I think the author means the following:
Given a rational number $r=p/q$ it has a decimal expansion. And the fact that $r$ is rational is equivalent to the decimal representation being periodic after a certain point. That is if $r=1/3$, then the decimal representation is $r=0.333\cdots$. As another example if you take $r=9/13$, we get $r=0.692307692307\cdots$. It is not necessary that the first few digits after the decimal place should repeat, say for example $r=52/225=0.23111\cdots$. Also for decimal expansions which terminate, we assume that $0$ repeats infinitely, that is if $r=1/2=0.5$, we consider $r=0.5000\cdots$.
In the book, $a$ denotes the integer part of $p/q$, $(\alpha_1,\cdots,\alpha_n)$ denotes the first $n$ non-repeating digits and $(\beta_1\cdots\beta_m)$ denotes repeating digits.
Now given a rational number we can find the corresponding $a$, $\alpha_i$'s and $\beta_j$'s. The reverse representation might mean that the reverse construction is also possible. That is given an integer $a$, $\alpha_i$'s and $\beta_j$ can we find the corresponding $r$.
They have given the formula as $$r=\pm a \pm \frac{\overline{\alpha_1\cdots\alpha_n\beta_1\cdots\beta_m}-\overline{\alpha_1\cdots\alpha_n}}{\overline{\underbrace{99\cdots 9}_{m}\underbrace{0\cdots 0}_{n}}}$$
Here by $\overline{123}$, let's say, I am guessing the author means the number 123 (One hundred and twenty three). I have also made a slight edit to the formula which I think is correct. Hope this helps.