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Exercise 1.3

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

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

(ii) ${1 \over {11}}$

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

(iii) $4{1 \over 8}$

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

(iv) ${3 \over {13}}$

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

(v) ${2 \over {11}}$

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

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

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.]

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) $0.\overline 6$

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) $0.4\overline 7$

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) $0.\overline {001}$

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.

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 ${1 \over {17}}$ ?

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?

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.

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

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

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) $\sqrt {23}$

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

(ii) $\sqrt {225}$

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

(iii) 0.3796

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

(iv) 7.478478…..

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