% ���l�v�Z�̏펯, �� 1 ��
% 2010/06/25 �L���@�� �쐬



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\Dauthor{�L���@��}
\Dtitle{���l�v�Z�̏펯}
\Ddate{2010/05/13}
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\begin{document}
\subsection*{��1�̉�@}
��1�̌덷��10�i��$x$��$\beta$�i�@�ւ̕ό`��
\begin{equation*}
x=f_1\beta ^{-1}+f_2\beta ^{-2}+f_3\beta ^{-3}+\cdots
\end{equation*} 
�ɂ�藝���ł���.\\
���̎����32�r�b�g�̏ꍇ16�i���ł�,
\begin{equation*}
(0.1)_{10}=(0.199999)_{16} \times 16^0
\end{equation*}
�ƂȂ�, 16�i�@�ŕ\�킷���Ƃ��ł���(�t�^�̌v�Z1���Q��).\\
������܂�10�i�@�ɖ߂�����, $\beta$ �i������10�i�ϊ��̎�\\
\begin{equation*}
(0.f_1 f_2\cdots f_m)_\beta =(f_1 f_2\cdots f_m)_\beta \times \beta ^{-m}
\end{equation*}
�ɂ��\\
\begin{equation*}
(0.199999)_{16}=(199999)_{16} \times 16^{-6}
\end{equation*}
�v�Z�����$0.099999964\cdots $�ƂȂ�.

\subsection*{��2�̉�@}
$\displaystyle\sum_{n=1}^{10000}0.01$�̌v�Z�ɂ�����
$n$���ڂ܂ł̕����a�̑傫����$0.01n$�ł���, $\varepsilon $�̌덷�����񐶂����Ƃ����
\[ \sum_{n=1}^{10000}0.01n\varepsilon  \cong  0.01 \times (10000^2\varepsilon  )/2=5 \times 10^5\varepsilon  \]
�ƂȂ��Ă����������Ȃ�.\\
16�i�؂�̂Ă̂Ƃ�, $\varepsilon=6\times10^{-8}\sim10^{-6}$�Ƃ���, ���̒l��$0.03\sim0.5$\\
2�i0��1���̂Ƃ�, $\varepsilon=3\times10-{-8}\sim6\times10^{-8}$�Ƃ���, ���̒l��$0.015\sim0.03$\\
�ƂȂ�덷�̐������ł���.

\subsection*{��3�̉�@}
16�i�؂�̂Ă̏ꍇ��$0.1=(0.199999)_{16}\times16^0$��10������
\[ (0.199999)_{16}\times16^0\times10=(0.FFFFFA)_{16} \]
�ƂȂ�, 1��菬�����̂�while�̏����͖�������Ă���.\\
������, $0.1=(0.CCCCCD)_{16}\times2^{-3}$��(����2�i�@��0��1�����Ȃ���)10������10�i�Ŗ�$1.000000192$�ƂȂ�̂ł����1���킸���ɑ傫��, while�̏����͖�������Ȃ��Ȃ�.(�t�^�̌v�Z2���Q��)

\newpage

\section*{�t�^}

\subsection*{�v�Z1}
�܂�$0.1=f_116^{-1}+f_216^{-2}+\cdots$���̎���16���������
\[ 1.6=f_1+f_216^{-1}+f_316^{-2}+\cdots \]
$f_i$��$0����\beta-1$�܂ł̐���, �܂�$f_i$�̍ő��15�ł���̂�$f_1$�����̍��̌v��1�ȉ��Ȃ̂�$f_1=1$�ƂȂ�
\[ f_216^{-1}+f_316^{-2}+\cdots=0.6 \]
������܂���L�ɏ]��, 16�������ē��l�̌v�Z�������
\[ f_2=f_3=\cdots=9 \]
����������
\[ (0.1)_{10}=(0.19999\cdots)_{16} \]
2�i�@�����l�Ɍv�Z�����
\[ 0.1=(0.CCCCCD)_{16}\times2^{-3} \]
�ƕ\�킷���Ƃ��ł���.\\

\subsection*{�v�Z2}
$(0.1)_{10}=(0.19999\cdots)_{16} $���

\begin{tabular}{c}
  0.110011001100110011001101\\
\underline{+0.110011001100110011001101}\\
  1.100110011001100110011010\\
\underline{+0.110011001100110011001101}\\
 10.011001100110011001100111\\
\underline{+0.110011001100110011001101}\\
 11.001100110011001100110100\\
\underline{+0.110011001100110011001101}\\
100.000000000000000000000001\\
\end{tabular}
\newpage
\begin{tabular}{l}
\underline{+0.110011001100110011001101}\\
100.110011001100110011001110\\
\underline{+0.110011001100110011001101}\\
101.100110011001100110011011\\
\underline{+0.110011001100110011001101}\\
110.011001100110011001101000\\
\underline{+0.110011001100110011001101}\\
111.001100110011001100110101\\
\underline{+0.110011001100110011001101}\\
1000.000000000000000000000010\\
\end{tabular}

�����10�i�ɒ�����, 1.000000192�ƂȂ�while�̏�������������Ȃ��Ȃ�.
\end{document}