# 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 2^{n}5^{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 2^{n}5^{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 = 5Ã—5Ã—5 = 5

^{3}As denominator is in the form 2

^{n}5^{m}only where n=0 and m=3.According to

Theorem 1.6,will have a

terminatingdecimal expansion.

(ii)By doing prime factorization of denominator, we get

8 = 2Ã—2Ã—2 = 2

^{3}As denominator is in the form 2

^{n}5^{m}only where n=3 and m=0.According to

Theorem 1.6,will have a

terminatingdecimal expansion.

(iii)By doing prime factorization of denominator, we get

455 = 5Ã—7Ã—13

As denominator is not in the form 2

^{n}5^{m}only.According to

contradiction of Theorem 1.6,will have a

non-terminatingdecimal expansion.

(iv)By doing prime factorization of denominator, we get

1600 = 2Ã—2Ã—2Ã—2Ã—2Ã—2Ã—5Ã—5 = 2

^{6}5^{2}As denominator is in the form 2

^{n}5^{m}only where n=6 and m=2.According to

Theorem 1.6, 1will have a

terminatingdecimal expansion.

(v)By doing prime factorization of denominator, we get

343 = 7Ã—7Ã—7 = 7

^{3}As denominator is not in the form 2

^{n}5^{m }only.According to

contradiction of Theorem 1.6,will have a

non-terminatingdecimal expansion.

(vi)Prime factorization of denominator, we have

= 2

^{3}5^{2}As denominator is in the form 2

^{n}5^{m}only where n=3 and m=2.According to

Theorem 1.6,will have a

terminatingdecimal expansion.

(vii)Prime factorization of denominator, we have

= 2

^{2}5^{7}7^{5}As denominator is not in the form 2

^{n}5^{m}only.According to

contradiction of Theorem 1.6,will have a

non-terminatingdecimal expansion.

(viii)by doing prime factorization of denominator, we get

5 = 5

^{1}As denominator is in the form 2

^{n}5^{m}only where n=0 and m=1.According to

Theorem 1.6,will have a

terminatingdecimal expansion.

(ix)by doing prime factorization of denominator, we get

50= 2Ã—5Ã—5 = 2

^{1}5^{2}As denominator is in the form 2

^{n}5^{m}where n=1 and m=2.According to

Theorem 1.6,will have a

terminatingdecimal expansion.

(x)by doing prime factorization of denominator, we get

210 = 2Ã—3Ã—5Ã—7

As denominator is not in the form 2

^{n}5^{m}only.According to

contradiction of Theorem 1.6,will have a

non-terminatingdecimal 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)43.123456789As 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 2^{n}5^{m}, where n, m are non-negative integers.= 43123456789 / 10

^{9}= 43123456789 / 2

^{9}Ã— 5^{9}

(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 2^{n}5^{m}, where n, m are non-negative integers.

(iii)43.[Tex] [/Tex]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 2^{n}5^{m }only, where n, m are non-negative integers

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