1.7: Special Numbers
- Page ID
- 96033
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\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}