# 共役な複素数の基本式

$\alpha$$\beta$複素数とし，それぞれの共役な複素数$\overline{\alpha }$$\overline{\beta }$とする．

• 和　$\overline{\alpha +\beta }=\overline{\alpha }+\overline{\beta }$      証明
• 差　$\overline{\alpha -\beta }=\overline{\alpha }-\overline{\beta }$      証明
• 積　$\overline{\alpha \beta }=\overline{\alpha }\overline{\beta }$      証明
• $\overline{\left(\frac{\alpha }{\beta }\right)}=\frac{\overline{\alpha }}{\overline{\beta }}\text{ }\left(\beta \ne 0\right)$      証明
• $\overline{\left(\overline{\alpha }\right)}=\alpha$

## ■和の証明

$\alpha =a+b\text{ }i$$\beta =c+d\text{ }i$ とする．

$\begin{array}{ll}\overline{\alpha +\beta }\hfill & =\overline{\left(a+b\text{ }i\right)+\left(c+d\text{ }i\right)}\hfill \\ \hfill & =\overline{\left(a+c\right)+\left(b+d\right)i}\hfill \\ \hfill & =\left(a+c\right)-\left(b+d\right)i\hfill \\ \hfill & =\left(a-b\text{ }i\right)+\left(c-d\text{ }i\right)\hfill \\ \hfill & =\overline{\alpha }+\overline{\beta }\hfill \end{array}$

## ■差の証明

$\alpha =a+b\text{ }i$$\beta =c+d\text{ }i$ とする．

$\begin{array}{ll}\overline{\alpha -\beta }\hfill & =\overline{\left(a+b\text{ }i\right)-\left(c+d\text{ }i\right)}\hfill \\ \hfill & =\overline{\left(a-c\right)+\left(b-d\right)i}\hfill \\ \hfill & =\left(a-c\right)-\left(b-d\right)i\hfill \\ \hfill & =\left(a-b\text{ }i\right)-\left(c-d\text{ }i\right)\hfill \\ \hfill & =\overline{\alpha }-\overline{\beta }\hfill \end{array}$

## ■積の証明

$\alpha =a+b\text{ }i$$\beta =c+d\text{ }i$ とする．

$\begin{array}{ll}\overline{\alpha \beta }\hfill & =\overline{\left(a+b\text{ }i\right)\left(c+d\text{ }i\right)}\hfill \\ \hfill & =\overline{\left(ac-bd\right)+\left(ad+bc\right)i}\hfill \\ \hfill & =\left(ac-bd\right)-\left(ad+bc\right)i\hfill \\ \hfill & =\left(a-b\text{ }i\right)\left(c-d\text{ }i\right)\hfill \\ \hfill & =\overline{\alpha }\overline{\beta }\hfill \end{array}$

## ■商の証明

$\begin{array}{ll}\overline{\left(\frac{\alpha }{\beta }\right)}·\overline{\beta }\hfill & =\overline{\frac{\alpha }{\beta }·\beta }\hfill \\ \hfill & =\overline{\alpha }\hfill \end{array}$

よって，$\overline{\left(\frac{\alpha }{\beta }\right)}=\frac{\overline{\alpha }}{\overline{\beta }}$

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