether it terminates.
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\(70 : 12 = 5\) remainder \(10\). First digit after the point: 5.
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\(100 : 12 = 8\) remainder \(4\). Next digit: 8.
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\(40 : 12 = 3\) remainder \(4\). Next digit: 3, and the remainder 4 has come back.
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The remainder 4 repeats, so the digit 3 repeats: \(0.58333\ldots\)
\(\frac{7}{12} = 0.58\overline{3}\), a repeating decimal, so it does not terminate.
Common mistake
Watch out
A calculator shows 0.4444444 for \(\frac{4}{9}\). Selma writes \(\frac{4}{9} = 0.44\).
Why it is wrong
The display is cut off by the screen. Since \(0.44 = \frac{44}{100} = \frac{11}{25}\), which is not \(\frac{4}{9}\), the two numbers differ. Long division gives remainder 4 every time, so the 4s never end: \(\frac{4}{9} = 0.\overline{4}\).
Related terms and lessons
- Grade 7 · Rational Numbers: Decimals, Fractions and Word ProblemsLessonWorksheetQuiz
- Grade 8 · Rational and Irrational NumbersLessonWorksheetQuiz
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