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}=0.36Terminating
(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 \ldotsLet 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 \ldotsAs 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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