复数的乘法为什么会定义成这副样子？

复数既然是数域的扩充, 那么所有运算规则应该是和实数域是一样的.

实数域多项式乘法可以写成 $\begin{equation}{\left({a}+{b} \right)}{\left({c}+{d} \right)}={a}{c}+{b}{c}+{a}{d}+{b}{d}\end{equation}$\begin{equation}{\left({a}+{b} \right)}{\left({c}+{d} \right)}={a}{c}+{b}{c}+{a}{d}+{b}{d}\end{equation} ;

那么复数的乘法和多项式乘法一样展开: $\begin{equation}{\left({a}+{b}{i} \right)}{\left({c}+{d}{i} \right)}={a}{c}+{b}{c}{i}+{a}{d}{i}+{b}{i}\cdot{d}{i}={a}{c}-{b}{d}+{\left({a}{d}+{b}{c} \right)}{i}\end{equation}$\begin{equation}{\left({a}+{b}{i} \right)}{\left({c}+{d}{i} \right)}={a}{c}+{b}{c}{i}+{a}{d}{i}+{b}{i}\cdot{d}{i}={a}{c}-{b}{d}+{\left({a}{d}+{b}{c} \right)}{i}\end{equation}

就是这么简单呀.