trunk/c_cis0/c_cis0.tex

%$Header$

\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}]
        {Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}

\label{ccis0}

\vworkexercisechapterheader{}
\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}}
We can show this result in two ways.  The first way, based on bit patterns, is to note
that adding an $m$-bit number, $u$, to its one's complement will result in a bit pattern
containing all 1's, i.e. $\forall i$, $u_{[i]} = 1$.  Adding 1 to this bit pattern will
always produce $\forall i$, $u_{[i]} = 0$ with a carry out.  Since the order of addition
does not matter, this establishes that adding $u$ to $\sim{}u+1$ will produce 0, thus showing
that $u$ and $\sim{}u+1$ are additive inverses.  This method, although valid, does not 
establish that $u$ and $\sim{}u+1$ actually represent additive inverses.  For example, if
$u=-2^{m-1}$, $u=\sim{}u+1$, and clearly a non-zero number cannot be an additive inverse of
itself.  Thus, it would be more comforting to show this result in a way that demonstrates the
actual values of the integers represented.

We present a second method now.  Assume that $u \neq -2^{m-1}$, since 
$-2^{m-1}$ cannot have an additive inverse in an $m$-bit signed integer.
If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met.  If $u<0$, then
$u_{[m-1]}=1$, and by
(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),


\end{vworkexercisesolution}
\vworkexercisechapterfooter

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\noindent\rule[-0.25in]{\textwidth}{1pt}
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\begin{verbatim}
$RCSfile: c_cis0.tex,v $
$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v $
$Revision: 1.3 $
$Author: dtashley $
$Date: 2002/09/12 23:30:20 $
\end{verbatim}
\end{tiny}
\noindent\rule[0.25in]{\textwidth}{1pt}
\end{figure}
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% $Log: c_cis0.tex,v $
% Revision 1.3  2002/09/12 23:30:20  dtashley
% Safety checkin as changing machines.
%
% Revision 1.2  2002/08/26 17:57:03  dtashley
% Additional solutions chapter added.  Precautionary checkin to be sure
% that I've captured all changes.
%
% Revision 1.1  2002/08/26 17:35:06  dtashley
% Initial checkin.
%
%End of file C_CIS0.TEX
1	dashley	140	%$Header$
2
3			\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}]
4			{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}
5
6			\label{ccis0}
7
8			\vworkexercisechapterheader{}
9			\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}}
10			We can show this result in two ways. The first way, based on bit patterns, is to note
11			that adding an $m$-bit number, $u$, to its one's complement will result in a bit pattern
12			containing all 1's, i.e. $\forall i$, $u_{[i]} = 1$. Adding 1 to this bit pattern will
13			always produce $\forall i$, $u_{[i]} = 0$ with a carry out. Since the order of addition
14			does not matter, this establishes that adding $u$ to $\sim{}u+1$ will produce 0, thus showing
15			that $u$ and $\sim{}u+1$ are additive inverses. This method, although valid, does not
16			establish that $u$ and $\sim{}u+1$ actually represent additive inverses. For example, if
17			$u=-2^{m-1}$, $u=\sim{}u+1$, and clearly a non-zero number cannot be an additive inverse of
18			itself. Thus, it would be more comforting to show this result in a way that demonstrates the
19			actual values of the integers represented.
20
21			We present a second method now. Assume that $u \neq -2^{m-1}$, since
22			$-2^{m-1}$ cannot have an additive inverse in an $m$-bit signed integer.
23			If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met. If $u<0$, then
24			$u_{[m-1]}=1$, and by
25			(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),
26
27
28			\end{vworkexercisesolution}
29			\vworkexercisechapterfooter
30
31			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
32			\vfill
33			\noindent\begin{figure}[!b]
34			\noindent\rule[-0.25in]{\textwidth}{1pt}
35			\begin{tiny}
36			\begin{verbatim}
37			$RCSfile: c_cis0.tex,v $
38			$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v $
39			$Revision: 1.3 $
40			$Author: dtashley $
41			$Date: 2002/09/12 23:30:20 $
42			\end{verbatim}
43			\end{tiny}
44			\noindent\rule[0.25in]{\textwidth}{1pt}
45			\end{figure}
46			%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
47			% $Log: c_cis0.tex,v $
48			% Revision 1.3 2002/09/12 23:30:20 dtashley
49			% Safety checkin as changing machines.
50			%
51			% Revision 1.2 2002/08/26 17:57:03 dtashley
52			% Additional solutions chapter added. Precautionary checkin to be sure
53			% that I've captured all changes.
54			%
55			% Revision 1.1 2002/08/26 17:35:06 dtashley
56			% Initial checkin.
57			%
58			%End of file C_CIS0.TEX