Last updated on July 5th, 2025
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.
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…
Now let’s learn how to represent recurring decimals. It can be done in two ways, as mentioned below:
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.
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
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.
The concept of recurring decimals is used in our daily life. Let’s see some of its applications:
Convert 0.34 to a fraction
0.34 = 34/99
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
Check whether 7/40 is a terminating or non-terminating decimal.
7/40 is a termnating decimal
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.
Convert 1.428 into a fraction.
1.428 can be expressed as 1428/999
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
Convert 9/11 into decimal.
9/11 = 0.81818…
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
Check whether ⅚ is a terminating or non-terminating decimal
5/6 is a non-terminating decimal
When we convert 5/6 to decimal form
That is ⅚ = 0.833…
⅚ is a non-terminating recurring decimal, since 3 repeats.
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.
: She loves to read number jokes and games.