Chapter 1: Real Numbers Exercise – 1.6

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)  23/8

(ii) 125/441

(iii) 35/50

 (iv) 77/210

Solution:

(i) The given number is 23/8

Here, 8 = 2³ and 2 is not a factor of 23.

So, the given number is in its simplest form.

Now, 8 = 2³ is of the form 2^m × 5ⁿ, where m = 3 and n = 0.

So, the given number has a terminating decimal expansion.

(ii) The given number is 125/441

Here, 441 = 3² × 7² and none of 3 and 7 is a factor of 125.

So, the given number is in its simplest form.

Now, 441 = 3² × 7² is not of the form 2^m × 5ⁿ

So, the given number has a non-terminating repeating decimal expansion.

(iii) The given number is 35/50 and HCF (35, 50) = 5.

Here, 7/10 is in its simplest form.

Now, 10 = 2 × 5 is of the form 2^m × 5ⁿ, where in = 1 and n = 1.

So, the given number has a terminating decimal expansion.

(iv) The given number is 77/210 and HCF (77, 210) = 7.

Here, 11/30 is in its simplest form. 30

Now, 30 = 2 × 3 × 5 is not of the form 2^m × 5ⁿ.

So, the given number has a non-terminating repeating decimal expansion.

(v) The given number isClearly, none of 2, 5 and 7 is a factor of 129.

So, the given number is in its simplest form.

Question: 2

Write down the decimal expansions of the following rational numbers by writing their denominators in the form of 2^m × 5ⁿ, where m, and n, are the non- negative integers.

(i)  3/8

(ii)  13/125

(iii)  7/80

(iv) 14588/625

Solution:

(i) The given number is 3/8

Clearly, 8 = 2³ is of the form 2^m × 5ⁿ, where m = 3 and n = 0.

So, the given number has terminating decimal expansion.

(ii) The given number is 13/125

Clearly, 125 = 5³ is of the form 2m × 5″, where m = 0 and n = 3.

So, the given number has terminating decimal expansion.

(iii) The given number is 7/80.

Clearly, 80 = 2⁴ × 5 is of the form 2^m × 5ⁿ, where m = 4 and n = 1.

So, the given number has terminating decimal expansion.

(iv) The given number is 14588/625

Clearly, 625 = 5⁴ is of the form 2^m × 5ⁿ, where m = 0 and n = 4.

So, the given number has terminating decimal expansion?

(v) The given number is  Clearly, 2² × 5⁷ is of the form 2^m × 5ⁿ, where in = 2 and n = 7.

So, the given number has terminating decimal expansion?

Question: 3

What can you say about the prime factorization of the denominators of the following rational?

(i) 43.123456789

(iv) 0.120120012000120000

Solution:

(i) Since 43.123456789 has terminating decimal expansion. So, its denominator is of the form 2^m × 5ⁿ, where m, n are non-negative integers.

(ii) Since has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iii) Since has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.

(iv) Since 0.120120012000120000 … has non-terminating decimal expansion. So, its denominator has factors other than 2 or 5.