NCERT Solutions: Real Numbers (Exercise 1.4)

# Real Numbers (Exercise 1.4) NCERT Solutions - Mathematics (Maths) Class 10

Q.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/(2352
(vii) 129/(225775
(viii) 6/15
(ix) 35/50
(x) 77/210
Solutions:
Note: If the denominator has only factors of 2 and 5 or in the form of 2m × 5n then it has terminating decimal expansion.
If the denominator has factors other than 2 and 5 then it has a non-terminating decimal expansion.
(i) 13/3125
Factorizing the denominator, we get,
3125 = 5 × 5 × 5 = 55
Since, the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion.
(ii) 17/8
Factorizing the denominator, we get,
8 = 2 × 2 × 2 = 23

Since, the denominator has only 2 as its factor, 17/8 has a terminating decimal expansion.
(iii) 64/455
Factorizing the denominator, we get,
455 = 5 × 7 × 13
Since, the denominator is not in the form of 2m × 5n, thus 64/455 has a non-terminating decimal expansion.
(iv) 15/ 1600
Factorizing the denominator, we get,
1600 = 2652
Since, the denominator is in the form of 2m × 5n, thus 15/1600 has a terminating decimal expansion.
(v) 29/343
Factorizing the denominator, we get,
343 = 7 × 7 × 7 = 73 Since, the denominator is not in the form of 2m × 5n thus 29/343 has a non-terminating decimal expansion.
(vi) 23/(2352)
Clearly, the denominator is in the form of 2m × 5n.
Hence, 23/ (2352) has a terminating decimal expansion.
(vii) 129/(225775)
As you can see, the denominator is not in the form of 2m × 5n.
Hence, 129/ (225775) has a non-terminating decimal expansion.
(viii) 6/15
6/15 = 2/5
Since, the denominator has only 5 as its factor, thus, 6/15 has a terminating decimal expansion.
(ix) 35/50
35/50 = 7/10
Factorising the denominator, we get,
10 = 2 5
Since, the denominator is in the form of 2m × 5n thus, 35/50 has a terminating decimal expansion.
(x) 77/210
77/210 = (7× 11)/ (30 × 7) = 11/30
Factorising the denominator, we get,
30 = 2 × 3 × 5
As you can see, the denominator is not in the form of 2m × 5n .Hence, 77/210 has a non-terminating decimal expansion.

Q.2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Solutions:
(i) 13/3125

13/3125 = 0.00416

(ii) 17/8

17/8 = 2.125

(iii) 64/455 has a Non terminating decimal expansion

(iv)15/ 1600

15/1600 = 0.009375

(v) 29/ 343 has a Non terminating decimal expansion

(vi)23/ (2352) = 23/(8×25)= 23/200

23/ (2352) = 0.115

(vii) 129/ (225775) has a Non terminating decimal expansion

(viii) 6/15 = 2/5

(ix) 35/50 = 7/10

35/50 = 0.7

(x) 77/210 has a non-terminating decimal expansion.

Q.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 what can you say about the prime factors of q?
(i) 43.123456789
(ii) 0.120120012000120000. . .

Solutions:
(i) 43.123456789
Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.

(ii) 0.120120012000120000. . .

Since, it has non-terminating and non- repeating decimal expansion, it is an irrational number.

Since it has non-terminating but repeating decimal expansion, it is a rational number in the form of p/q and q has factors other than 2 and 5.

The document Real Numbers (Exercise 1.4) NCERT Solutions | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## Mathematics (Maths) Class 10

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## FAQs on Real Numbers (Exercise 1.4) NCERT Solutions - Mathematics (Maths) Class 10

 1. What are real numbers?
Ans. Real numbers are the set of all rational and irrational numbers. They include numbers such as integers, fractions, decimals, square roots, and pi.
 2. What is the difference between rational and irrational numbers?
Ans. Rational numbers are numbers that can be expressed as a fraction, where both the numerator and denominator are integers. Irrational numbers are numbers that cannot be expressed as a fraction and have an infinite number of non-repeating decimals.
 3. How do you determine if a number is irrational?
Ans. A number is irrational if it cannot be expressed as a fraction and has an infinite number of non-repeating decimals. One way to determine if a number is irrational is to try and express it as a fraction. If the number cannot be expressed as a fraction, then it is irrational.
 4. What is the significance of real numbers in mathematics?
Ans. Real numbers are the foundation of mathematics and are used in various mathematical concepts, including algebra, geometry, and calculus. They are used to represent and solve real-world problems, such as distance, time, and money.
 5. Can real numbers be negative?
Ans. Yes, real numbers can be negative. Real numbers include all rational and irrational numbers, which can be either positive or negative. Examples of negative real numbers include -3, -2.5, and -√2.

## Mathematics (Maths) Class 10

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