Saturday, December 15, 2018

Real Numbers Exercise 1.4

Real Numbers Exercise 1.4

 

Page No: 17

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) 17/8 

(iii) 64/455 

(iv) 15/1600 

(v) 29/343 

(vi) 23/2× 52 

(vii) 129/2× 5× 75 

(viii) 6/15 

(ix) 35/50 

(x) 77/210

Answer

(i)    13/3125
Factorize the denominator we get
3125   =5 × 5 × 5 × 5 × 5      = 55

So denominator is in form of 5m so it is terminating .

(ii)  17/8         
Factorize the denominator we get
8          =2 × 2 × 2                   = 23
So denominator is in form of 2m so it is terminating .

(iii) 64/455     
Factorize the denominator we get
455     =5 × 7 × 13 
There are 7 and 13 also in denominator so denominator is not in form of 2m × 5n . so it is not terminating.

(iv) 15/1600   
Factorize the denominator we get
1600   =2 × 2 × 2 ×2 × 2 × 2 × 5 × 5  = 26 × 52
so denominator is in form of 2m × 5n
Hence it is terminating.

(v)  29/343     
Factorize the denominator we get
343     = 7 × 7 × 7                  = 73
There are 7 also in denominator so denominator is not in form of 2m × 5n
Hence it is non-terminating.

(vi) 23/(23 × 52)     
Denominator is in form of 2m × 5n
Hence it is terminating.

(vii)      129/(22 × 57 × 75  )
Denominator has 7 in denominator so denominator is not in form of 2m × 5n
Hence it is none terminating.

(viii)    6/15 
divide nominator and denominator both by 3 we get 2/5
Denominator is in form of 5m so it is terminating.

(ix)   35/50  divide denominator and nominator both by 5 we get 7/10
Factorize the denominator we get
10=2 × 5
So denominator is in form of 2m × 5n so it is terminating.

(x)        77/210
simplify it by dividing nominator and denominator both by 7 we get 11/30      
Factorize the denominator we get
30=2 × 3 × 5 
Denominator has 3 also in denominator so denominator is not in form of 2m × 5n

Hence it is none terminating.

Page No: 18

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

Answer

(i) 13/3125 = 13/55 = 13×25/55×25 = 416/105 = 0.00416

(ii) 17/8 = 17/23 = 17×53/23×53 = 17×53/103 = 2125/103 = 2.125

(iv) 15/1600 = 15/24×102 = 15×54/24×54×102 = 9375/106 = 0.009375

(vi) 23/2352 = 23×53×22/23 52×53×22 = 11500/105 = 0.115

(viii) 6/15 = 2/5 = 2×2/5×2 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7.

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 p , q you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000...

(iii) 43.123456789

Answer

(i) Since this number has a terminating decimal expansion, it is a rational number of the form p/q, and q is of the form 2m × 5n.

(ii) The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
(iii) Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q, and q is not of the form 2m × 5n.

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