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Writing recurring decimals as fractions

Writing recurring decimals as fractions

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Summary

Writing recurring decimals as fractions

​​In a nutshell

Not all fractions can be expressed as a terminating decimals (decimals with only a finite number of digits). In some cases, their decimal form may be infinitely long and so you cannot write down the entire number. 



Recurring decimals

Definition

A recurring decimal is a number containing an infinitely repeating digit or series of digits. The repeating digit(s) are indicated with a  ˙\dot{}  above the numbers.


Example 1

The following are examples of recurring decimals:


0.1111...=0.1˙, 0.345345...=0.3˙45˙, 0.298454545...=0.2984˙5˙0.1111... = 0.\dot{1},\text{ } 0.345345...=0.\dot{3}4\dot{5},\text{ } 0.298454545... = 0.298\dot{4}\dot{5}



Converting between recurring decimals and fractions

Fraction to recurring decimal

The process of converting a fraction to a recurring decimal is the same as before, except ensure that the  ˙\dot{}  goes above the correct numbers to indicate which are repeating.


Recurring decimal to fraction - higher only

​​Procedure

1.
Multiply the recurring decimal by a multiple of 1010, for example 10,100,or 100010 , 100, or \space1000 etc. such that the recurring digits overlap in the same position.​
2.
Subtract the recurring decimal part from another, leaving a 9x, 99x9x, \text{ } 99x etc.​
3.
Rearrange to obtain the number as a fraction.
4.
Simplify the fraction (if required).


Example 2

Write 0.4545...=0.4˙5˙0.4545... = 0.\dot{4}\dot{5} as a fraction in its simplest form.


x=0.454545...1100x=45.454545...299x=4521x=4599\begin{aligned}x &= 0.454545... &\qquad \textcircled{1} \\100x &= 45.454545... &\qquad \textcircled{2} \\ 99x &= 45 &\qquad \textcircled{2}- \textcircled{1}\\x &= \dfrac {45}{99}\end {aligned}


Therefore, x=511\underline{ x= \dfrac {5}{11}}.



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FAQs - Frequently Asked Questions

Can all decimals be written as fractions?

Are recurring decimals rational numbers?

How do you identify a recurring decimal?

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