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

Tue Aug 13 02:35:39 2019 UTC (4 years, 6 months ago) by dashley
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 1 dashley 140 %$Header$ 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 dashley 277 $HeadURL$ 38 $Revision$ 39 $Date$ 40 $Author$ 41 dashley 140 \end{verbatim} 42 \end{tiny} 43 \noindent\rule[0.25in]{\textwidth}{1pt} 44 \end{figure} 45 dashley 277 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 46 dashley 140 % 47 %End of file C_CIS0.TEX