1.7: Special Numbers

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\begin{array}{ll}\text{Smallest exponent: }\quad \mathrm{e}=00000000 \text{, represents denormal numbers} (1.f \rightarrow 0.f) \end{array}

\begin{array}{ll}\text{Largest exponent: } \quad &\mathrm{e}=11111111 \text{, represents } \pm \infty \text{, if } \mathrm{f}=0 \\
&\mathrm{e}=11111111 \text{, represents } \mathrm{NaN} \text{, if } \mathrm{f} \neq 0 \end{array}

\begin{array}{ll}\text { Number Range: } & \mathrm{e}=11111111=2^{8}-1= \quad \quad &\text{reserved}\\
& \mathrm{e}=00000000=0 \quad \quad &\text{reserved}\\
\text { so, } & \mathrm{p}=\mathrm{e}-127 \text { is } \\
& 1-127 \leq \mathrm{p} \leq 254-127 \\ & -126 \leq \mathrm{p} \leq 127\end{array}

\begin{array}{ll}\text{Smallest positive normal number} \\
&= 1.00000000 \cdots \cdot .0000 \times 2^{-126} \\
&\simeq 1.2 \times 10^{-38} \\
&\text{bin: 00000000100000000000000000000000} \\
&\text{hex: 00800000}\\
&\text{MATLAB: realmin('single')}
\end{array}

\begin{array}{ll}\text{Largest positive number} \\
&=1.11111111 \cdots \cdots \cdot 1111 \times 2^{127} \\
&=\left(1+\left(1-2^{-23}\right)\right) \times 2^{127} \\
&\simeq 2^{128} \simeq 3.4 \times 10^{38} \\
&\text{bin: 01111111011111111111111111111111} \\
&\text{hex: 7f7fffff} \\
&\text{MATLAB: realmax('single')} \\
\end{array}

\begin{array}{ll}\text{Zero} \\
&\text{bin: 0000 0000 0000 0000 0000 0000 0000 0000} \\
&\text{hex: 00000000}
\end{array}

\begin{array}{ll}\text{Subnormal numbers} \\
&\text{Allow 1.f } \rightarrow \text{ (in software)} \\
&\text{Smallest positive number }=0.00000000 \cdots \cdots 0001 \times 2^{-126}\\
&=2^{-23} \times 2^{-126} \simeq 1.4 \times 10^{-45}
\end{array}