## Thomas' Calculus 13th Edition

$\displaystyle \frac{1}{9}=0.\overline{1},\quad \displaystyle \frac{2}{9}=0.\overline{2},\quad\frac{3}{9}=0.\overline{3},\quad\frac{8}{9}=0.\overline{8}$ $\displaystyle \frac{9}{9}=1=0.\overline{9}$
$\displaystyle \frac{1}{9}$ is another way of writing $1\div 9\quad$ So, we write 1 as a decimal number, $1.000000...$ and divide: $\begin{array}{lllllll} & & 0. & 1 & 1 & 1 & 1...\\ & -- & -- & -- & -- & -- & --\\ 9 & ) & 1. & 0 & 0 & 0 & 0\\ & & & 9 & & & \\ & & -- & -- & & & \\ & & & 1 & 0 & & \\ & & & & 9 & & \\ & & & -- & -- & & \\ & & & & 1 & 0 & \\ & & & & & 9 & \\ & & & & & ... & \end{array}$ The numeral "1" repeats from the first decimal place indefinitely, and we write $\displaystyle \frac{1}{9}=0.\overline{1}$ Similarly, $\displaystyle \frac{2}{9}=0.\overline{2}$ $\displaystyle \frac{3}{9}=0.\overline{3}$ $\displaystyle \frac{8}{9}=0.\overline{8}$ $\displaystyle \frac{9}{9}=1=0.\overline{9}$