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Exercise 1.3 - Chapter 1 Number Systems class 9 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Write the following in decimal form and say what kind of decimal expansion each has:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_cbf7830.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_6d6daa02.gif  (iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_3dea260f.gif

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_m755223f6.gif  (v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_m37f296eb.gif  (vi) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_38245c8d.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_m1d07425c.gif

Terminating

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_3ae39919.gif  https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_1271ba11.gif

Non-terminating repeating

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_21938256.gif

Terminating

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_5057e2f0.gif  https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_6fa31d44.gif

Non-terminating repeating

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_m59d445c5.gif https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_ma4c8715.gif

Non-terminating repeating

(vi) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1588/Chapter%201_html_m5c17f564.gif

Terminating

 

Question 2:

You know thathttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1590/Chapter%201_html_3dcff47e.gif . Can you predict what the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1590/Chapter%201_html_m4abcd010.gif  are, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1590/Chapter%201_html_144fa404.gif  carefully.]

Answer:

Yes. It can be done as follows.

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1590/Chapter%201_html_5b00254.gif

 

Question 3:

Express the following in the formhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_65af025c.gif , where p and q are integers and q ≠ 0.

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_26af32c7.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_18e1e719.gif  (iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_13d474b7.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_755a107d.gif

Let x = 0.666…

10x = 6.666…

10= 6 + x

9x = 6

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_78f36f24.gif

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_35961f00.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_m19b1762.gif

Let x = 0.777…

10x = 7.777…

10= 7 + x

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_m7072fe0c.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_16bbbda.gif

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_6dbaec94.gif

Let x = 0.001001…

1000x = 1.001001…

1000x = 1 + x

999x = 1

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1593/Chapter%201_html_mab70c11.gif

 

Question 4:

Express 0.99999…in the formhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1596/Chapter%201_html_65af025c.gif . Are you surprised by your Answer? With your teacher and classmates discuss why the Answer makes sense.

Answer:

Let x = 0.9999…

10x = 9.9999…

10x = 9 + x

9x = 9

x = 1

 

Question 5:

What can the maximum number of digits be in the repeating block of digits in the decimal expansion ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1598/Chapter%201_html_7ff9823d.gif ? Perform the division to check your Answer.

Answer:

It can be observed that,

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1598/Chapter%201_html_126272b2.gif

There are 16 digits in the repeating block of the decimal expansion ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1598/Chapter%201_html_7ff9823d.gif .

 

Question 6:

Look at several examples of rational numbers in the form https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1600/Chapter%201_html_65af025c.gif (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 https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1600/Chapter%201_html_65af025c.gif  is either of 2, 4, 5, 8, 10, and so on…

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1600/Chapter%201_html_m3d57311b.gif

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.

 

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…

 

Question 8:

Find three different irrational numbers between the rational numbershttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1605/Chapter%201_html_m6fc434b6.gif  andhttps://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1605/Chapter%201_html_24c171ae.gif .

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1605/Chapter%201_html_4d6800a4.gif

3 irrational numbers are as follows.

0.73073007300073000073…

0.75075007500075000075…

0.79079007900079000079…

 

Question 9:

Classify the following numbers as rational or irrational:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_4679cb65.gif  (ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_m23601954.gif  (iii) 0.3796

(iv) 7.478478 (v) 1.101001000100001…

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_2ca61f38.gif

As the decimal expansion of this number is non-terminating non-recurring, therefore, it is an irrational number.

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_4f7b2a48.gif

It is a rational number as it can be represented in https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_65af025c.gif  form.

(iii) 0.3796

As the decimal expansion of this number is terminating, therefore, it is a rational number.

(iv) 7.478478 … https://img-nm.mnimgs.com/img/study_content/curr/1/9/7/98/1608/Chapter%201_html_m4d4763fd.gif

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.

Also Read : Exercise-1.4-Chapter-1-Number-Systems-class-9-ncert-solutions-Maths

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