NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4 are part of NCERT Solutions for Class 10 Maths. Here are we have given Chapter 1 Real Numbers Class 10 NCERT Solutions Ex 1.4.
- Real Numbers Class 10 Ex 1.1
- Real Numbers Class 10 Ex 1.2
- Real Numbers Class 10 Ex 1.3
- Real Numbers Class 10 Ex 1.4
| Board | CBSE |
| Textbook | NCERT |
| Class | Class 10 |
| Subject | Maths |
| Chapter | Chapter 1 |
| Chapter Name | Real Numbers |
| Exercise | Ex 1.4 |
| Number of Questions Solved | 3 |
| Category | NCERT Solutions |
NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Ex 1.4
Page No: 17
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:
(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
Solution:
3125 = 55
The denominator is of the form 5m.
8 = 23
The denominator is of the form 2m.
455 = 5 x 7 x 13
Since the denominator is not in the form 2m x 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.
1600 = 26 × 52
The denominator is of the form 2m x 5n.
343 = 73
Since the denominator is not in the form 2m x 5n, and it has 7 as its factor,
The denominator is of the form 2m x 5n. Hence, the decimal expansion of is 
terminating.
(vii)
Since the denominator is not of the form 2m 5n, and it also has 7 as its
The denominator is of the form 5n.
10 = 2 x 5
The denominator is of the form 2m x 5n.
30 = 2 x 3 x 5
Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors,
Concept Insight: The concept used in this problem is that The decimal expansion of rational number p/q where p and q are coprime numbers, terminates if and only if the prime factorization of q is of the form 2n5m, where n and m are non negative integers. Do not forget that 0 is also a non negative integer so n or m can take value 0.
Generally, mistake is committed in identifying terminating decimals when either of the two prime numbers 2 or 5 is appearing in the prime factorization.
Page No: 18
Question 2
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solution:
Concept Insight:
This is based on performing the long division and expressing the rational number in the decimal form learned in lower classes.
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 p, q you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000…
(iii) 43.123456789
Solution:
(i) 43.123456789
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 x 5n,
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) 43.123456789
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 x 5n i.e., the prime factors of q will also have a factor other than 2 or 5.
Concept Insight: The concept used in this problem is that, If the decimal expansion of rational number p\q, [where p and q are coprime numbers] terminates, then prime factorization of q is of the form 2n5m, where n and m are non negative integers.
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