Recurring Decimal
2026-02-28 09:08 Diff
https://brightchamps.com/en-us/math/numbers/recurring-decimal

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Last updated on December 6, 2025

A recurring decimal is the decimal in which the digits after the decimal point repeat in a fixed pattern. For example, in 2.354354…, the repeating block is 354. Such the decimals are also called repeating decimals and differ from terminating or non-repeating decimals.

What is a Recurring Decimal?

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Decimals represent numbers as fractions of a whole. For example, 2.356, where 2 is the whole number and 0.356 is the fractional part, and it is separated by a decimal point(.).

A recurring decimal, or a repeating decimal, is a type of decimal where the digits after the decimal point repeat. It is a non-terminating decimal because the digits after the decimal point repeat indefinitely. 

Example:
Consider the number: 0.727272…

Did you notice that the repeating pattern?

The digits 72 repeat again and again. So, 72 is the recurring, which means repeating part. We can write it as: 0.72, the bar shows the digits that repeat.
 

How to Represent Recurring Decimals?

Now let’s learn how to represent recurring decimals. It can be done in two ways, as mentioned below: 
 

  • A bar is placed over the repeating digits to represent recurring decimals. For example, \(25.66666…\) can be represented as 25.66̅ where 6 keeps repeating. Another example is \(0.727272...\) as 0.72̅  where 72 repeats forever.       
     
  • Another method is a dot notation, where a dot is placed above the recurring digit(s). For example, \(0.3333... \)is written as \(0.\dot{3}\) and \(0.313131...\) can be represented as \(0.\dot{3}1\dot{3}1…\) 
     

How to Represent Recurring Decimals as Rational Numbers?

A rational number is written in the form of \(\frac{p}{q} \). Decimals can be expressed as rational numbers using the long division method. Rational numbers can have decimal representations that are either terminating or non-terminating repeating decimals.

For example, \(\frac{1}{2} = 0.5 \) is a terminating decimal, as the division process ends without repeating the digits. In non-terminating but repeating decimals, the digits repeat, for example, \(⅓ = 0.33333…\) It can be represented as 0.3 bar or 0.3. 

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Conversion of Recurring Decimal to Fraction

We have learned how to express recurring decimals as rational numbers. Now let’s see how to convert recurring decimals to fractions.

Step 1: To convert recurring decimal to fraction, first, let’s consider the recurring decimal as x

Step 2: Let n be the number of recurring digits.

Step 3: Multiply x by 10n.

Step 4: Subtract the original equation from the equation obtained in Step 3 to eliminate the repeating part.

Step 5: Then find the value of x and simplify the fraction.

For example, convert \(0.23232323..…\) into a fraction


Step 1: Here, x = \( 0.23232323…\)


Step 2: The repeating digits are 23 so, n = 2


Step 3: \(x \times 10^2 = 0.23232323\ldots \times 10^2 \)


As \(10^2 = 100 \)


\(100x = 23.232323…\)


\( x = 0.232323…\)


Step 4: \( 100x - x = 23.232323 - 0.232323\)


\(99x = 23\)


So, \(x = \frac{23}{99} \)
So, \( 0.23232323…\) in fraction can be represented as \(\frac{23}{99}\)

Tips and Tricks for Mastering Recurring decimals

Recurring decimals helps children understand patterns in numbers and improves their accuracy in calculations. These tips and tricks make learning repeating decimals simple and fun.

  • Identify the repeating part of a decimal to recognize the pattern quickly.
     
  • Convert simple fractions like \(\frac{1}{3} \) or \(\frac{2}{7} \) to decimals to practice repeating sequences.
     
  • Use visual aids like number lines or grids to see recurring decimals clearly.
     
  • Write the repeating part with a bar (vinculum) to make it easier to read and remember.
     
  • Practice with real-life examples, like dividing money or objects, to understand repeating decimals in everyday life.
     

  • Before introducing recurring decimals, ensure children are comfortable with decimals that end. This helps them understand the difference more easily.
     

  • Explain that every recurring decimal is a rational number.
     

  • Provide a worksheet where students identify repeating parts, write bar notation, and convert to fractions.
     

  • Ask students to write fractions such as 1/3, 2/9, or 5/6, then convert them to decimals.
    This shows directly how recurring decimals come from fractions.
     

Common Mistakes and How to Avoid Them in Recurring Decimal

After learning about recurring decimals, we must understand how to use them without making mistakes. Below are some commonly made mistakes while working with decimals. Knowing about them will keep us from making such mistakes.
 

Real-Life Applications of Recurring Decimal

The concept of recurring decimals is used in our daily life. Let’s see some of its applications: 
 

  • Recurring decimals are used in banks or other financial institutions to calculate interest rates.
     
  • Students use recurring decimals while solving basic mathematic problems.
     
  • Scientific calculations involving constants like the speed of light or gravitational acceleration often include recurring decimals.
     
  • Recurring decimals can appear when measuring lengths, weights, or volumes that don’t convert exactly into decimal form, requiring precise calculations.
     
  • They also arise when representing fractions in code or running simulations that involve repeating patterns in calculations.

Download Worksheets

Problem 1

Convert 0.34 to a fraction

Okay, lets begin

\(0.34 = \frac{34}{100} \)

Explanation

\(0.3434\ldots = \frac{34}{99} \);
 

0.34 (terminating)
 

= \(\frac{34}{100} \).

Well explained 👍

Problem 2

Check whether 7/40 is a terminating or non-terminating decimal.

Okay, lets begin

\(\frac{7}{40} \) is a terminating decimal.

Explanation

 A fraction is terminating if it can be expressed as \(\frac{p}{2^n \times 5^m} \).

The prime factorization of 40 is \(2^3 \times 5 \)

So, it can be expressed as \(\frac{7}{2^3 \times 51} \)

Therefore, \(\frac{7}{40} \) is a terminating decimal. 

Well explained 👍

Problem 3

Convert 1.428 into a fraction.

Okay, lets begin

1.428 can be expressed as \(\frac{1428}{999} \)
 

Explanation

If repeating 1.428428…,
 

then 1.428… = \(\frac{1428}{999} \)
 

If terminating 1.428, then \(1.428 = \frac{1428}{1000} \)
 

= \(\frac{357}{250} \).

Well explained 👍

Problem 4

Convert 9/11 into decimal.

Okay, lets begin

 \(\frac{9}{11} = 0.81818\ldots \)

Explanation

To convert \(\frac{9}{11} \) to decimal we divide 9 by 11

So, \(\frac{9}{11} = 0.818181 \)

Since 81 is repeated, it can be written as 0.81

Well explained 👍

Problem 5

Check whether ⅚ is a terminating or non-terminating decimal

Okay, lets begin

\(\frac{5}{6} \) is a non-terminating decimal

Explanation

When we convert \(\frac{5}{6} \) to decimal form 

that is \(\frac{5}{6} = 0.833\ldots \)

\(\frac{5}{6} \) is a non-terminating recurring decimal, since 3 repeats.

Well explained 👍

FAQs on Recurring Decimal

1.What is a recurring decimal?

The recurring decimal is a type of decimal, here the digits after the decimal points repeat again and again. For example, 0.5555…, 2.353535…. 

2.Is 0.7777… a recurring decimal?

Yes, 0.7777… is a recurring decimal as the digit 7 is repeating

3.Is 9.37 a recurring decimal?

No, 9.37 is a non-recurring decimal, as the digits after the decimal point are not repeated. 

4.How do you represent a recurring decimal?

The recurring decimal is represented using a bar over a repeating digit, (). For example, 0.525252… = 0.52

5.What is the difference between recurring and terminating decimal?

In a recurring decimal, the digits after the decimal points are repeating, for example, 2.232323…. Whereas in terminating decimals the digits after the decimal point end after a few digits, for example, .23, 2.255. 

6.How can I help my child understand recurring decimals at home?

Using visual aids such as number lines or objects can make it easier for them to see the repeating pattern and understand the concept.

7.Can recurring decimals affect my child’s everyday math learning?

Yes, recurring decimals appear when converting fractions to decimals or during division problems. Understanding them helps children handle money, measurements, and other real-life calculations accurately.

Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.