I'm working out some details to implement a division algorithm, I'm following the explanation given in this book (chapter 5) for who is interested.Anyway I need to work out how many bits are necessary to represent a value $\omega$ bounded by$$| \omega | \leq \rho r^{k+1} y$$where$$\begin{array}{l}\frac{1}{2} < \rho \leq 1 \\r = 2^l, \text{where $l\geq 1$ is some integer} \\0 \leq y \leq 2^{k} - 1, \text{ $k$ positive integer}\end{array}$$My approach is finding the number of bits to represent $| \omega |$ and then adding one bit to represent the sign of $\omega$, in two complement.There fore$$\left\lceil \log_2(|\omega|) \right\rceil \leq \left\lceil \log_2(\rho r^{k+1} y) \right\rceil = \left\lceil \log_2(\rho) + (k+1)\log_2 r + \log_2(y) \right\rceil \leq \left\lceil (k+1)l  + \log_2(2^k-1))\right\rceil \leq \left\lceil (k+1)l  + k \right\rceil = \left\lceil (k+1)(l+1) - 1 \right\rceil = (k+1)(l+1) - 1$$Therefore in two complement I would need a total of $(k+1)(l+1)$ bits to represent my value $\omega$.Is this correct?
How many bits to represent a quantity $\omega$ bounded in a particular way?
arithmetic