# 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