Question Number 128684 by Study last updated on 09/Jan/21
$${how}\:{we}\:{can}\:{convert}\:\mathrm{0}.\overset{−} {\mathrm{9}}\:{to}\:\frac{{q}}{{p}}? \\ $$
Answered by liberty last updated on 09/Jan/21
$$\:\mathrm{let}\:\mathrm{r}\:=\:\mathrm{0},\mathrm{99999999999}... \\ $$$$\:\:\:\:\:\:\:\mathrm{10r}\:=\:\mathrm{9}.\mathrm{9999999999}... \\ $$$$\mathrm{substract}\:\Rightarrow\:\mathrm{9r}\:=\:\mathrm{9}\:\Rightarrow\:\mathrm{r}\:=\:\mathrm{1}\:=\:\frac{\mathrm{1}}{\mathrm{1}}\:=\:\frac{\mathrm{2}}{\mathrm{2}}=\frac{\mathrm{n}}{\mathrm{n}}\:;\:\mathrm{n}\neq\:\mathrm{0} \\ $$$$ \\ $$