共轭复数：共轭复数的性质及复数模的运算性质

定义：复数 z = a + b i ( a , b ∈ R ) z=a+bi(a,b\in R) z=a+bi(a,b∈R)的共轭复数记作 z ‾ \overline {z} z,也就是
z ‾ = a − b i \overline {z}=a-bi z=a−bi

一.共轭复数的性质：

1. ∣ z ∣ = ∣ z ‾ ∣ |z|=| \overline z| ∣z∣=∣z∣
2. z + z ‾ = 2 a , a ∈ R z+\overline z=2a,a \in R z+z=2a,a∈R
3. z − z ‾ = 2 b i z-\overline z=2bi z−z=2bi
4. z z ‾ = ∣ z ∣ 2 z\overline z=|z|^2 zz=∣z∣2
5. z ‾ 1 + z ‾ 2 = z ‾ 1 + z 2 \overline z_1+\overline z_2=\overline z_1+z_2 z1​+z2​=z1​+z2​
6. z ‾ 1 − z ‾ 2 = z ‾ 1 − z 2 \overline z_1-\overline z_2=\overline z_1-z_2 z1​−z2​=z1​−z2​
7. z ‾ 1 ∗ z ‾ 2 = z ‾ 1 ∗ z 2 \overline z_1*\overline z_2=\overline z_1*z_2 z1​∗z2​=z1​∗z2​
8. z ‾ 1 z ‾ 2 = z 1 z 2 ‾ \frac {\overline z_1}{\overline z_2}=\overline \frac {z_1} {z_2} z2​z1​​=z2​z1​​​

二.复数模的运算性质

1. ∣ ∣ z 1 ∣ − ∣ z 2 ∣ ∣ ≤ ∣ z 1 ± z 2 ∣ ≤ ∣ z 1 ∣ + ∣ z 2 ∣ ||z_1|-|z_2|| \leq|z_1\pm z_2| \leq|z_1|+|z_2| ∣∣z1​∣−∣z2​∣∣≤∣z1​±z2​∣≤∣z1​∣+∣z2​∣
2. ∣ z 1 ∗ z 2 ∣ = ∣ z 1 ∣ ∗ ∣ z 2 ∣ |z_1*z_2|=|z_1|*|z_2| ∣z1​∗z2​∣=∣z1​∣∗∣z2​∣ 推广： ∣ z n ∣ = ∣ z ∣ n |z_{n}|=|z|^{n} ∣zn​∣=∣z∣n
3. ∣ z 1 z 2 ∣ = ∣ z 1 ∣ ∣ z 2 ∣ |\frac {z_1}{z_2}|=\frac {|z_1|}{|z_2|} ∣z2​z1​​∣=∣z2​∣∣z1​∣​

转载自：https://www.sogou.com/link?url=DSOYnZeCC_owkDvmYG0gMz-JrNZwwuWKk9Ad4XOkWlNOvwtkBl1WG_Jj0o8CWtA5xZOdkxQo0KgEYJnfG7pWJc3jAG-edc3Kri2jTBr9OJ8.