# Diff of /pubs/books/ucbka/trunk/c_cis0/c_cis0.tex

revision 139 by dashley, Thu Oct 6 03:15:02 2016 UTC revision 140 by dashley, Mon Jul 3 01:59:16 2017 UTC
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1  %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v 1.3 2002/09/12 23:30:20 dtashley Exp$  %$Header$
2
3  \chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}]  \chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}]
4          {Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}          {Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}
5
6  \label{ccis0}  \label{ccis0}
7
9  \begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}}  \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  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  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  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  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  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  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  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  $u=-2^{m-1}$, $u=\sim{}u+1$, 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  itself.  Thus, it would be more comforting to show this result in a way that demonstrates the
19  actual values of the integers represented.  actual values of the integers represented.
20
21  We present a second method now.  Assume that $u \neq -2^{m-1}$, since  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.  $-2^{m-1}$ 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  If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met.  If $u<0$, then
24  $u_{[m-1]}=1$, and by  $u_{[m-1]}=1$, and by
25  (\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),  (\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),
26
27
28  \end{vworkexercisesolution}  \end{vworkexercisesolution}
29  \vworkexercisechapterfooter  \vworkexercisechapterfooter
30
31  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
32  \vfill  \vfill
33  \noindent\begin{figure}[!b]  \noindent\begin{figure}[!b]
34  \noindent\rule[-0.25in]{\textwidth}{1pt}  \noindent\rule[-0.25in]{\textwidth}{1pt}
35  \begin{tiny}  \begin{tiny}
36  \begin{verbatim}  \begin{verbatim}
37  $RCSfile: c_cis0.tex,v$  $RCSfile: c_cis0.tex,v$
38  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v$  $Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v$
39  $Revision: 1.3$  $Revision: 1.3$
40  $Author: dtashley$  $Author: dtashley$
41  $Date: 2002/09/12 23:30:20$  $Date: 2002/09/12 23:30:20$
42  \end{verbatim}  \end{verbatim}
43  \end{tiny}  \end{tiny}
44  \noindent\rule[0.25in]{\textwidth}{1pt}  \noindent\rule[0.25in]{\textwidth}{1pt}
45  \end{figure}  \end{figure}
46  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
47  % $Log: c_cis0.tex,v$  % $Log: c_cis0.tex,v$
48  % Revision 1.3  2002/09/12 23:30:20  dtashley  % Revision 1.3  2002/09/12 23:30:20  dtashley
49  % Safety checkin as changing machines.  % Safety checkin as changing machines.
50  %  %
51  % Revision 1.2  2002/08/26 17:57:03  dtashley  % Revision 1.2  2002/08/26 17:57:03  dtashley