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45 lines
2.0 KiB
TeX
45 lines
2.0 KiB
TeX
\subsubsection{Fractional Binary Numbers}
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We can represent any real number (with a finite decimal representation) as:
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$$
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d=\sum_{i=-n}^{m}10^i\cdot d_i \qquad\qquad \underbrace{d_m d_{m-1} \cdots d_1 d_0\ .\ d_{-1} d_{-2} \cdots d_{-(n-1)} d_{-n}}_{d_i \text{ is the } i \text{-th digit of } d \text{ (neg. indices indicate decimals)}}
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$$
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We can use the same idea for Base $2$ as well:
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$$
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b=\sum_{i=-n}^{m} 2^i \cdot b_i \qquad\qquad b_m b_{m-1} \cdots b_1 b_0\ .\ b_{-1} b_{-2} \cdots b_{-(n-1)} b_{-n}
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$$
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To get an intuition for this representation, looking at some examples is helpful:
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\begin{multicols}{2}
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A few observations:
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\begin{enumerate}
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\item Shifting the dot right: Division by $2$
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\item Shifting the dot left: Multiply by $2$
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\item Numbers of the form $0.111\ldots$ are just below $1.0$
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\item Some numbers representable in finite Base $10$ are infinite in Base $2$, e.g. $\frac{1}{5} = 0.20_{10}$
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\end{enumerate}
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\newcolumn
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\renewcommand{\arraystretch}{1.2}
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\begin{center}
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\begin{tabular}{lcl}
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\textbf{Binary} & \textbf{Fraction} & \textbf{Decimal} \\
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\hline
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$0.0$ & $\frac{0}{2}$ & $0.0$ \\
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$0.01$ & $\frac{1}{4}$ & $0.25$ \\
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$0.010$ & $\frac{2}{8}$ & $0.25$ \\
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$0.0011$ & $\frac{3}{16}$ & $0.1875$ \\
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$0.00110$ & $\frac{6}{32}$ & $0.1875$ \\
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$0.001101$ & $\frac{13}{64}$ & $0.203125$ \\
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$0.0011010$ & $\frac{26}{128}$ & $0.203125$ \\
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$0.00110101$ & $\frac{51}{256}$ & $0.19921875$ \\
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\end{tabular}
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\end{center}
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\renewcommand{\arraystretch}{1.0}
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\end{multicols}
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A major issue with this representation is that very large (respectively very small) numbers require a large representation.\\
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E.g $a_{10} = 5 \cdot 2^{100}$ has the representation $a_2 = 101\underbrace{000000000000000\ldots}_{100 \text{ Zeros}}\ $. Floating Point is designed to address this.
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