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Recurring Decimal

Last updated date: 09th Apr 2024
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Repeating Decimal

Have you ever come across decimal numbers and wondered why not all such numbers have a fixed number of digits after the decimal point? Or how do I convert these numbers into fractions? These are some questions that keep students bothered for a long time. 

Vedantu is here to help you. To solve such problems first, you need to know what Repeating Decimal is. 

Recurring Decimal numbers are those numbers that keep on repeating the same value after a decimal point. These numbers are also called Repeating Decimals. For example:

1/3 = 0.33333..... (3 repeats forever)

1/7 = 0.142857142857142857....... (14285714 repeat forever)

77/600= 0.128333333...... (3 repeat forever)

To display a repeating digit in a decimal number, often we put a dot or a line over the repeating digit as shown below:

For Example:

1/3=0.33333...=0.\[\dot{3}\] = 0.\[\bar{3}\]

(Image will be uploaded soon)

Non - Recurring Numbers

Non - recurring numbers are those in Mathematics that do not repeat their values after a decimal point. They are also called non-terminating decimals and non-Repeating Decimal numbers. For Example:

√2 = 1.41421356237309504…...

√7 = 2.647568759…...

π = 3.1415926535897932384626….....

e = 2.7182818284590452353602….....


Terminating and Recurring Decimals

When performing the conversion from fractions to decimals, you can formulate whether the decimal will terminate or recur.


Types of Decimal Numbers

Decimal numbers are classified into three types i.e.:

  1. Terminating Decimals: these decimals have a finite number of digits followed by the decimal point.

For example, 0.5, 1.456, 123.456, etc.

  1. Recurring Decimals: These consist of one or more repeating numbers or sequences of numbers followed by the decimal point, which keeps on infinitely.

For example, 5.232323…., 21.123123…, 0.1111….

  1. Irrational: These Decimals go on forever, are never-ending and also never form a repeating pattern. These numbers are known as irrational numbers and cannot be written in the form of a fraction.

For example, 0.45445544…., etc.


Converting Recurring Decimals to Fractions

Conversion of Recurring Decimals into fractions is very useful for students throughout their academic life and also thereafter. It forms part of their basic Mathematical aptitude in various competitive exams in the future. 

Let us understand and perform an example to understand how we a Recurring Decimal to a fraction.

Let’s check the steps involved in converting Repeating Decimals to fractions recurring (rational). The steps involved are as given:-

Step I: Let ‘x’ be the Repeating Decimal number that we want to convert into a rational number.

Step II: observe the Repeating Decimal to identify the repeating digits.

Step III: Carefully place the repeating digits to the left of the decimal point.

Step IV: Place the repeating digits to the right of the decimal point.

Step V: Now deduct the left sides of the two equations. Then, do the subtraction on the right side of the two equations. As we subtract, just ensure the differences of both sides will be positive.


Solved Examples

Example: Convert 0.7 (one recurring digit) into a fraction.

Solution: Follow the below steps to convert Recurring Decimals to fractions:

Let x= Recurring Decimal

Let n = the number of recurring digits

Multiply the Recurring Decimal by 10

Subtracting (1) from (3) to remove the recurring part

Solve for x, expressing the answer as the fraction in the simplest form

Now, to solve for the given example

x = 0.777777....

10x = 7.77777

10x –x = 7

9x = 7

X = 7/9

Therefore, x = 7/9 is the required rational number.

Example: Convert 1.256 (two recurring digit) into a fraction


x = 0.125656....

10x = 125.6565....

100x – x = 125.6565... -1.256565

99x = 124.4

x = 124.4/99 = 1244/990 = 622/495

Example of Conversion of Repeating Decimal to Fraction

Convert the numerical digit 4.567878….. into a rational fraction.


Converting the given decimal number into a rational fraction can be performed by undertaking the following conversion steps:

Step I: Let x = 4.56787878…

Step II: After analyzing the expression, we identified that the repeating digits are ‘78’.

Step III: Now have to place the repeating digits ‘78’ to the left of the decimal point. To do so, we are required to move the decimal point to the right by 4 places. This can be accomplished by multiplying the given number by’10,000’.

10,000x = 45678.787878

Step IV: Now we would require moving the repeating digits to the left of the decimal point in the original decimal number. For this purpose, we will have to multiply the original number by ‘100’.

100x = 456.787878

Step V: here, the two equations become:

10,000x = 45678.787878, and

100x = 456.787878

Step VI: Now we need to subtract both the left and right-hand sides of the two equations and equate them in such a manner that the equality remains the same.

10,000x - 100x = 45678.787878 - 456.787878

⟹ 9,900x = 45,222

⟹ x = 45222/9900

This rational fraction can further be reduced to

x = 75371650 (dividing both denominator and numerator by 6)

Thus, the rational conversion of the provided decimal number comes out to be 75371650.

All the conversion of this type can be carried out by using the above-mentioned steps carefully.


Fun Facts About Recurring Decimal

  • Some numbers cannot be expressed as a fraction. For example √2.

  • Ancient Egyptians only used unit fractions.

  • Solving Recurring Decimals can touch infinity. To understand this you need to do more complicated Mathematical calculations.


In this topic, you have learned about recurring numbers, recurring numbers, types of recurring numbers and conversion of decimal numbers into fractions. Apart from this, you have also been given a sufficient number of solved examples to illustrate each concept. 

After studying these you are in a position to further with other topics in Mathematics that are more complex. But you need to worry about it because at every stage Vedantu is here to help you.

FAQs on Recurring Decimal

1. What is Meant by a Rational Number?

A rational number is a number in the form of p q where ‘p’ and q’ are the integers and ‘q’ is not equal to zero (0). Both ‘p’ and ‘q’ could be positive as well as negative. A rational number can also be converted to both terminating and non-terminating decimal numbers. Remember that, non- terminating decimal numbers can be further categorized into two types that are recurring decimal and non- recurring decimal numbers.

2. How to Simplify, Solve and Express 15.02˙ as a Rational Number?

Firstly, we would require writing down the whole number in an order. For example, the whole number we got by writing the digits in its order = 1502. The whole number formed by the non recurring digits in order = 150. The number of digits after the decimal point = 2. The number of digits after the decimal point that do not repeat = 1. Hence, 15.02˙ = 1502−150102 − 101=1352100 − 10 = 135290.

3. What is a Repeating Decimal?

A repeating decimal, also known as the recurring decimal, is a number whose decimal display eventually becomes periodic (i.e., the same sequence of digits repeats indefinitely, infinite number of times).

4. How useful is Recurring Decimal in Mathematics?

The topic of Recurring Decimal is very important in Mathematical calculations. In higher standards, you will come across problems where, to proceed further with the calculations, you will be expected to convert the numbers from decimals to fractions because it is easier to solve fractions. Apart from this, in some places, you will have to cinder only a few digits after the decimal place. For this, you need to know how to round off numbers to the nearest subsequent number.

5. Can Recurring Decimal be useful in bank exams?

Yes Recurring Decimal can surely help you in Bank PO and Clerical exams. In the quant section of these exams, many questions are asked where the candidates are expected to convert the number from decimal to fractions or vice versa. Many times, questions are very easy and only conversions cause problems for the students. They get stuck because of not having a command over this topic. Thus Vedantu recommends you study Recurring Decimals religiously.

6. How many types of decimal numbers have been covered in Recurring Decimal?

In Mathematics, decimal numbers are defined as a number whose whole number part and fractional part is separated by a decimal point. Vedantu has covered three types of decimal numbers in Recurring Decimal which are as follows:

  • Terminating decimal numbers: These numbers have a fixed number of digits after the decimal point

  • Recurring Decimal numbers: As the name suggests, these numbers have infinite digits after the decimal point

  • Non-Recurring Decimal numbers: These are nin0 terminating as well as non-repeating numbers. They have an infinite number of digits after their decimal places and the digits do not follow a specific order.

7. Can I rely on Recurring Decimal by Vedantu?

Yes you can completely rely on Vedantu’s Recurring Decimal. Vedantu is committed to providing its students with nothing less than the best. To fulfill this commitment Vedantu makes sure all the study materials are brought to you only after thorough research and analysis. The same has been applied here. All the questions brought for you are prepared by subject matters experts having years of expertise in Mathematics. Many past top scorers have solved these questions and they recommend it to you as well.

8. How many sums do I need to practice to get a proper hold over the topic of Recurring Decimal?

Maths is a subject in which any topic can be mastered by solving several problems. You can do the same for Recurring Decimal. start by solving the questions given above. Even in these, ensure you proceed in ascending order of difficulty. After this, you can pick up sums from RD Sharma’s books. Even in other subjects whenever you come across such calculations try to solve them on your own without using a calculator. These small steps will help you master this fundamental concept of Mathematics in a short period.