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

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 $\frac{\mathbf{P}}{\mathbf{q}}$, what can you say about the prime factors of $\mathrm{g}$ ?

(i) $43.123456789$

(ii) $0.120120012000120000 \ldots$

(iii) $43 . \overline{23456789}$

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 $2^{\mathrm{m}} \times 5^{\mathrm{n}}$.