Repeating Decimal
In short: Repeating decimal names a number whose digits fall into a looping block after the point, which happens when dividing the top of a fraction by its bottom never leaves a zero remainder.
7th GradeNumbers and Operations7.NS.A.2d
What is Repeating Decimal?
Dividing 7 by 11 on paper gives a repeating decimal, because the digits 6 and 3 keep trading places without end; a bar over the looping digits records the pattern as \(0.\overline{63}\).
Any ratio of two integers either stops or loops. Division can only produce a limited list of remainders, so one of them eventually comes back and the digits cycle from there. Such numbers are rational even though their decimal form never stops.
How to spot it
- Paper division hands you the same remainder twice, so the quotient digits start to loop.
- A calculator screen shows 0.3333333 or 0.142857142857, where a block plainly comes back around.
- The bottom of the fraction, in lowest terms, has a prime factor other than 2 or 5.
- You are told to convert a ratio into a decimal and mark the loop with a bar.
The method
- Run long division on the fraction and jot down every remainder next to its digit.
- Halt once a remainder shows up for a second time; the digits produced between the two matches are the loop.
- Write the quotient and draw a bar over the looping digits only.
- Cross-check: the lowest-terms bottom number should hold a prime other than 2 or 5.
Repeating Decimal: solved example
Problem
Turn \(\frac{5}{6}\) into a decimal and mark which digits loop.
Solution
Divide: 5 ÷ 6 gives 0.8333 and the 3s never stop. After the 8 the remainder is 2, and 20 divided by 6 gives 3 with remainder 2 again, forever.
Only the 3 repeats, so \(\frac{5}{6} = \)\(0.8\overline{3}\), with the bar over the 3 only and the 8 kept outside.
Try it
Question 1
Convert \(\frac{4}{9}\) with a bar over the looping digit.
Show the answer
The quotient is 0.444 with the 4s continuing forever, so it is \(0.\overline{4}\).
Question 2
Why does \(\frac{2}{15}\) repeat while \(\frac{3}{8}\) does not?
Show the answer
The denominator 15 contains the prime 3, so the decimal repeats: \(0.1\overline{3}\). The denominator 8 is only 2s, so \(\frac{3}{8} = 0.375\) stops.
Repeating Decimal vs. Irrational Number
A repeating decimal has endless digits that follow a predictable cycle, so a fraction always matches it. An irrational number such as \(\pi\) is endless too, yet its digits never fall into a cycle and no fraction equals it. Endless alone is not the test; the loop decides whether a decimal is rational.
Keep going
Lessons
Worksheets and quizzes
- Rational and Irrational Numbers – Worksheet
- Rational and Irrational Numbers – Quiz
- Solving Real-World Problems with Rational Numbers – Worksheet
- Solving Real-World Problems with Rational Numbers – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Numbers and Operations