The following real numbers have decimal expansions as given below.

Question : 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 \(p\over q\), what can you say about the prime factors of q ?

(i)  43.123456789

(ii)  0.120120012000120000…..

(iii)  43.\(\overline{123456789}\)

Solution :

(i)  43.123456789 is terminating.

So, it represents a rational number.

Thus, 43.123456789 = \(43123456789\over 1000000000\) = \(p\over q\). Thus, q = \(10^9\).

(ii)  0.120120012000120000….. is non-terminating and non-repeating. So, it is an irrational.

(iii)  43.123456789 is non-terminating but repeating. So, it is a rational.

Thus, 43.\(\overline{123456789}\) = \(4312345646\over 999999999\) = \(p\over q\)

Thus, q = 999999999

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