Ex 1.4 NCERT Solutions - Real Numbers Class 10 Notes | EduRev

Exercise 1.4
Ques 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:
(i) 13/ 3125      
(ii) 17/18        
(iii) 64/455       
(iv) 15/1600       
(v) 29/343  
(vi) 23/2³5³       
(vii) 129/2²5⁷7⁵        
(viii) 16/15     
(ix) 35/50        
(x)  77/ 210 
Sol: 
(i) 13/3125  
∵  3125 =  5 × 5 × 5 × 5 × 5 = 5⁵
=  2⁰ × 5⁵,    |∵ 2⁰ = 1
which is of the form (2ⁿ · 5^m)
∴ 13/3125 is a terminating decimal.
i.e., 13/3125 will have a terminating decimal expansion.

(ii) 17/8  
∵ 8 = 2 × 2 × 2 = 2³ = 1 × 2³
= 5⁰ × 2³, which is of the form 5^m · 2ⁿ
∴ 17/8 will have a terminating decimal expansion.

(iii) 64/455
∵ 455 = 5 × 7 × 13, which is not of the form 2ⁿ · 5^m
∴ 64/455 will have a non-terminating repeating decimal expansion.

(iv) 15/1600
∵  1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
= 2⁶ × 5², which is of the form 2ⁿ · 5^m
∴ 15/1600 will have a terminating decimal expansion.

(v) 29/343  
∵ 343 = 7 × 7 × 7 = 73, which is not of the form 2ⁿ · 5^m.
∴ 29/343 will have a non-terminating repeating decimal expansion.

(vi)  23/2³5² 
Let p/q = 23/2³5²  
i.e., q =  2³ · 5², which is of the form 2n · 5m.
∴  23/2³5² will have a terminating decimal expansion.

(vii) 129/2²5⁷7⁵   
Let p/q = 129/2²5⁷7⁵  
i.e., q = 2² · 5⁷ · 7⁵, which is not of the form 2ⁿ · 5^m.
∴ 129/2²5⁷7⁵   will have a non-terminating repeating decimal expansion.

(viii) 6/15   
∵  =   which is of the form 2^m· 5ⁿ.
∴  6/15 will have a terminating decimal expansion.

(ix) 35/50   
∵ 50 = 2 × 5 × 5 = 21 × 52, which is of the form 2ⁿ · 5^m.
∴ 35/50 will have a terminating decimal expansion.

(x) 77/210    
∵  210 =  2 × 3 × 5 × 7 = 2¹ · 3¹ · 5¹ · 7¹,
which is not of the form of 2ⁿ · 5^m.
∴  77/ 210 will have a non-terminating repeating decimal expansion.

Fig: Decimal expression of rational numbers

Ques 2: Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
(i)  13/ 3125      
(ii) 17/18        
(iii) 64/455       
(iv) 15/1600       
(v) 29/343   
(vi)       
(vii)         
(viii) 6/15      
(ix) 35/50        
(x) 77/ 210 
Sol: 
(i) 13/3125  
We have
Multiplying and dividing both sides, by 2⁵, we have:

(ii) 17/18   

Multiplying and dividing by 5³, we have

(iii) 64/455 represents non-terminating repeating decimal expansion.

(iv) 15/1600

Multiplying and dividing by 5⁴, we have

(v) 29/343 , represents a non-terminating repeating decimal expansion.

(vi) 

=   |Multiplying and dividing by 5. =  = 0.115

(vii)  represents a non-terminating repeating decimal expansion.

(viii) 6/15  
6/15 =  2/5
 |Multiplying and dividing by 2,

= 4/10   = 0.4

(ix) 35/50
35/50 = 0.7

(x) 77/ 210 , represents a non-terminating repeating decimal expansion.

Ques 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, what can you say about the prime factors of q?
(i) 43.123456789    
(ii) 0.120120012000120000...    
(iii)    
Sol: 
(i) 43.123456789
∵ The given decimal expansion terminates.
∴ It is a rational of the form p/q
⇒ p/q = 43.123456789

Here, p  = 43.123456789  and
q = 2⁹ × 5⁹
∴ Prime factors of q are 2⁹ and 5⁹.

(ii) 0.120120012000120000 .....
∵ The given decimal expansion is neither terminating nor non-terminating repeating,
∴ It is not a rational number.

(iii)   
∵ The given decimal expansion is non-terminating repeating.
∴ It is a rational number.
Let p/q = x = 43.123456789 ......(1)
Multiplying both sides by 1000000000, we have
1000000000 x = 43.123456789  ...(2)
Subtracting (1) from (2), we have:
(1000000000 x) – x = (43.123456789) – 43.123456789
⇒ 999999999 x  = 43.123456746

           
Here, p =  4791495194
and  q  = 111111111, which is not of form 2ⁿ· 5^m
i.e., the prime factors of q are not of form 2ⁿ· 5^m.