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Real numbers Exercise 1.4

1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a nonterminating repeating decimal expansion: Solution:

(i) 13/3125

13/3125 = 13/55

The denominator is of the form 55.

Hence, the decimal expansion of is 13/3125 terminating.

(ii) 17/8

17/8 = 17/23

The denominator is of the form 23.

Hence, the decimal expansion of is 17/8 terminating.

(iii) 64/455

64/455 = 64/(5 × 7 × 13)

Since the denominator is not in the form 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv) 15/1600

15/1600 = 15/(26 × 52)

The denominator is of the form 2m × 5n.

Hence, the decimal expansion of 15/1600 is terminating.

(v) 29/343

29/343 = 29/73

Since the denominator is not in the form 2m × 5n, and it contains 7as its factors, its decimal expansion will be non-terminating repeating.

(v) 23/(23 x 52)

The denominator is of the form 2m × 5n. Hence, the decimal expansion of 23/(23 x 52) is terminating

(vi) 129/(22x57x75)

Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of 129/(22x57x75) is non-terminating repeating.

(vii)6/15 = (2 x 3)/(3 x 5)

The denominator is of the form 5n. Hence, the decimal expansion of 6/15 is terminating.

(viii) 35/50 = (5 x 7)/(5 x 10) = 7/(2 x 5)

The denominator is of the form 2m × 5n. Hence, the decimal expansion of 35/50 is terminating.

(ix) 77/210 = (11 x 7)/(30 x 7) = 11/30 = 11/(3 x 5 x 2)

Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors, the decimal expansion of 77/210 is non-terminating repeating.

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

Solution:

(i) 13/3125 Therefore, 13/3125 = 0.00416

(ii) 17/8 Therefore, 17/8 = 2.125

(iii) 15/1600 Therefore, 15/1600 = 0.009375

(iv) 23/200 Therefore, 23/200 = 0.115

(v) 2/5 Therefore, 2/5 = 0.4

(vi) 35/50 Therefore, 35/50 = 0.7

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 factor of q?

(i) 43.123456789

(ii) 0.120120012000120000…

(iii) 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 × 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) 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 × 5n i.e., the prime factors of q will also have a factor other than 2 or 5.