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

Initial commit after migrating from CVS.

 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