trunk/c_cis0/c_cis0.tex

%$Header$

\chapter[Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}]
        {Solutions: \ccilzeroxrefcomma{}Chapter \ref{ccil0}, \ccilzerolongtitle{}}

\label{ccis0}

\vworkexercisechapterheader{}
\begin{vworkexercisesolution}{\ref{exe:ccil0:sexe0:01}}
We can show this result in two ways.  The first way, based on bit patterns, is to note
that adding an $m$-bit number, $u$, to its one's complement will result in a bit pattern
containing all 1's, i.e. $\forall i$, $u_{[i]} = 1$.  Adding 1 to this bit pattern will
always produce $\forall i$, $u_{[i]} = 0$ with a carry out.  Since the order of addition
does not matter, this establishes that adding $u$ to $\sim{}u+1$ will produce 0, thus showing
that $u$ and $\sim{}u+1$ are additive inverses.  This method, although valid, does not 
establish that $u$ and $\sim{}u+1$ actually represent additive inverses.  For example, if
$u=-2^{m-1}$, $u=\sim{}u+1$, and clearly a non-zero number cannot be an additive inverse of
itself.  Thus, it would be more comforting to show this result in a way that demonstrates the
actual values of the integers represented.

We present a second method now.  Assume that $u \neq -2^{m-1}$, since 
$-2^{m-1}$ cannot have an additive inverse in an $m$-bit signed integer.
If $u=0$, $\sim{}u+1=0$, so the relationship is clearly met.  If $u<0$, then
$u_{[m-1]}=1$, and by
(\ccilzeroxrefhyphen\ref{eq:ccil0:sroi0:sros0:00}),


\end{vworkexercisesolution}
\vworkexercisechapterfooter

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1	%$Header$
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
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33	\noindent\begin{figure}[!b]
34	\noindent\rule[-0.25in]{\textwidth}{1pt}
35	\begin{tiny}
36	\begin{verbatim}
37	$HeadURL$
38	$Revision$
39	$Date$
40	$Author$
41	\end{verbatim}
42	\end{tiny}
43	\noindent\rule[0.25in]{\textwidth}{1pt}
44	\end{figure}
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46	%
47	%End of file C_CIS0.TEX
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