Find

Find 

$$\left \begin{array}{rcl} 10 \times 0.\overline{9} &=& 9.\overline{9} \\ 1 \times 0.\overline{9} &=& 0.\overline{9} \end{array} \right\} - \\ \hline$$

$$\begin{array}{rcl} (10-1)\times 0.\overline{9} &=& 9.\overline{9} - 0.\overline{9} \\ 9 \times 0.\overline{9} &=& 9 \\ 9 \times 0.\overline{9} &=& 9 \qquad | \qquad : 9 \\ 0.\overline{9} &=& \dfrac{9}{9} \\ 0.\overline{9} &=& 1 \\ \end{array}$$

 = 1 - 1 = 0

Note that........

(1.00000....) - (.9999......)   = .(00000).....1  = .01    where the "bar" is over the "0'

{sorry.....I dont know how to insert the bar  !!!}

$$\\0.\bar3=\frac{1}{3}\\\\ 3*0.\bar3=3*\frac{1}{3}\\\\ therefore\\ 0.\bar9=1\\\\\\ 1-0.\bar9 = 1-1 = 0$$