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%$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 $ |
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\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}] |
\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}] |
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{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}} |
{Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}} |
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\label{ccis0} |
\label{ccis0} |
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\vworkexercisechapterheader{} |
\vworkexercisechapterheader{} |
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\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}} |
\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}} |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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 |
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$u=-2^{m-1}$, $u=\sim{}u+1$, 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 |
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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 |
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actual values of the integers represented. |
actual values of the integers represented. |
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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 |
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$-2^{m-1}$ 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. |
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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 |
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$u_{[m-1]}=1$, and by |
$u_{[m-1]}=1$, and by |
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(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}), |
(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}), |
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\end{vworkexercisesolution} |
\end{vworkexercisesolution} |
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\vworkexercisechapterfooter |
\vworkexercisechapterfooter |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\vfill |
\vfill |
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\noindent\begin{figure}[!b] |
\noindent\begin{figure}[!b] |
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\noindent\rule[-0.25in]{\textwidth}{1pt} |
\noindent\rule[-0.25in]{\textwidth}{1pt} |
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\begin{tiny} |
\begin{tiny} |
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\begin{verbatim} |
\begin{verbatim} |
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$RCSfile: c_cis0.tex,v $ |
$HeadURL$ |
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$Source: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/esrgubka/c_cis0/c_cis0.tex,v $ |
$Revision$ |
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$Revision: 1.3 $ |
$Date$ |
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$Author: dtashley $ |
$Author$ |
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$Date: 2002/09/12 23:30:20 $ |
\end{verbatim} |
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\end{verbatim} |
\end{tiny} |
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\end{tiny} |
\noindent\rule[0.25in]{\textwidth}{1pt} |
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\noindent\rule[0.25in]{\textwidth}{1pt} |
\end{figure} |
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\end{figure} |
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
% |
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% $Log: c_cis0.tex,v $ |
%End of file C_CIS0.TEX |
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% Revision 1.3 2002/09/12 23:30:20 dtashley |
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% Safety checkin as changing machines. |
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% |
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% Revision 1.2 2002/08/26 17:57:03 dtashley |
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% Additional solutions chapter added. Precautionary checkin to be sure |
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% that I've captured all changes. |
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% |
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% Revision 1.1 2002/08/26 17:35:06 dtashley |
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% Initial checkin. |
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% |
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%End of file C_CIS0.TEX |
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