For any positive integer $b2,$ the base $b$ representation of a $($positive$)$ real number $$ is the digit string $(d_n{d}_{n-1}cdots{d}_1{d}_0*d_{-1}{d}_{-2}{d}_{-3}$cdots${)}_b$ and is interpreted as $=d_n{b}^n+d_{n-1}{b}^{n-1}+$cdots$+d_1{b}^1+b_0{b}^0+d_{-1}{b}^{-1}+d_{-2}{b}^{-2}+d_{-3}{b}^{-3}+$dots, where each of the digits $d_i$ are integers with $d_n>0b-10d_i.$

$We$ also set $d_n>0$ and don't allow the representation $to$ end $in$ infinitely many digits that equal $b-1to$ ensure that the representation $is$ unique for each $.$