Intro to Complex Numbers

Complex numbers have Real and Imaginary parts $z=x+yi$. The addition and multiplication are intuitive, while division is a little tricky:

$$\frac{z_1}{z_2}=\frac{x_1+y_{1}i}{x_2+y_{2}i}=\frac{(x_1+y_{1}i)(x_2-y_{2}i)}{(x_2+y_{2}i)(x_2-y_{2}i)}=\frac{(x_1+y_{1}i)(x_2-y_{2}i)}{x_2^2+y_2^2}$$

The best way to visualize complex numbers is by using a complex plane, where they become vectors. With this representation it’s trivial to prove the two inequalities:

$$||z_1|-|z_2||\le |z_1+z_2|\le |z_1|+|z_2|$$

Since we can represent a complex number $z$ in two dimension using $x,y$ coordinates, we also can use Polar Coordinates.

Polar Coordinates

We use $r,\theta$, so $x=r\cos (\theta )$, $y=r\sin (\theta )$, where $r=\sqrt{x^2+y^2}$, but calculating $\theta$ is a little tricky. Given $x,y$ we can find a particular ${\theta }_0$. However, then ${\theta }_0+2\pi n$ for $n\in Z$ also works perfectly - we have infinite possibilities for $\theta$. To speak about one theta, we look inside a particular interval. For example, $[-\pi ,\pi ]$ is the most popular choice, and we denote it as $\theta =Arg(z)=ar{g}_{-\pi }(z)$.

Euler’s Formula

There is a better way to write $z$ than $z=r(\cos (\theta )+i\sin (\theta ))$. Recall the Taylor expansion for $e^x$:

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...$$

What if x is imaginary? Like $x=i\theta$.

$$e^{i\theta }=1+i\theta +\frac{(i\theta {)}^2}{2!}+\frac{(i\theta {)}^3}{3!}+...$$

Looking closely we recognize the cosine and sine expansions:

$$e^{i\theta }=(1+\frac{(i\theta {)}^2}{2!}+\frac{(i\theta {)}^4}{4!}+...)+(i\theta +\frac{(i\theta {)}^3}{3!}+\frac{(i\theta {)}^5}{5!}+...)$$ $$e^{i\theta }=(1-\frac{(\theta {)}^2}{2!}+\frac{(\theta {)}^4}{4!}-...)+i(\theta -\frac{(\theta {)}^3}{3!}+\frac{(i\theta {)}^5}{5!}+...)$$ $$e^{i\theta }=\cos (\theta )+i\sin (\theta )$$

Which means that our initial expression can be written compactly as $z=r{e}^{i\theta }$. The elegant way to think about this expansion is in the picture below:

Using this form of $z$ it’s much easier to multiply and divide. Thinking with vectors, what operation multiplication and division represent in the complex plane?