102 LearnersLast updated on July 5th, 2025
Recurring Decimal
A recurring decimal is a decimal where the digits after the decimal point are repeated after a fixed interval. That is 2.354354354…., here the recurring decimals are 354. A recurring decimal is also known as a repeating decimal. Based on the digits after the decimal point, decimals can be categorized into repeating, non-repeating, end, or unending decimals.
What is a Recurring Decimal?
The way of representing numbers with fractions of a whole is decimals. 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 a digit after the decimal point recurs indefinitely. For example, 23.456456456…, 2.23232323…., and 0.898989…
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 0.3̇ and 0.313131... can be represented as 0.3
How to Represent Recurring Decimals as Rational Numbers?
A rational number is written in the form of p/q. Decimals can be expressed as rational numbers using the long division method. There are two types of decimal representations of rational numbers: terminating decimals and non-terminating, repeating decimals.
For example, ½ = 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.
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 × 102 = 0.23232323… × 102
As 102 = 100
100x = 23.232323…
Step 4: 100x - x = 23.232323 - 0.232323
99x = 23
So, x = 23/99
So, 0.23232323… in fraction can be represented as 23/99
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.
Mistake 1
Confusing recurring and terminating decimals
The recurring and terminating decimals are the different types of decimals and students tend to confuse them. So to avoid this error students should understand what is recurring and terminating decimals. In recurring decimals, the digits after the decimal point repeat again and again. In terminating decimals, the digits after the decimal point are finite.
Mistake 2
Using the wrong notation for recurring decimals
When using the notation to represent the recurring decimals, students should be careful. Because sometimes to represent 23.36666… students represent it as 23.36, instead of 23.36. So make sure that the bar notation is adding only over the digits that are repeated, for example, 23.2555555… as 23.25.
Mistake 3
Errors when converting the decimal to a fraction
When converting a decimal to a fraction, errors are common among students, as they assume the value that is 0.3 3/10 instead of ⅓. So, students should use the algebraic method instead of assuming the value.
Mistake 4
Forgetting to simplify the fraction
After converting the decimal to a fraction, students sometimes fail to simplify the fraction. It is important to simplify the fraction after the conversion. For example, 0.75 can be converted to 75/99. This should not be our final answer as 75/99 can further be simplified as 25/33.
Mistake 5
Thinking that all the fractions are recurring decimals
Students think that all fractions are recurring decimals which is wrong. Only decimals where the digits after the decimal point repeat infinitely are called recurring decimals. For instance, 1/8 = 0.125 which is not a recurring decimal but a terminating decimal.
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 may appear in banks or other financial institutions while calculating interest rates.
- Students when doing basic calculations in mathematics such as converting fractions to decimals or performing arithmetic operations, the result can be expressed in the form of recurring decimals.
- They may also appear in scientific calculations involving constants like the speed of light or gravitational acceleration.
Solved Examples of Recurring Decimal
Problem 1
Convert 0.34 to a fraction
0.34 = 34/99
Explanation
Step 1: Here, x = 0.343434…
Step 2: The repeating digits are 34 so, n = 2
Step 3: x × 102 = 0.343434… × 102
As 102 = 100
100x = 34.343434…
Step 4: 100x - x = 34.343434 - 0.343434
99x = 34
So, x = 34/99
Problem 2
Check whether 7/40 is a terminating or non-terminating decimal.
7/40 is a termnating decimal
Explanation
A fraction is terminating if it can be expressed as p/2n × 5m
The prime factorization of 40 is 23 × 5
So, it can be expressed as 7/(23 × 51)
Therefore, 7/40 is a terminating decimal.
Problem 3
Convert 1.428 into a fraction.
1.428 can be expressed as 1428/999
Explanation
Step 1: Here, x = 1.428…
Step 2: The repeating digits are 428 so, n = 3
Step 3: x × 103 = 1.428428… × 103
As 103 = 1000
1000x = 1428.428428…
Step 4: 1000x - x = 1428.428428 - 0.428428
999x = 1428
So, x = 1428/999
Problem 4
Convert 9/11 into decimal.
9/11 = 0.81818…
Explanation
To convert 9/11 to decimal we divide 9 by 11
So, 9/11 = 0.818181
Since 81 is repeated, it can be written as 0.81
Problem 5
Check whether ⅚ is a terminating or non-terminating decimal
5/6 is a non-terminating decimal
Explanation
When we convert 5/6 to decimal form
That is ⅚ = 0.833…
⅚ is a non-terminating recurring decimal, since 3 repeats.
FAQs on Recurring Decimal
1.What is a recurring decimal?
2.Is 0.7777… a recurring decimal?
3.Is 9.37 a recurring decimal?
4.How do you represent a recurring decimal?
5.What is the difference between recurring and terminating decimal?
6.How can children in Canada use numbers in everyday life to understand Recurring Decimal ?
7.What are some fun ways kids in Canada can practice Recurring Decimal with numbers?
8.What role do numbers and Recurring Decimal play in helping children in Canada develop problem-solving skills?
9.How can families in Canada create number-rich environments to improve Recurring Decimal skills?
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.
