Class 10 NCERT Solutions- Chapter 1 Real Numbers - Exercise 1.4

NCERT Theorem 1.5 : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form,  where p and q are coprime, and the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

NCERT Theorem 1.6 : Let x =  be a rational number, such that the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers. Then x has a decimal expansion which terminates. 

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) 

(ii)  

(iii) 

(iv) 

(v) 

(vi) 

(vii) 

(viii) 

(ix) 

(x) 

Solution:

(i) 

By doing prime factorization of denominator, we get

3125 = 5×5×5 = 5³

As denominator is in the form 2ⁿ5^m only where n=0 and m=3.

According to Theorem 1.6, 

will have a terminating decimal expansion.

(ii) 

By doing prime factorization of denominator, we get

8 = 2×2×2 = 2³

As denominator is in the form 2ⁿ5^m only where n=3 and m=0.

According to Theorem 1.6, 

will have a terminating decimal expansion.

(iii) 

By doing prime factorization of denominator, we get

455 = 5×7×13

As denominator is not in the form 2ⁿ5^m only.

According to contradiction of Theorem 1.6, 

will have a non-terminating decimal expansion.

(iv) 

By doing prime factorization of denominator, we get

1600 = 2×2×2×2×2×2×5×5 = 2⁶5²

As denominator is in the form 2ⁿ5^m only where n=6 and m=2.

According to Theorem 1.6, 1

will have a terminating decimal expansion.

(v) 

By doing prime factorization of denominator, we get

343 = 7×7×7 = 7³

As denominator is not in the form 2ⁿ5^m only.

According to contradiction of Theorem 1.6,

will have a non-terminating decimal expansion.

(vi) 

Prime factorization of denominator, we have

= 2³5²

As denominator is in the form 2ⁿ5^m only where n=3 and m=2.

According to Theorem 1.6, 

will have a terminating decimal expansion.

(vii) 

Prime factorization of denominator, we have

= 2²5⁷7⁵

As denominator is not in the form 2ⁿ5^m only.

According to contradiction of Theorem 1.6,

will have a non-terminating decimal expansion.

(viii) 

by doing prime factorization of denominator, we get

5 = 5¹

As denominator is in the form 2ⁿ5^m only where n=0 and m=1.

According to Theorem 1.6, 

will have a terminating decimal expansion.

(ix) 

by doing prime factorization of denominator, we get

50= 2×5×5 = 2¹5²

As denominator is in the form 2ⁿ5^m where n=1 and m=2.

According to Theorem 1.6,

will have a terminating decimal expansion.

(x) 

by doing prime factorization of denominator, we get

210 = 2×3×5×7

As denominator is not in the form 2ⁿ5^m only.

According to contradiction of Theorem 1.6,

will have a non-terminating decimal expansion.

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

(i) 

(ii) 

(iv) 

(vi) 

(viii) 

(ix) 

Solution:

(i)

= 0.00416

(ii)

= 2.125

(iv)

= 0.009375

(vi)

= 0.0115

(viii)

= 0.4

(ix)

= 0.7

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,  what can you say about the prime factors of q?

(i) 43.123456789 

(ii) 0.120120012000120000. . . 

(iii) 43.123456789

Solution:

(i) 43.123456789

As this is a rational number whose decimal expansion terminates. Then it can be expressed in the form,  where p and q are coprime, and the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

= 43123456789 / 10⁹

= 43123456789 / 2⁹ × 5⁹

(ii) 0.120120012000120000..............

As given decimal number expansion is non-terminating and non-repeating, then it is not a rational number. Then it can't be expressed in the form, where p and q are coprime, and the prime factorization of q is of the form 2ⁿ5^m, where n, m are non-negative integers.

(iii) 43.

As given decimal number expansion is non-terminating and repeating, then it is a rational number. Then it can be expressed in the form, where p and q are coprime, but the prime factorization of q is not in the form of 2ⁿ5^m only, where n, m are non-negative integers

Summary

Exercise 1.4 of NCERT Class 10 Chapter 1 - Real Numbers focuses on revisiting rational numbers and their decimal expansions, as well as introducing irrational numbers. It reinforces the concept that rational numbers have either terminating or non-terminating repeating decimal expansions, while irrational numbers have non-terminating, non-repeating decimal expansions. The exercise also covers operations with irrational numbers, proving irrationality, and understanding the properties of rational and irrational numbers on the number line.