# Question About Floating Point System

I have begun reading Numerical Analysis by Walter Gautschi. On page $$3$$, the author introduces the floating point number system as follows: a floating point number is a number representible as $$$$\pm (.b_{-1}b_{-2}...b_{-t})_2 \cdot 2^{\pm(c_{s-1}c_{s-2}...c_0.)_2} \tag {1}$$$$ where $$s, t$$ are fixed. We denote this number system by $$\mathbb{R}(t, s)$$.

Then, Walter says that a number $$x \in \mathbb{R}(t, s)$$ is normalized if $$b_{-1}=1$$, and that we $$"$$assume all numbers in $$\mathbb{R}(t, s)$$ are normalized $$"$$.

My Question: Consider consider the number $$x = +(.00100) \cdot 2^{-(111.)}$$ Is this number an element of $$\mathbb{R}(5, 3)$$? According to the definition, I want to say yes. But when the author says $$"$$we assume all numbers in $$\mathbb{R}(t, s)$$ are normalized,$$"$$ I don't know if he is implicitly redefining $$\mathbb{R}(t, s)$$ to be numbers of the form $$(1)$$ with the additional requirement that $$b_{-1}=1$$. If we try to impose this requirement on $$x$$, I believe $$x$$ would come out to $$x=+(.10000) \cdot 2^{-1001}$$ which shouldn't be part of $$\mathbb{R}(5, 3)$$

Background I am a math major taking my first numerical analysis. I do not have any prior experience with computer science or numerical mathematics.

After reading further, I found the answer. The answer is that the $$x$$ given in the question is not in $$\mathbb{R}(5, 3)$$. The way I figured this out is as follows:
On the next page, the author states that $$\min_{x \in \mathbb{R}(t, s)} |x| = 2^{-2^s}$$ This is consistent with $$(.100 \cdots 0) \cdot2^{-(11\cdots 1)}$$ being the least positive number in $$\mathbb{R}(t, s)$$, and not $$(.000 \cdots 01) \cdot2^{-(11\cdots 1)}.$$