Question

Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and q ≠ 0.
$0.\overline{001}$

Answer 1

Let $x=0.\overline{001}$

i.e. $x=0.001001001..$(1)

Multiplying (1) by $1000$, we get

$1000x=1.001001\dots$(2)

Subtracting (1) from (2), we get

$1000x–x=(1.001001\dots )–(0.001001\dots )$

$\Rightarrow 999x=1$

$\Rightarrow x=\frac{1}{999}$

Thus $0.\overline{001}=\frac{1}{999}$