Chapter 01 Number Systems Exercise-03
EXERCISE 1.3
1. Write the following in decimal form and say what kind of decimal expansion each has :
(i) $\frac{36}{100}$ (ii) $\frac{1}{11}$ (iii) $4 \frac{1}{8}$
(iv) $\frac{3}{13}$ (v) $\frac{2}{11}$ (vi) $\frac{329}{400}$
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Solution
(i) $\frac{36}{100}=0.36$
Terminating
(ii) $\frac{1}{11}=0.090909 \ldots \ldots=0 . \overline{09}$
Non-terminating repeating
(iii) $4 \frac{1}{8}=\frac{33}{8}=4.125$
Terminating
(iv) $\frac{3}{13}=0.230769230769 \ldots=0 . \overline{230769}$
Non-terminating repeating
(v) $\frac{2}{11}=0.18181818$ $=0 . \overline{18}$
Non-terminating repeating
(vi) $\frac{329}{400}=0.8225$
Terminating
2. You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}$, $\frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how?
[Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
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Solution
Yes. It can be done as follows.
$ \begin{aligned} & \frac{2}{7}=2 \times \frac{1}{7}=2 \times 0 . \overline{142857}=0 . \overline{285714} \\ & \frac{3}{7}=3 \times \frac{1}{7}=3 \times 0 . \overline{142857}=0 . \overline{428571} \quad 10 x=6+x \\ & \frac{4}{7}=4 \times \frac{1}{7}=4 \times 0 . \overline{142857}=0 . \overline{571428} \\ & \frac{5}{7}=5 \times \frac{1}{7}=5 \times 0 . \overline{142857}=0 . \overline{714285} \\ & \frac{6}{7}=6 \times \frac{1}{7}=6 \times 0 . \overline{142857}=0 . \overline{857142} \end{aligned} $
, where $p$ and $q$ are integers and $q$
$\neq 0$.
3. Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(i) $0 . \overline{6}$ (ii) $0.4 \overline{7}$ (iii) $0 . \overline{001}$
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Solution
(i)
$0 . \overline{6}=0.666 \ldots$
Let $x=0.666 \ldots$
$10 x=6.666 \ldots$
$999 x=1$
$x=\frac{1}{999}$
4. Express $0.99999 \ldots$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
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Solution
Let $x=0.9999 \ldots$
$10 x=9.9999 \ldots$
$10 x=9+x$
$9 x=9 x$
=1
5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$ ? Perform the division to check your answer.
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Solution
It can be observed that,
$\frac{1}{17}=0 . \overline{0588235294117647}$
There are 16 digits in the repeating block of the decimal expansion of $\frac{1}{17}$.
6. Look at several examples of rational numbers in the form $\frac{p}{q}(q \neq 0)$, 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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Solution
Terminating decimal expansion will occur when denominator $q$ of rational number $\frac{p}{q}$ is either of $2,4,5,8,10$, and so on…
$\frac{9}{4}=2.25$
$\frac{11}{8}=1.375$
$\frac{27}{5}=5.4$
It can be observed that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.
7. Write three numbers whose decimal expansions are non-terminating non-recurring.
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Solution
3 numbers whose decimal expansions are non-terminating non-recurring are as follows.
$0.505005000500005000005 \ldots$
$0.7207200720007200007200000 \ldots 0.080080008000080000080000008 \ldots$
8. Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
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Solution
$ \frac{5}{7}=0 . \overline{714285} $
$\frac{9}{11}=0 . \overline{81}$
3 irrational numbers are as follows.
$0.73073007300073000073 \ldots$
$0.75075007500075000075 \ldots \quad 0.79079007900079000079 \ldots$
9. Classify the following numbers as rational or irrational :
(i) $\sqrt{23}$ (ii) $\sqrt{225}$ (iii) 0.3796 (iv) $7.478478 \ldots$ (v) $1.101001000100001 \ldots$
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Solution
As the decimal expansion of this number is non-terminating non-recurring, therefore, it
is an irrational number.
(ii)
$ \sqrt{225}=15=\frac{15}{1} $
It is a rational number as it can be represented in $\frac{p}{q}$ form.
(iii) 0.3796
As the decimal expansion of this number is terminating, therefore, it is a rational number.
(iv) $7.478478 \ldots=7 . \overline{478}$
As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.
(v) $1.10100100010000 \ldots$
As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.
