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. 

 

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.
 

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 \(\frac{p}{q} \). 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. 

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 \(\frac{23}{99}\)

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.
   

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.