複素数の除法

\begin{align*}
& \textcolor{orange}{\phase{\footnotesize\ 分母分子に共役な複素数をかける！}}\\
\dfrac{2+9\,i}{1+2\,i} &= \dfrac{(2+9\,i)\colorbox{mistyrose}{$(1-2\,i)$}}{(1+2\,i)\colorbox{mistyrose}{($1-2\,i$)}}\\\\
&= \dfrac{2-4\,i+9\,i-18\,\colorbox{lightcyan}{$i^2$}}{1-4\,\colorbox{lightcyan}{$i^2$}}\\\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ i^2=(-1)に変える！}}\\
&= \dfrac{2+5\,i -18 \colorbox{lightcyan}{$\times (-1)$}}{1-4 \colorbox{lightcyan}{$\times (-1)$}}\\\\
&= \dfrac{2+5\,i+18}{1+4}\\\\
&= \dfrac{20+5\,i}{5}\\\\
&= \dfrac{5(4+i)}{5} = \textcolor{red}{\underline{\textcolor{black}{4+i}}}
\end{align*}

\begin{align*}
& \textcolor{orange}{\phase{\footnotesize\ 分母分子に共役な複素数をかける！}}\\
\dfrac{1+2\,i}{2+3\,i} &= \dfrac{(1+2\,i)\colorbox{mistyrose}{$(2-3\,i)$}}{(2+3\,i)\colorbox{mistyrose}{($2-3\,i$)}}\\\\
&= \dfrac{2-3\,i+4\,i-6\,\colorbox{lightcyan}{$i^2$}}{4-9\,\colorbox{lightcyan}{$i^2$}}\\\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ i^2=(-1)に変える！}}\\
&= \dfrac{2+\,i -6 \colorbox{lightcyan}{$\times (-1)$}}{4-9 \colorbox{lightcyan}{$\times (-1)$}}\\\\
&= \dfrac{2+\,i+6}{4+9}\\\\
&= \dfrac{8+\,i}{13} = \textcolor{red}{\underline{\textcolor{black}{\dfrac{8}{13}+\dfrac{1}{13}i}}}\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ a+b\,i の形で答えよう！}}
\end{align*}

\begin{align*}
& \textcolor{orange}{\phase{\footnotesize\ 分母分子に共役な複素数をかける！}}\\
\dfrac{1-\,i}{1+\,i} &= \dfrac{(1-\,i)\colorbox{mistyrose}{$(1-\,i)$}}{(1+\,i)\colorbox{mistyrose}{($1-\,i$)}}\\\\
&= \dfrac{1-2\,i+\,\colorbox{lightcyan}{$i^2$}}{1-\,\colorbox{lightcyan}{$i^2$}}\\\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ i^2=(-1)に変える！}}\\
&= \dfrac{1-2\,i + \colorbox{lightcyan}{$(-1)$}}{1- \colorbox{lightcyan}{$(-1)$}}\\\\
&= \dfrac{1-2\,i-1}{1+1}\\\\
&= \dfrac{-2\,i}{2} = \textcolor{red}{\underline{\textcolor{black}{-\,i}}}
\end{align*}

\begin{align*}
& \textcolor{orange}{\phase{\footnotesize\ 分母分子に共役な複素数をかける！}}\\
\dfrac{5\,i}{2-\,i} &= \dfrac{5\,i\colorbox{mistyrose}{$(2+\,i)$}}{(2-\,i)\colorbox{mistyrose}{($2+\,i$)}}\\\\
&= \dfrac{10\,i+5\,\colorbox{lightcyan}{$i^2$}}{4-\,\colorbox{lightcyan}{$i^2$}}\\\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ i^2=(-1)に変える！}}\\
&= \dfrac{10\,i + 5\colorbox{lightcyan}{$(-1)$}}{4- \colorbox{lightcyan}{$(-1)$}}\\\\
&= \dfrac{10\,i-5}{4+1}\\\\
&= \dfrac{-5+10\,i}{5}\\\\
&= \dfrac{5(-1+2\,i)}{5} = \textcolor{red}{\underline{\textcolor{black}{-1+2\,i}}}
\end{align*}

\begin{align*}
& \textcolor{orange}{\phase{\footnotesize\ 分母分子に\ i\ をかける！}}\\
\dfrac{1}{i} &= \dfrac{1\colorbox{mistyrose}{$\times\,i$}}{i\colorbox{mistyrose}{$\times\,i$}} = \dfrac{i}{\colorbox{lightcyan}{$i^2$}}\\\\
& 　　　　\textcolor{orange}{\phase{\footnotesize\ i^2=(-1)に変える！}}\\
&= \dfrac{i}{\colorbox{lightcyan}{$-1$}}\\\\
&= -\,i
\end{align*}