Operations with Complex Numbers

Complex Conjugation

Definition: Complex Conjugate

The complex conjugate of a complex number $z=a+\mathrm{i}b$ is the number

$$\bar{z}\stackrel{\text{def}}{=}a-\mathrm{i}b$$

Theorem: Distributivity of Complex Conjugation

The complex conjugation of is distributive over addition and multiplication:

$$\overline{z_1+z_2}={\bar{z}}_1+{\bar{z}}_2$$ $$\overline{z_1\cdot z_2}={\bar{z}}_1\cdot {\bar{z}}_2$$

PROOF

TODO

Arithmetic with Complex Numbers

Definition: Addition and Subtraction of Complex Numbers

The addition and subtraction of two complex numbers $z_1=a+\mathrm{i}b$ and $z_2=c+\mathrm{i}d$ are defined as

$$z_1\pm z_2\stackrel{\text{def}}{=}(a\pm c)+\mathrm{i}(b\pm d)$$

Definition: Multiplication of Complex Numbers

The multiplication of two complex numbers $z_1=a+\mathrm{i}b$ and $z_2=c+\mathrm{i}d$ is defined as

$$z_1\cdot z_2\stackrel{\text{def}}{=}(ac-bd)+\mathrm{i}(ad+bc)$$

Theorem: Multiplication in Polar Form

If $z_1$ and $z_2$ have the polar forms $z_1=r_1(\cos {\phi }_1+\mathrm{i}\sin {\phi }_1)$ and $z_2=r_2(\cos {\phi }_2+\mathrm{i}\sin {\phi }_2)$, then

$$z_1\cdot z_2=r_1{r}_2(\cos ({\phi }_1+{\phi }_2)+\mathrm{i}\sin ({\phi }_1+{\phi }_2))$$

PROOF

TODO

Theorem: Multiplication in Exponential Form

If $z_1$ and $z_2$ have the exponential forms $z_1=r_1{\mathrm{e}}^{\mathrm{i}{\phi }_1}$ and $z_2=r_2{\mathrm{e}}^{\mathrm{i}{\phi }_2}$, then

$$z_1\cdot z_2=r_1{r}_2{\mathrm{e}}^{\mathrm{i}({\phi }_1+{\phi }_2)}$$

PROOF

TODO

Definition: Division of Complex Numbers

The division of two complex numbers $z_1=a+\mathrm{i}b$ and $z_2=c+\mathrm{i}d$ is

$$\frac{z_1}{z_2}\stackrel{\text{def}}{=}\frac{z_1\cdot {\bar{z}}_2}{|z_2{|}^2}$$

Theorem: The Field of the Complex Numbers

The set of all complex numbers $\mathbb{C}$ forms a field together with the addition and multiplication defined on it.

PROOF

TODO