My textbook defines the floating point number system as the numbers given by $$x=(-1)^s\cdot \beta^{-k}\sum_{j=0}^{N-2}a_j\beta^j$$ where $N$ is the number of memory positions for a number $x\in\mathbb R$ with digits $a_j$ and base $\beta$. Also, $s=0$ whenever $x>0$ and $s=1$ otherwise (this I understand).
I do not understand how this formula is developed at all. can anyone help?