NCERT Solutions 10th Maths Chapter 1 Real Number Exercise 1.4

NCERT Solutions For Class 10 Maths Chapter 1 Exercise 1.4 we will restart our exploration of the world of real numbers. We will study Euclid’s division algorithm and the Fundamental Theorem of Arithmetic. Also, we will see decimal representation of real numbers.

Exercise 1.4
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) 23/23 × 52
(vii) 129/22 × 57 × 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.

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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.