I'm reading the book Numerical Analysis by Gautschi, and in the first chapter the author makes the following assertion:
The set of floating-point numbers on a computer is denoted $\mathbb R(t,s)$ such that: $$ x \in \mathbb R(t,s) \iff x = f \cdot 2^e $$ where, $f = \pm (.b_{-1}\cdots b_{-t})_2$ and $e = \pm (c_{s-1}\cdots c_0.)_2.$ Here every $b_i$ and $c_j$ are binarary digits.
Then, the author states that a number $x$ is said to be normalized if in its fraction $f$ we have $b_{-1}=1$.
Finally, he claims that from this definition, we can easily see that for the normalized numbers we have $$ \min_{x \in \mathbb R(t,s)}|x| = 2^{-2^s}. $$
I don't follow why the minimum is given by the formula above. If I understood correctly, the minimum should be $1/2 \cdot 2^{-2^s}$, since the minimum of $f$ is $1/2$.
Can anyone explain why the minimum is the one given by the author ?