Restructure chapters, delete a bunch of words, add more words, do some things, panic...

[ipdf/sam.git] / chapters / Background / FixedPoint.tex

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+A positive real number $z$ may be written as the sum of smaller integers ``digits'' $d_i < z$ multiplied by powers of a base $\beta$. 
+\begin{align}
+       z &= \displaystyle\sum_{i=-\infty}^{\infty} d_i \beta^{i}\label{fixedpointZ}
+\end{align}
+Where each digit $d_i < \beta$ the base. A set of $\beta$ unique symbols are used to represent values of $d_i$.
+A seperate sign '-' can be used to represent negative integers using equation \eqref{fixedpointZ}.
+
+To express a real number using equation \eqref{fixedpointZ} in practice we are limited to a finite number of terms between $i = -m$ and $i = n$. Fixed point representations are capable of representing a discrete set of numbers $0 \leq |z| \leq \beta^{n+1}-\beta^{-m}$ seperated by $\Delta z = \beta^{-m} \leq 1$. In the case $m = 0$, only integers can be represented.
+
+Example integer representation in base 10 (decimal) and base 2 (binary):
+\begin{align*}
+       5682_{10} &= 5\times10^3 + 6\times10^2 + 8\times10^1 + 2\times10^0 \\
+       1011000110010_2 &= 1\times2^{12} + 0\times2^{11} + \text{ ...} + 0\times2^0
+\end{align*}
+
+
+
+
+
+%, but could be represented by the combination of a numerator $7 = 111_2$ and denominator $3 = 11_2$. Lastly, some values such as $e \approx 2.718\text{...}$ can only be expressed exactly using a symbolical system --- in this case as the result of an infinite summation $e = \displaystyle\sum_n^{\infty} 1/n!$
+