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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

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