X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fsam.git;a=blobdiff_plain;f=chapters%2FBackground%2FFixedPoint.tex;fp=chapters%2FBackground%2FFixedPoint.tex;h=4ef2f6ca7e75b7f613cc31859b59102723a5c3eb;hp=352e9d303fae6b899d9e680b287e07867ee6f64f;hb=ae8d5f837db032eb4d9e9666f5026fab7e3e8e4a;hpb=253d241eb8279be539ad72a0283d3f1575b537ab

diff --git a/chapters/Background/FixedPoint.tex b/chapters/Background/FixedPoint.tex
index 352e9d3..4ef2f6c 100644
--- a/chapters/Background/FixedPoint.tex
+++ b/chapters/Background/FixedPoint.tex
@@ -1,8 +1,8 @@
-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 &= d_0 \beta^0 + d_1 \beta^1 + d_2 \beta^2 + \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 $d_i$.
+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
+%{\bf FIXME} Add Maths reference (Cantor's Diagonal argument) without going into all the Pure maths details