Q.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 factors of q?

(i) \[\mathbf{43}.\mathbf{123456789}\]

(ii) \[\mathbf{0}.\mathbf{120120012000120000}\ldots \ldots \]

$\left( III \right)$ $43.\overline{123456789}$

(i) \[\mathbf{43}.\mathbf{123456789}\]

Since ,it has a terminating decimal expansion, Therefore  it is a rational number in the form of ${}^{p}/{}_{q}$  and q has factors of 2 and 5 only.

(ii) \[\mathbf{0}.\mathbf{120120012000120000}\ldots \ldots \]

Since, it has non-terminating and non- repeating decimal expansion, therefore it is an irrational number.

$\left( III \right)$ $43.\overline{123456789}$

Since ,it has non-terminating but repeating decimal expansion,Therefore it is a rational number in the form of ${}^{p}/{}_{q}$  and q has factors other than 2 and 5 .where factors of q are not of the form ${{2}^{n}}\times {{5}^{m}}$ where n , m are non-negative integers