Question

Find the rational number which is represented by the following decimal numbers: $\mathrm{0.999999...}$

Answer 1

Let $r=\mathrm{0.999999...}$, then $10r$ will be as follows:

$10r=10\times 0.999999=\mathrm{9.99999...}$

Now consider $10r-r$ as shown below:

$10r-r=9.99999-0.99999=9.00000\Rightarrow 9r=9\Rightarrow r=\frac{9}{9}\Rightarrow r=1$

Hence, the required rational number is $1$.