Recurring Decimal

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. 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 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 \(0.\dot{3}1\dot{3}1…\) 
   

A rational number is written in the form of 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, 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…
 x = 0.232323…

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

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.