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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 | |

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 |

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