We would like a unique way to represent \(x\). One approach is to pick the one which gives the smallest exponent possible (while still not smaller than \(e_{\textit{min}}\)).
This is called the normalized representation.
A normal number is one where \(1\le|m|<\beta\), or \(\beta^{p-1}\le|M|<\beta^{p}\).
If \(e=e_{\textit{min}}\), then \(x\) is said to be subnormal, or denormal. Then \(|m|<1\) or \(|M|\le\beta^{p-1}-1\)
The smallest possible denormal number is \(\alpha=\beta^{e_{\textit{min}}-p+1}\).
The smallest possible normal number is \(\beta^{e_{\textit{min}}}\)
The largest finite floating point number is:
\begin{equation*}
\Omega=(\beta-\beta^{1-p})\beta^{e_{\textit{max}}}
\end{equation*}