Drafted into submission

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

diff --git a/chapters/Background/FixedPoint.tex b/chapters/Background/FixedPoint.tex
index 707dab2..bb1dea3 100644 (file)
--- a/chapters/Background/FixedPoint.tex
+++ b/chapters/Background/FixedPoint.tex
@@ -1,9 +1,9 @@
-A positive real number $z$ may be written as the sum of smaller integers ``digits'' $d_i < z$ multiplied by powers of a base $\beta$. 
+A positive real number $z$ may be written as the sum of smaller integers ``digits'' $d_i$ multiplied by powers of a base $\beta$. 
 \begin{align}
-       z &= \displaystyle\sum_{i=-\infty}^{\infty} d_i \beta^{i}\label{fixedpointZ}
+       z &= \text{... } + d_{-1} \beta^{-1} + d_0 \beta^0 + d_1 \beta^1 + \text{ ...} = \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}.
+Where each digit $d_i < \beta$. A set of $\beta$ unique symbols are used to represent values of $d_i$.
+A seperate sign '-' can be used to represent negative reals 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.
 
@@ -13,7 +13,7 @@ Example integer representation in base 10 (decimal) and base 2 (binary):
        1011000110010_2 &= 1\times2^{12} + 0\times2^{11} + \text{ ...} + 0\times2^0
 \end{align*}
 
-
+%{\bf FIXME} Add Maths reference (Cantor's Diagonal argument) without going into all the Pure maths details