Mathematics · Ch 2 — Complex Numbers

Summary

2.9

In this chapter we studied:

Rectangular form: a complex number is $x+iy$ (or $x+yi$ ), $x,y\in\mathbb R$ .
Equality: $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ are equal $\iff \operatorname{Re}(z_1)=\operatorname{Re}(z_2)$ and $\operatorname{Im}(z_1)=\operatorname{Im}(z_2)$ , i.e. $x_1=x_2$ and $y_1=y_2$ .
Conjugate: the conjugate of $x+iy$ is $x-iy$ .

Properties of complex conjugates:

\text{(1) }\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\quad\text{(6) }\operatorname{Im}(z)=\dfrac{z-\overline z}{2i}

\text{(2) }\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}\quad\text{(7) }\overline{z^n}=(\overline z)^n,\ n\text{ an integer}

\text{(3) }\overline{z_1z_2}=\overline{z_1}\,\overline{z_2}\quad\text{(8) }z\text{ is real iff }z=\overline z

\text{(4) }\overline{\left(\dfrac{z_1}{z_2}\right)}=\dfrac{\overline{z_1}}{\overline{z_2}},\ z_2\ne0\quad\text{(9) }z\text{ is purely imaginary iff }z=-\overline z

\text{(5) }\operatorname{Re}(z)=\dfrac{z+\overline z}{2}\quad\text{(10) }\overline{\overline z}=z

Modulus: if $z=x+iy$ , $\sqrt{x^2+y^2}$ is called the modulus of $z$ , denoted $|z|$ .

Properties of modulus:

\text{(1) }|z|=|\overline z|\quad\text{(5) }\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|},\ z_2\ne0

\text{(2) }|z_1+z_2|\le|z_1|+|z_2|\ \text{(Triangle inequality)}\quad\text{(6) }|z^n|=|z|^n,\ n\text{ an integer}

\text{(3) }|z_1z_2|=|z_1||z_2|\quad\text{(7) }\operatorname{Re}(z)\le|z|

$\text{(4) }|z_1-z_2|\ge\big||z_1|-|z_2|\big|\quad\text{(8) }\operatorname{Im}(z)\le|z|$ …