The following real numbers have decimal expansions as given below. In each case, decide

Solution:

(i) $43.123456789$

Since, the decimal expansion terminates, so the given real number is rational and therefore of the form $\frac{\mathbf{P}}{\mathbf{q}} \cdot 43.123456789$

$=\frac{43123456789}{1000000000}$

$=\frac{43123456789}{10^{9}}$

$=\frac{43123456789}{(2 \times 5)^{9}}$

$=\frac{43123456789}{2^{9} 5^{9}}$

Hence, $\mathrm{q}=2^{9} 5^{9}$

The prime factorization of $\mathrm{q}$ is of the form $2^{\mathrm{n}} 5^{\mathrm{m}}$, where $\mathrm{n}=9, \mathrm{~m}=9$.

(ii) $0.120120012000120000 \ldots$

Since, the decimal expansion is neither terminating nor non-terminating repeating, therefore, the given real number is not rational.

(iii) $43 . \overline{123456789}$

Since, the decimal expansion is non-terminating and repeating, therefore, the given real number is rational.

As the number is non-terminating so $\mathrm{q}$ is not of the form $2^{\mathrm{m}} \times 5^{\mathrm{n}}$.