In this section I have explained you how to convert rational numbers to decimals.
There are two types of rational no.
1) finite or terminating decimal 2) Non terminating decimals.
Conversion of rational no. to decimals
1) Finite or terminating decimals : The rational no. with a finite decimal part or for which the long division terminates ( stops) after a definite number of steps are known as finite or terminating decimals.
Example 1 :
Express 7 / 8 rational numbers to decimals form by long division method.
7 / 8 = 0.875
Example 2 :
Express - 17 / 8 in the decimal form by long division method.
- 17 / 8 = - 2.125
Example 3 :
Express 8 / 3 in the decimal form by long division method.
8 / 3 = 2.6666
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⇒ 2.6
From the above we observe that remainder start repeating in the same order.
Such a decimal numbers are known as non-terminating decimals.
2) Non terminating decimals : There are two types of decimal representations.
a) A decimal in which all the digits after the decimal point are repeated. These type of decimals are known as pure recurring decimals.
Example :
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o.6, 0.16
are pure recurring decimals.
Conversion of a pure recurring decimal to the form p / q.
1) Obtain the repeating decimal and put it equal to x.(say)
2) Write the number without using bar and equal to x.
3) Determine the number of digits having bar on their heads or number of digits before the bar for mixed recurring decimal.
4) If the repeating number is 1 then multiply by 10; if repeating number is 10 then multiply by 100 and so on.
5) Subtract the equation formed by step 2 and step 4.
6) Then find the value of x in the simplest form. Example 1:
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Express 0.2 in p / q form.
Solution :
Let x = 0.2222 ------> (1)
Multiply equation (1) by 10 as there is only one number is repeating.
10 x = 2.2222 ------> (2)
Subtract equation (1) from (2)
9x = 2 ( dividing both side by 9)
x = 2 / 9 Example 2 :
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Express 0.585 in p / q form.
Solution : Let x = 0.585585585 --------> (1)
As there is bar on three digits, so multiply equation by 1000.
1000 x = 585.585585 ------------> (2)
Subtract equation (1) from (2), we get
999 x = 585
x = 585 / 999 ( dividing both sides by 999)
x = 65 / 111 ( writing it in lowest form) Example 3 :
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Express 0.123 in p / q form.
Solution : Let x = 0.1233 ------------> (1)
As there is a bar on one digit so multiply equation (1) by 10
10 x = 12.333 --------------> (2)
Subtract equation (1) from (2)
9 x = 1.11
x = 1.11 / 9
x = 111 / 900 Rational number
