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1 %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v 1.3 2002/09/12 23:30:20 dtashley Exp $
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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