# Repeating decimals and rational numbers

In this chapter we will understand if repeating decimals are rational numbers.

Then we will learn methods to convert the given recurring decimal into rational number form.

Let us starts with the basics.

## What are recurring decimals ?

The decimal number whose digit repeats infinite number of times and never ends are called recurring decimals.

For example;

(a) 0.333333 . . . .

In the above decimal the digit 3 is repeated infinite times.

The above decimal can be expressed as \mathtt{0.\overline{3}}

(b) 13.65656565 . . . .

Here the number 65 is repeating infinite times.

The above decimal can be expressed as \mathtt{13.\overline{65}} .

### Are repeating decimals part of rational numbers ?

Yes !!!

Because we can convert the given recurring decimal into rational number form p / q.

For example;

\mathtt{\ \frac{1}{3} \Longrightarrow \ 0.33333\ .\ .\ .}\\\ \\ \mathtt{\ \frac{100}{6} \Longrightarrow \ 16.6666\ .\ .\ .}\\\ \\ \mathtt{\frac{11}{9} \Longrightarrow \ 1.2222\ .\ .\ .}

## Converting recurring decimal into rational number

Let the given recurring decimal is 0.33333 . . . .

To convert the repeating decimal into rational number, follow the below steps;

(a) Write the number in form of equation.

x = 0.3333 . . . .

(b) Identify the recurring digit and take it before the decimal point.

Here, digit 3 is repeated again and again.

Multiply the equation by 10 to take digit 3 before decimal point.

10x = 3.33333. . . . .

(c) Subtract the 2nd equation from the first.

Solving the rest of equation.

\mathtt{\ 9x\ =\ 3}\\\ \\ \mathtt{x=\frac{3}{9}}\\\ \\ \mathtt{x=\frac{1}{3}}

Hence, recurring decimal is converted into rational number 1/3.

Example 02
Convert 0.25252525 . . . . into rational number

Solution
To convert the repeating decimal into fraction, follow the below steps;

(a) Express in equation form

x = 0.25252525 . . . .

(b) Identify the repeating digit and take it before decimal point.

Here digit 25 is repeated again and again.

To take the repeating digit before decimal point, multiply it by 100.

100x = 25.25252525 . . . .

(c) Subtract 2nd equation with 1st equation.

Solving the expression further.

\mathtt{99x\ =\ 25}\\\ \\ \mathtt{x=\frac{25}{99}}

Hence, 25/99 is the required rational number.

Example 03
Convert 2.73737373. . . . into rational number.

Solution

(a) Write the number in equation form

x = 2.737373 . . .

(b) Identify the repeating digits and take it in from of decimal point.

Here 73 is the repeating digit.

To take 73 in front of decimal point, multiply the equation by 100.

100x = 273.737373 . . .

(c) Subtract both the equations.

Solving the rest of the equation.

\mathtt{\ 99x\ =\ 271}\\\ \\ \mathtt{x=\frac{271}{99}}

Hence, 271 / 99 is the required rational number.

Example 04
Convert 1.256565656 . . . . . into rational number.

Solution

(a) Write the decimal in form of equation.

x = 1.256565656 . . . .

(b) First write the expression such that only recurring numbers are present on right side of decimal.

Multiply the equation by 10.

10x = 12.56565656 . . . .

This is the first equation.

(c) Identify the recurring digits and take it before the decimal point.

Here digits 56 are repeated again and again.

Multiply the first equation by 100, so that number 56 come before decimal point.

100 (10x) = 100 (12.56565656 . . .)

1000x = 1256.56565656 . . . . .

(d) Subtract both the equations.

Solving the rest of the equation.

\mathtt{990x\ =\ 1244}\\\ \\ \mathtt{x=\frac{1244}{990}}

Hence, 1244 / 990 is the required rational number.

Example 05
Convert 0.324324324 . . . . into rational number.

Solution

(a) Write the number in equation form.

x = 0.324324324 . . . .

(b) Identify the recurring digits and take it before decimal point.

Here digits 324 are repeating again and again.

Multiply by 1000 to take it before decimal point.

1000x = 324.324324324 . . . .

(c) Subtract both the equations.

Solving the rest of equation.

\mathtt{999x\ =\ 324}\\\ \\ \mathtt{x\ =\ \frac{324}{999}}\\\ \\ \mathtt{x=\frac{324\div 9}{999\div 9}}\\\ \\ \mathtt{x\ =\ \frac{36}{111}}

Hence, 36/111 is the required rational number.

Next chapter : Important laws of rational numbers