Question

Convert $0.\overline{9}$ into a rational number.

Answer 1

Let $x=0.\overline{9}$. Then $x=\mathrm{0.99999...}$
Multiplying by $10$ on both sides, we get
$10x=\mathrm{9.99999...}=9+\mathrm{0.9999...}=9+x$
$\Rightarrow =9x=9$
$\Rightarrow x=1.$ That is, $0.\overline{9}=1$
$(\because 1$ is rational number$)$.