Question

Express $0.\overline{001}$ in $\frac{p}{q}$ form

Answer 1

Expressing $0.\overline{001}$ in $\frac{p}{q}$ form:

Let $x=0.\overline{001}=0.001001001.......\left(i\right)$

Since the numbers are repeating after every $3$ digits, multiply $x$ by $1000$ (i.e., ${10}^3$)

$1000x=1.\overline{001}...\left(ii\right)$

Subtract equation (i) from equation (ii)

$$\begin{gathered} \Rightarrow 1000x-x=1.\overline{001}-0.\overline{001} \\ \Rightarrow 999x=1 \\ \Rightarrow x=\frac{1}{999} \end{gathered}$$

Thus, $\overline{0.001}$ can be expressed as $\frac{1}{999}$.