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Exercise 1.4 - Chapter 1 Real Numbers class 10 ncert solutions Maths PDF Download [2022-2023]


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Question 1:

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m55e2a861.gif

Answer:

(i) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_32d68b64.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_5e269fa3.gif

The denominator is of the form 5m.

Hence, the decimal expansion ofhttps://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_32d68b64.gifis terminating.

(ii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_7189a13.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_705a748f.gif

The denominator is of the form 2m.

Hence, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_7189a13.gifis terminating.

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_6968de0a.gif

455 = 5 × 7 × 13

Since the denominator is not in the form 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m563aad61.gif

1600 = 26 × 52

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m563aad61.gif is terminating.

(v) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_2602c67e.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_23050bd9.gif

Since the denominator is not in the form 2m × 5n, and it has 7 as its factor, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_2602c67e.gif is non-terminating repeating.

(vi) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m74b8081a.gif

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m74b8081a.gifis terminating.

(vii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m20e3d08a.gif

Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m20e3d08a.gif is non-terminating repeating.

(viii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_6918d3a9.gif

The denominator is of the form 5n.

Hence, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m6b3974a8.gif is terminating.

(ix) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m2588c91.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_106832cd.gif

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m35d4aadf.gif is terminating.

(x) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_m2b43cb05.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_60da22d2.gif

Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors, the decimal expansion of https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1885/Chapter%201_html_5054639e.gif is non-terminating repeating.

 

Question 2:

Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_m4c1e25e4.gif

 

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_72b2fa51.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_213f45ce.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_m583f7e24.gif

(viii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_m61a67e07.gif https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_m367b2519.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1892/Chapter%201_html_782abef1.gif

 

Question 3:

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_65af025c.gif , what can you say about the prime factor of q?

(i) 43.123456789 (ii) 0.120120012000120000… (iii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_76bf9085.gif

Answer:

(i) 43.123456789

Since this number has a terminating decimal expansion, it is a rational number of the formhttps://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_65af025c.gif and q is of the form https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_m7249bbc8.gif

i.e., the prime factors of q will be either 2 or 5 or both.

(ii) 0.120120012000120000 …

The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.

(iii) https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_76bf9085.gif

Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_65af025c.gif and q is not of the form https://img-nm.mnimgs.com/img/study_content/curr/1/10/9/128/1901/Chapter%201_html_m7249bbc8.gif i.e., the prime factors of q will also have a factor other than 2 or 5.

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