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Exercise - 1.4

Real Numbers

**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) 13/3125, (ii) 17/8, (iii) 64/455, (iv) 15/1600, (v) 29/343, (vi) 23/2³5², (vii) 129/2²5^{7}7^{5}, (viii) 6/15, (ix) 35/50, (x) 77/210

(i) 13/3125 3125 = 5^{5}The denominator is of the form 5^{m}. Hence, the decimal expansion of 13/3125 is terminating.

(ii) 17/8 17/8 = 2³ The denominator is of the form 2^{m}. Hence, the decimal expansion of 17/8 is terminating.

(iii) 64/455 455 = 5 x 7 x 13 Since the denominator is not in the form 2^{m}× 5^{n}, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.

(iv) 15/1600 1600 = 2^{6}× 5² The denominator is of the form 2^{m}× 5^{n}. Hence, the decimal expansion of 15/1600 is terminating.

(v) 29/343 343 = 7³ Since the denominator is not in the form 2^{m}× 5^{n}, and it also contains 7 as its factors, its decimal expansion will be non-terminating repeating.

(vi) 23/2³5² The denominator is of the form 2^{m}× 5^{n}. Hence, the decimal expansion of 23/2³5² is terminating.

(vii) 129/2²5^{7}7^{5}Since the denominator is not in the form 2^{m}× 5^{n}, and it also contains 7 as its factors, its decimal expansion will be non-terminating repeating.

(viii) 6/15 6/15 = (2 x 3)/(3 x 5) = 2/5 The denominator is of the form 5^{m}. Hence, the decimal expansion of 6/15 is terminating.

(ix) 35/50 35/50 = (5 x 7)/(2 x 5 x 5) = 7/(2 x 5) The denominator is of the form 2^{m}× 5^{n}. Hence, the decimal expansion of 35/50 is terminating.

(x) 77/210 77/210 = (11 x 7)/(30 x 7) = 11/30 = 11/(2 x 3 x 5) Since the denominator is not in the form 2^{m}× 5^{n}, and it also contains 3 as its factors, its decimal expansion will be non-terminating repeating.

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

(i) 13/3125, (ii) 17/8, (iv) 15/1600, (vi) 23/2³5², (viii) 6/15, (ix) 35/50

**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 factor of q?
(i) 43.123456789, (ii) 0.120120012000120000…, (iii) 43.123456789

(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 2^{m}× 5^{n}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) 43.123456789 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 2^{m}× 5^{n}i.e., the prime factors of q will also have a factor other than 2 or 5.

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