(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