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# 复数运算复习

$\left({a}_{1}+{b}_{1}i\right)+\left({a}_{2}+{b}_{2}i\right)=\left({a}_{1}+{a}_{2}\right)+\left({b}_{1}+{b}_{2}\right)i$

$\left({a}_{1}+{b}_{1}i\right)-\left({a}_{2}+{b}_{2}i\right)=\left({a}_{1}-{a}_{2}\right)+\left({b}_{1}-{b}_{2}\right)i$

$\left({a}_{1}+{b}_{1}i\right)\cdot \left({a}_{2}+{b}_{2}i\right)=\left({a}_{1}{a}_{2}-{b}_{1}{b}_{2}\right)+\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}\right)i$

## 练习1：复数加减

### 例1：复数相加

$\begin{array}{rl}& \phantom{=}\left(3+4i\right)+\left(6-10i\right)\\ \\ & =\left(3+6\right)+\left(4-10\right)i\\ \\ & =9-6i\end{array}$

### 例2：复数相减

$\begin{array}{rl}& \phantom{=}\left(3+4i\right)-\left(6-10i\right)\\ \\ & =\left(3-6\right)+\left(4-\left(-10\right)\right)i\\ \\ & =-3+14i\end{array}$

$\left(7-10i\right)-\left(3+30i\right)=$

## 练习 2：复数相乘

$\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd$

### 例题1

$\begin{array}{rl}& \phantom{=}2\cdot \left(-3+4i\right)\\ \\ & =2\cdot \left(-3\right)+2\cdot 4i\\ \\ & =-6+8i\end{array}$

### 例题 2

$\begin{array}{rl}& \phantom{=}3i\cdot \left(1-5i\right)\\ \\ & =3i\cdot 1+3i\cdot \left(-5\right)i\\ \\ & =3i-15{i}^{2}\\ \\ & =3i-15\left(-1\right)\\ \\ & =15+3i\end{array}$

### 例题 3

$\begin{array}{rl}& \phantom{=}\left(2+3i\right)\cdot \left(1-5i\right)\\ \\ & =2\cdot 1+2\cdot \left(-5\right)i+3i\cdot 1+3i\cdot \left(-5\right)i\\ \\ & =2-10i+3i-15{i}^{2}\\ \\ & =2-7i-15\left(-1\right)\\ \\ & =17-7i\end{array}$

$8\cdot \left(11i+2\right)=$