NUMBER SYSTEMS – Exercise 1.3 – (MATHEMATICS) – 9th Class

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Exercise 1.3

1.  Write the following in decimal form and say what kind of decimal expansion each has:
(i) {{36} \over {100}}

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Ans. –  {{36} \over {100}} = 0.36, terminating decimal expansion.

(ii) {1 \over {11}}

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Ans. – {1 \over {11}} =0.090909……=0.\overline {09} , non-terminating repeating decimal expansion.

(iii) 4{1 \over 8}

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Ans. –  4{1 \over 8} = {{33} \over 8} = 4.125, terminating decimal expansion.

(iv) {3 \over {13}}

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Ans. – {3 \over {13}} = 0.230769230769....... = 0.\overline {230769} , non-terminating repeating decimal expansion.

(v) {2 \over {11}}

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Ans. – {2 \over {11}} = 0.181818... = 0.\overline {18} , non-terminating repeating decimal expansion.

(vi) {{329} \over {400}}

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Ans. – {{329} \over {400}} = 0.8225, terminating decimal expansion.

2. You know that {1 \over 7}  = {1 \over 7} = 0.\overline {142857} . Can you predict what the decimal expansions of {2 \over 7},{3 \over 7},{4 \over 7},{5 \over 7},{6 \over 7} are without actually doing the long division? If so, how?
[Hint: Study the remainders while finding the value of {1 \over 7} carefully.]

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Sol. Yes, we can predict the required decimal expansions.
We are given, {1 \over 7} = 0.\overline {142857}
On dividing 1 by 7, we find that the remainders repeat after six divisions, therefore, the quotient has a repeating  block of six digits in the decimal expansion of {1 \over 7} . So, to obtain decimal expansions of  {2 \over 7},{3 \over 7},{4 \over 7},{5 \over 7},and{6 \over 7}; and . we
multiply 142857 by 2, 3, 4, 5 and 6 respectively, to get the integral part and in the decimal part, we take block of six repeating digits in each case. Hence, we get
{2 \over 7} = 2 \times {1 \over 7} = 0.\overline {285714}
{3 \over 7} = 3 \times {1 \over 7} = 0.\overline {428571}
{4 \over 7} = 4 \times {1 \over 7} = 0.\overline {571428}
{5 \over 7} = 5 \times {1 \over 7} = 0.\overline {714285}
and{6 \over 7} = 6 \times {1 \over 7} = 0.\overline {857142}

3. Express the following in the form {p \over q} , where p and q are integers and q ≠ 0:
(i) 0.\overline 6

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Ans. –  let x = 0.\overline 6
or x = 0.666..                 ….(i)
10x = 6.666…                …..(ii)                [On multiplying (i) by 10 ]
⇒  9x = 6                                                        [ Subtracting (i) from (ii) ]
x = {2 \over 3}

(ii) 0.4\overline 7

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Ans. –  Let    x = 0.0.4\overline 7
or     x  =  0.4777….                                       ……(i)
Multiplying (i) by 10, we get
10x = 47.777…..                                             …….(ii)
Again multiplying (ii) by 10, we get
100x = 4.777…                                                 …….(iii)
Subtracting equation (ii) from equation (iii), we get
90x  = 43
∴  x = {{43} \over {90}}

(iii) 0.\overline {001}

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Ans. –   Let x = 0.\overline {001}
or                 x = 0.001001….                                 ….(i)
⇒               1000x = 1.001001….                              ….(ii)
.                                                                                            [On multiplying (i) by 1000]
⇒              999x = 1                                                            [On subtracting (i) from (ii)]
∴                x = {1 \over {999}}

4. Express 0.99999… in the form {p \over q}. Are you surprised by your answer? With your teacher and classmates, discuss why the answer makes sense.

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Sol.   Let  x     = 0.99999…                                             …(i)
⇒           10 x    =  9.99999…                            …(ii) [On multiplying (i) by 10]
⇒             9 x    = 9 [On subtracting (i) from (ii)]
∴                     x =       1
Yes , we are surprised by our answer.

5. What can the maximum number of digits be in the repeating block of digits in the expansion of {1 \over {17}} ?
Perform the division to check your answer.

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Sol.


Thus,   {1 \over {17}} = 0.\overline {0588235294117647}
Hence, the required number of digits in the repeating block is 16.


6. Look at several examples of rational numbers in the form {p \over q}(q \ne 0)
where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

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Sol.  Examples are  {3 \over 4},{4 \over 5},{7 \over 8},{9 \over {10}},etc.
q have only powers of 2 or powers of  5 or both.


7. Write three mumbers whose decimal expansions are non-terminating  non-recurring.

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Sol. (i) 2.01011011101111011111……  (ii) 0.03003000300003…….
(iii) 4.12112111211112

8. Find three different irrational numbers between the rational numbers  {5 \over 7}and{9 \over {11}}.

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Sol.    Given: {5 \over 7} =  0.7\overline {14285} and  {9 \over {11}}0.\overline {81}

We can have irrational numbers as .
0.72072007200072……;
0.801001800018……….;
0.74301010010001…….;


9. Classify the following numbers as rational or irrational:
(i) \sqrt {23}

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Ans. –  \sqrt {23}  As it is square root of a prime number, so, irrational number.

(ii) \sqrt {225}

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Ans. – \sqrt {225}  = 15, rational number.

(iii) 0.3796

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Ans. – 0.3796, terminating decimal, so rational number.

(iv) 7.478478…..

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Ans. – 7.478478…..= 7.\overline {478} , non-terminating repeating (recurring), so rational number.

(v)  1.101001000100001….

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Ans. –  1.101001000100001……. non – terminating non-repeating, so irrational number.

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