The number $0.\overline{9}$ (which means 0.99999 ...) is a rational number. Using algebra (let $x = 0.\overline{9}$, multiply by 10, and subtract), explain why $0.\overline{9}$ is exactly equal to 1. [Page No. 62]

Let $x = 0.9999_{-}$ ...(i)

Multiply by 10: $10x = 9.9999_{-}$ ...(ii)

Subtract eq(i) from (ii):

$10x - x = 9.9999_{-} - 0.9999_{-}$

$9x = 9$

$\Rightarrow x = 1$

Therefore, $0.(9) = 1$ .