NCERT10th MathematicsExercise 1.4Topic: 1 Real Numbers
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EXERCISE 1.4
Q1. 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/(2352) (vii) 129/(225775) (viii) 6/15 (ix) 35/50 (x) 77/210
Explanation:
1) Let x = p/q be a rational number, such that the prime factorisation of q is of form 2n 5m, where n and m are non-negative integers. Then x has a decimal expansion which terminates.
Solution:
(i) 13/3125
1) Here the denominator is 3125.
2) Find the factors of 3125, we get,
3125 = 5 x 625
3125 = 5 x 5 x 1253125 = 5 x 5 x 5 x 253125 = 5 x 5 x 5 x 5 x 53125 = 55
3) Here our expression is 13/55, and the denominator is 2n x 5m where n=0 and m = 5, so the decimal expansion of 13/3125 is terminating.
(ii) 17/8
1) Here the denominator is 23, which is of the form 2n x 5m where n=3 and m = 0.
2) So the decimal expansion of 17/8 is terminating.
(iii) 64/455
1) Here the denominator is 455.
2) Find the factors of 455, we get,
455 = 5 x 91
455 = 5 x 7 x 13
3) Here our expression is 13/(5 x 7 x 13), and the denominator is (5 x 7 x 13), which is not of the form 2n 5m, so the decimal expansion of 64/455 is a non-terminating repeating decimal expansion.
(iv) 15/1600
1) Here the denominator is 1600.
2) Find the factors of 1600, we get,
1600 = 16 x 100
1600 = 16 x 4 x 25
1600 = 26 x 52
3) Here our expression is 15/(26 x 52) and the denominator is 2n x 5m where n=6 and m = 2, so the decimal expansion of 15/1600 is terminating.
(v) 29/343
1) Here the denominator can be written as 73.
2) The decimal expansion of 29/73, the denominator is (73), which is not of form 2n 5m, so the decimal expansion of 29/343 is a non-terminating repeating decimal expansion.
(vi) 23/(2352)
1) Here the denominator is of the form 2n x 5m where n=3 and m = 2.
2) The decimal expansion of 23/(2352) is terminating.
(vii) 129/(225775)
1) Here the denominator is not of the form 2n x 5m and has 75 as a factor.
2) So the decimal expansion of 129/(225775) is a non-terminating repeating decimal expansion.
(viii) 6/15
1) Here the denominator is 15.
15 = 3 x 5
2) Here the denominator is (3 x 5), which is not of form 2n 5m, so the decimal expansion of 6/15 is a non-terminating repeating decimal expansion.
(ix) 35/50
1) Here the denominator is 50.
2) Find the factors of 1600, we get,
50 = 2 x 25
50 = 21 x 52
3) Here our expression is 35/(21 x 52) and the denominator is of the form 2n x 5m where n=1 and m = 2, so the decimal expansion of 35/50 is terminating.
(x) 77/210
1) Here the denominator is 210.
210 = 21 x 10
210 = 3 x 7 x 2 x 5
2) Here the denominator is (3 x 7 x 2 x 5), which is not of the form 2n 5m, so the decimal expansion of 77/210 is a non-terminating repeating decimal expansion.
Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solution:
(i) 13/3125
13/3125 = (13 x 2)/(3125 x 2)
= (26)/(6250)= (2.6)/(625)= (2.6 x 2)/(625 x 2)= (5.2)/(1250)= (0.52)/(125)= (0.52 x 2)/(125 x 2)= (1.04)/(250)= (0.104)/(25)= (0.104 x 4)/(25 x 4)= (0.416)/(100)= (0.00416)
So 13/3125 = 0.00416
(ii) 17/8
(iii) 64/455. It is non-terminating.
(iv) 15/1600
15/1600 = (15)/(16 x 100)
= (0.15)/(16)= (0.15 x 5)/(16 x 5)= (0.75)/(80)= (0.075)/(8)= (0.0375)/(4)= (0.01875)/(2)= (0.009375)
So 15/1600 = 0.009375
(v) 29/343. It is non-terminating.
(vi) 23/(2352)
23/(2352) = (23)/(212252)
= (23)/(2 x 102)= (23)/(2 x 100)
= (11.5)/(100)= (0.115)
So 23/(2352) = 0.115
(vii) 129/(225775). It is non-terminating.
(viii) 6/15
6/15 = (6)/(15)
= (2)/(5)= (2 x 2)/(5 x 2)= (4)/(10)= (0.4)
So 6/15 = 0.4
(ix) 35/50
35/50 = (35 x 2)/(50 x 2)
= (70)/(100)
= (0.7)
So 35/50 = 0.7
(x) 77/210. It is non-terminating.
Q3. 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 what can you say about the prime factors of q?
Explanation:
1) Let x = p/q be a rational number, such that the prime factorisation of q is of form 2n 5m, where n and m are non-negative integers. Then x has a decimal expansion which terminates.
Solution:
(i) 43.123456789
1) 43.123456789 = (43123456789)/(109)
= (43123456789)/(29 x 59)
2) Here the denominator is of the form 2n 5m, so the number is rational.
(ii) 0.120120012000120000
1) Here the decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
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(iii) 43.123456789
1) Here the decimal expansion is non-terminating and recurring, so the given number is a rational number of the form p/q where q is not of form 2n 5m. The prime factors of q will also have a factor other than 2 or 5.
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