Terminating and Non-terminating Decimals

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Terminating and Non-terminating Decimals

Every rational number or fraction can be expressed in terms of decimal numbers by the actual division. And, those decimal numbers can either be terminating or non-terminating. 



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What are terminating decimals?


Terminating decimal is a fraction decimal having a finite number of digits after decimal point. In this case division ends after a finite number of stages. For example : 1/5,  1/4, 1/8 etc. are terminating decimals because,

1/5 = 0.2 , 1/4 = 0.25, 1/8 = 0.125


What are non-terminating decimals?


Non-terminating decimal is a fraction decimal having infinite number of digits after decimal point. In this case division never ends but recur the same digit or group of digits again and again. So they are also called recurring decimals. For example: 1/3, 2/3, 4/7 etc. are non-terminating decimals because,

1/3 = 0.3333333……, 2/3 = 0.6666666……, 2/11 = 0.181818……

Here, a number or group of numbers are repeated and do not end after a finite number of stages, such number are denoted by a dot (.) or a bar(–) above it.

0.333333… = 0."3" ̅ , 0.666666… = 0."6" ̅ , 0.181818… = 0."18" ̅

Types of Non-terminating Decimals:


There are two types of non-terminating (recurring) decimals: 

a.   Simple recurring decimals: In this case the recurring starts just after the decinal point. For example, 0.333…

b.   Mixed recurring decimals: In this case the recurring does not start just after the decimal point. For example, 0.5181818…


 

Conversion of Decimal into Fraction


For terminating decimal we can convert a decimal into fraction simply by removing decimal and dividing by 10 if there is 1 decimal place, by 100 if there is 2 decimal places and by 1000 if there is 3 decimal places and so on. For example:


Example 1. Convert the decimal numbers 2.5, 0.28, 0.125 into fraction. Solution: Here, 2.5 = "25" /"10" = "5" /"2" 0.28 = "28" /"100" = "7" /"25" 0.125 = "125" /"1000" = "1" /"8"

For non-terminating decimal we can convert a decimal into fraction by the following method as given in the example below:



Example 2. Convert the non-terminating decimal 0."3" ̅ into fraction. Solution: Let, x = 0."3" ̅ ………….. (i) Multiplying both sides by 10, we get 10x = 3."3" ̅ ……………. (ii) Subtracting equation (i) from (ii), we get 10x – x = 3."3" ̅ – 0."3" ̅ or, 9x = 3 or, x = "3" /"9" or, x = "1" /"3" ∴ 0."3" ̅ = "1" /"3"


Example 3. Convert the non-terminating decimal 0.("18" ) ̅ into fraction. Solution: Let, x = 0.("18" ) ̅ ………….. (i) Multiplying both sides by 100, we get 100x = 18.("18" ) ̅ ……………. (ii) Subtracting equation (i) from (ii), we get 100x – x = 18.("18" ) ̅ – 0.("18" ) ̅ or, 99x = 18 or, x = "18" /"99" or, x = "2" /"11" ∴ 0.("18" ) ̅ = "2" /"11"
Example 4. Convert the non-terminating decimal 0.("657" ) ̅ into fraction. Solution: Let, x = 0.("657" ) ̅ ………….. (i) Multiplying both sides by 1000, we get 1000x = 657.("657" ) ̅ ……………. (ii) Subtracting equation (i) from (ii), we get 1000x – x = 657.("567" ) ̅ – 0.("657" ) ̅ or, 999x = 657 or, x = "657" /"999" or, x = "73" /"111" ∴ 0.("657" ) ̅ = "73" /"111"


Example 5. Convert the non-terminating decimal 2.("13" ) ̅ into fraction. Solution: Let, x = 2.("13" ) ̅ ………….. (i) Multiplying both sides by 100, we get 100x = 213.("13" ) ̅ ……………. (ii) Subtracting equation (i) from (ii), we get 100x – x = 213.("13" ) ̅ – 2.("13" ) ̅ or, 99x = 211 or, x = "211" /"99" ∴ 2.("13" ) ̅ = "211" /"99"


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