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 pq, what can you say about the prime factors of q?(i) 43.123456789 (ii) 0.120120012000120000. . . (iii) 43.123456789¯

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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 $\frac{p}{q}$ , what can you say about the prime factors of q?
(i) $43.123456789$ (ii) $0.120120012000120000 . . .$ (iii) $43 . \bar{123456789}$

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Detailed Solution

(i) Since the given number has a terminating decimal expansion, it is a rational number of the form of $\frac{p}{q}$ and q will have factors of 2 and 5 only.

(ii) Since the given number has a non-terminating decimal expansion, it is an irrational number.

(iii) Since the given number has a non-terminating but repeating decimal expansion, it is a rational number of the form of $\frac{p}{q}$ and q will have factors of 2 and 5 only.

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