Repeating Decimals

How to repeating decimals, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Repeating decimals appear regularly on the GED Mathematical Reasoning test, especially when you convert fractions to decimals or classify rational numbers. A repeating decimal is any decimal whose digits repeat in a predictable pattern without end. Recognizing and working with repeating decimals is a key skill for rational number fluency.

What Is a Repeating Decimal?

A repeating decimal (also called a recurring decimal) is a decimal in which one or more digits repeat in a cycle forever. It is written with a bar (vinculum) placed over the repeating digit(s). For example:

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• \(\color{blue}{\frac{1}{3} = 0.333\ldots = 0.&\#773;3}\)
• \(\color{blue}{\frac{2}{9} = 0.222\ldots = 0.&\#773;2}\)
• \(\color{blue}{\frac{1}{7} = 0.142857142857\ldots = 0.142857}\)

Repeating decimals are rational numbers. They can always be written as fractions.

Terminating vs. Repeating Decimals

Terminating decimals

A decimal that ends (has a finite number of digits) after the decimal point. These come from fractions whose denominator (in lowest terms) has only factors of 2 and/or 5.

• \(\color{blue}{\frac{3}{4} = 0.75}\) (terminates)
• \(\color{blue}{\frac{7}{20} = 0.35}\) (terminates)

Repeating decimals

A decimal that repeats forever. These come from fractions whose denominator (in lowest terms) has a prime factor other than 2 or 5.

• \(\color{blue}{\frac{1}{3} = 0.&\#773;3}\) (3 is neither 2 nor 5)
• \(\color{blue}{\frac{5}{6} = 0.8&\#773;3}\) (\(\color{blue}{6 = 2 \times 3}\); the factor 3 causes the repeat)
• \(\color{blue}{\frac{7}{11} = 0.63}\) (11 is neither 2 nor 5)

Step-by-Step Summary

1. Convert the fraction to a decimal by dividing the numerator by the denominator.
2. If the long division terminates, it is a terminating decimal.
3. If a remainder repeats, the decimal repeats; write the repeating block under a bar.
4. To convert a repeating decimal back to a fraction: let \(\color{blue}{x = 0.&\#773;d}\), multiply both sides to shift the decimal, subtract, and solve.

Watch: Converting a Fraction to a Repeating Decimal (Video Lesson)

Khan Academy walks through converting \(\color{blue}{\frac{19}{27}}\) to a repeating decimal with long division:

Repeating Decimals – Worked Examples

Example 1: Convert \(\color{blue}{\frac{1}{3}}\) to a decimal. Is it repeating?

Divide: \(\color{blue}{1.000 \div 3}\). Remainders cycle: \(\color{blue}{1, 1, 1, \ldots}\)
\(\color{blue}{\frac{1}{3} = 0.333\ldots = 0.&\#773;3}\). Yes, it is repeating.

Example 2: Convert \(\color{blue}{\frac{2}{9}}\) to a decimal.

Divide: \(\color{blue}{2.000 \div 9}\). Remainders cycle: \(\color{blue}{2, 2, 2, \ldots}\)
\(\color{blue}{\frac{2}{9} = 0.222\ldots = 0.&\#773;2}\).

Example 3: Convert \(\color{blue}{\frac{5}{6}}\) to a decimal.

Divide: \(\color{blue}{5 \div 6 = 0.8333\ldots}\) The digit 8 does not repeat, but 3 does.
\(\color{blue}{\frac{5}{6} = 0.8&\#773;3}\).

Example 4: Convert the repeating decimal \(\color{blue}{0.&\#773;3}\) back to a fraction.

Let \(\color{blue}{x = 0.333\ldots}\)
Then \(\color{blue}{10x = 3.333\ldots}\)
Subtract: \(\color{blue}{10x – x = 3.333\ldots – 0.333\ldots = 3}\)
\(\color{blue}{9x = 3}\), so \(\color{blue}{x = \frac{3}{9} = \frac{1}{3}}\).

More Practice: Converting Repeating Decimals to Fractions (Video)

Khan Academy shows the algebraic method for converting any repeating decimal to a fraction:

Exercises for Repeating Decimals

1. Convert \(\color{blue}{\frac{1}{6}}\) to a decimal. Is it repeating?
2. Convert \(\color{blue}{\frac{7}{9}}\) to a decimal.
3. Convert \(\color{blue}{\frac{5}{11}}\) to a decimal.
4. Is \(\color{blue}{\frac{3}{8}}\) a terminating or repeating decimal? Justify your answer.
5. Convert \(\color{blue}{0.&\#773;6}\) to a fraction in lowest terms.
6. Convert \(\color{blue}{0.27}\) (i.e., \(\color{blue}{0.272727\ldots}\)) to a fraction.

Answers

1. \(\color{blue}{\frac{1}{6} = 0.1&\#773;6}\); yes, it is repeating (\(\color{blue}{6 = 2 \times 3}\); factor 3 causes the repeat).
2. \(\color{blue}{\frac{7}{9} = 0.&\#773;7}\)
3. \(\color{blue}{\frac{5}{11} = 0.45}\)
4. Terminating: \(\color{blue}{\frac{3}{8} = 0.375}\). The denominator \(\color{blue}{8 = 2^{3}}\) has only the factor 2.
5. Let \(\color{blue}{x = 0.666\ldots}\); \(\color{blue}{10x = 6.666\ldots}\); \(\color{blue}{9x = 6}\); \(\color{blue}{x = \frac{6}{9} = \frac{2}{3}}\).
6. Let \(\color{blue}{x = 0.272727\ldots}\); \(\color{blue}{100x = 27.272727\ldots}\); \(\color{blue}{99x = 27}\); \(\color{blue}{x = \frac{27}{99} = \frac{3}{11}}\).

Frequently Asked Questions

What is a repeating decimal in math?

A repeating decimal is a decimal in which one digit or a group of digits repeats forever in the same pattern. For example, 1/3 = 0.333…, 4/11 = 0.3636…, and 1/7 = 0.142857142857…. The repeating part is called the repetend. Every repeating decimal comes from dividing one integer by another.

What is the difference between repeating and terminating decimals?

A terminating decimal ends after a finite number of digits, like 3/8 = 0.375. A repeating decimal goes on forever with a repeating pattern, like 5/6 = 0.8333…. You can predict which one you will get: simplify the fraction, and if the denominator has only 2s and 5s as prime factors, the decimal terminates. Otherwise it repeats.

Are repeating decimals rational or irrational?

Repeating decimals are always rational, because every one of them can be written as a fraction of two integers. For instance, 0.777… = 7/9 and 0.1212… = 12/99 = 4/33. Irrational numbers such as π and √2 have decimals that never end and never settle into a repeating pattern.

How do you write repeating decimals as fractions?

Let x equal the decimal, multiply by 10, 100 or 1,000 to shift one full repeating block past the decimal point, then subtract the original equation. For x = 0.454545…, 100x = 45.4545…, so 99x = 45 and x = 45/99 = 5/11. With one repeating digit, like 0.666…, multiply by 10 instead: 9x = 6, so x = 2/3.

How do you show repeating decimals?

The standard way is to draw a bar, called a vinculum, over the digit or digits that repeat. You would write 0.5 with a bar over the 5 for 0.555…, and 0.27 with a bar over both digits for 0.272727…. Another common way is to write a few repeats followed by three dots, such as 0.2727….

What is a non-repeating decimal?

A non-repeating decimal never forms a repeating pattern. If it also never ends, it is non-terminating and non-repeating, which makes it irrational. Examples are π = 3.14159… and √3 = 1.73205…. Terminating decimals like 0.4 also do not repeat, but they are rational because they can be written as fractions, here 2/5.

Related Topics

• Convert Between Fractions and Decimals
• Using a Venn Diagram to Classify Rational Numbers
• Simplifying Fractions
• Convert Between Decimals and Mixed Numbers
• Order of Decimals Mixed Numbers and Fractions