EXERCISE 1.3
EVIDYARTHI
Q 1 (i, ii, iii), Ex 1.3, Page No 14, Number Systems, Class 9 Maths Solutions
Q 1 (iv, v, vi), Ex 1.3, Page No 14, Number Systems, Class 9th Maths
MKR
Decimal Expansion of Real Numbers Class - 9th CBSE NCERT
Class - 9th, Ex - 1.3, Q 1 ( NUMBER SYSTEM ) CBSE NCERT
Page No 14:
Question 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) $\frac{329}{400}$
Answer:
(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) $\frac{2}{11}=0.18181818 \ldots \ldots .=0 . \overline{18}$
Non-terminating repeating
(vi) $\frac{329}{400}=0.8225$
Terminating
EVIDYARTHI
Q 2, Ex 1.3, Page No 14, Number Systems, CBSE Maths Class 9th
MKR
Class - 9th, Ex - 1.3, Q 2 ( NUMBER SYSTEM ) CBSE NCERT
Question 2:
You know that $\frac{1}{7}=0 . \overline{142857}$.Can you predict what the decimal expansion 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.]
Answer:
Yes. It can be done as follows.$\frac{2}{7}=2 \times \frac{1}{7}$ $=2 \times 0 . \overline{142857}=0 . \overline{285714}$
$\frac{3}{7}=3 \times \frac{1}{7}$ $=3 \times 0 . \overline{142857}=0 . \overline{428571}$
$\frac{4}{7}=4 \times \frac{1}{7}$ $=4 \times 0 . \overline{142857}=0 . \overline{571428}$
$\frac{5}{7}=5 \times \frac{1}{7}$ $=5 \times 0 . \overline{142857}=0 . \overline{714285}$
$\frac{6}{7}=6 \times \frac{1}{7}$ $=6 \times 0 . \overline{142857}=0 . \overline{857142}$
EVIDYARTHI
Q 3, Ex 1.3, Page No 14, Number Systems, Class 9th Maths
MKR
Class - 9th, Ex - 1.3, Q 3 ( NUMBER SYSTEM ) CBSE NCERT
Question 3:
Express the following in the form $\frac{p}{q}$, where p and q are integers and q ≠ 0.(i) $0 . \overline{6}$
(ii) $0.4 \overline{7}$
(iii) $0 . \overline{001}$
Answer:
(i) $0 . \overline{6}=0.666 \ldots$Let x = 0.666…
10x = 6.666…
10x = 6 + x
9x = 6
$x=\frac{2}{3}$
(ii) $0 . \overline{47}=0.4777 \ldots$
$=\frac{4}{10}+\frac{0.777}{10}$
Let x = 0.777…
10x = 7.777…
10x = 7 + x
$x=\frac{7}{9}$
$\frac{4}{10}+\frac{0.777 \ldots}{10}=\frac{4}{10}+\frac{7}{90}$
$=\frac{36+7}{90}=\frac{43}{90}$
(iii) $0 . \overline{001}=0.001001 \ldots$
Let x = 0.001001…
1000x = 1.001001…
1000x = 1 + x
999x = 1
$x=\frac{1}{999}$
EVIDYARTHI
Q 4, Ex 1.3, Page No 14, Number Systems, Maths CBSE Class 9th R D Sharma Solutions
MKR
Class - 9th, Ex - 1.3, Q 4 ( NUMBER SYSTEM ) CBSE NCERT
Question 4:
Answer:
Let x = 0.9999…10x = 9.9999…
10x = 9 + x
9x = 9
x = 1
EVIDYARTHI
Q 5, Ex 1.3, Page No 14, Number Systems, Mathematics Class 9th
MKRClass - 9th, Ex - 1.3, Q 5 ( NUMBER SYSTEM ) CBSE NCERT
Question 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.Answer:
It can be observed that,$\frac{1}{17}=0 . \overline{0588235294117647}$
There are 16 digits in the repeating block of the decimal expansion of $\frac{1}{17}$
MKR
Class - 9th, Ex - 1.3, Q 6 ( NUMBER SYSTEM ) CBSE NCERT
Question 6:
Look at several examples of rational numbers in the form $\frac{p}{q}$(q ≠ 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?Answer:
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.
EVIDYARTHI
Q 7, Ex 1.3,Page No 14, Number Systems, Class 9th Maths Solutions
MKR
Class - 9th, Ex - 1.3, Q 7 ( NUMBER SYSTEM ) CBSE NCERT
Question 7:
Write three numbers whose decimal expansions are non-terminating non-recurring.Answer:
3 numbers whose decimal expansions are non-terminating non-recurring are as follows.0.505005000500005000005…
0.7207200720007200007200000…
0.080080008000080000080000008…
EVIDYARTHI
Q 8, Ex 1.3, Page No 14, Number Systems, CBSE Class 9th Maths
MKR
Class - 9th, Ex - 1.3, Q 8 ( NUMBER SYSTEM ) CBSE NCERT
Question 8:
Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$Answer:
$\frac{5}{7}=0 . \overline{714285}$$\frac{9}{11}=0 . \overline{81}$
3 irrational numbers are as follows.
0.73073007300073000073…
0.75075007500075000075…
0.79079007900079000079…
EVIDYARTHI
Q 9, Ex 1.3, Page No 14, Number Systems, Class 9th CBSE MathsQuestion 9:
Classify the following numbers as rational or irrational:(i) $\sqrt{23}$
(ii) $\sqrt{225}$
(iii) 0.3796
(iv) 7.478478
(v) 1.101001000100001…
Answer:
(i) $\sqrt{23}=4.79583152331 \ldots$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 …$=7 . \overline{478}$
As the decimal expansion of this number is non-terminating recurring, therefore, it is a rational number.
(v) 1.10100100010000 …
As the decimal expansion of this number is non-terminating non-repeating, therefore, it is an irrational number.
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