Answer
$\frac{8}{9}$
Work Step by Step
$0.8888\ldots =0.8+0.08+0.008+0.0008+\cdots $
So, the value of $\left| r \right|$ is,
$\begin{align}
& \left| r \right|=\left| \frac{0.08}{0.8} \right| \\
& =\left| 0.1 \right| \\
& =0.1
\end{align}$
Find the limit of the infinite geometry series by using the formula
${{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}$,
$\begin{align}
& {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r} \\
& =\frac{0.8}{1-0.1} \\
& =\frac{0.8}{0.9} \\
& =\frac{8}{9}
\end{align}$
Thus, the fraction notation of the decimal number $0.8888\ldots $ is $\frac{8}{9}$.
