Complex Number

A complex number is any number that is an element of the complex set, $\mathbb{C}$. They can be represented in various forms and are useful in integrating with complex numbers, getting solutions of polynomials and much more.

The imaginary unit - $i$ §

Every complex number must contain the imaginary unit $i$. The imaginary unit always satisfies the equation $i^2=-1$

Cartesian Definition §

Let $z$ be a complex number, $z\in \mathbb{C}$.

The Cartesian form of $z$ is:

$z=x+yi$

• $x$ is the real component of $z$, i.e. the portion of $z$ that is an element of $\mathbb{R}$.
• $y$ is the imaginary part of $z$ as it is multiplied by $i$.

$Re(z)$ and $Im(z)$ are functions used to obtain the real and imaginary parts of $z$ respectively. For example, $Re(9+3i)=9$ and $Im(7-5i)=-5$

Operations on $\text{Re() and Im()}$

Just like any other operator, $\text{Re() and Im()}$ are defined with certain rules. The main thing to take note is that real numbers do not affect them. This includes addition, subtraction, multiplication, division, etc. as long as it is done within the $\mathbb{R}$ set.
This also extends to derivatives and antiderivatives. I.e. $\int \text{Re(}f(x))dx=\text{Re}\left(\int f(x)dx\right)$ and $\frac{d}{dx}(\text{Im}(f(x)))=\text{Im}\left(\frac{d}{dx}(f(x))\right)$ if $f(x)\in \mathbb{R}$

Using the Cartesian definition, an imaginary number can easily be graphed with the x-axis being $Re(z)$ and the y-axis being $Im(z)$. These types of graphs are known as Argand diagrams.

Polar Form §

If you look back to the unit circle from trigonometry, each coordinate point on the circle $(x,y)$ can be represented as $(\cos (\theta ),\sin (\theta ))$. Multiply this by a radius value $r$ and we can plot any point on the real set.

Using a similar approach, complex numbers can be represented in Polar form:

$z=r\text{cis}(\theta )=r(\cos (\theta )+i\sin (\theta ))$

• $r$ : The magnitude of the complex number, $r=|z|=\sqrt{x^2+y^2}$
• $\theta$ : The angle of the complex number, $\theta =\arctan \left(\frac{y}{x}\right)$

Since there are multiple values of $\theta$ ( $cis(\theta )=cis(\theta +2\pi )=cis(\theta +4\pi )=\dots$) , it is restricted to the domain $(-\pi ,\pi ]$. This is the principle argument of theta.

Complex Exponential §

The complex exponential is based on Euler’s formula connecting trigonometric functions with the complex set.

$e^{i\theta }=\cos (\theta )+i\sin (\theta )=r{e}^{i\theta }$ where $\theta \in \mathbb{R}$

It can be used to represent complex numbers in polar form using an exponential.

• $e^{0i}=1$

• $e^{i\theta }\times e^{i\varphi }=e^{i(\theta +\varphi )}$

• $e^{i\theta }/e^{i\varphi }=e^{i(\theta -\varphi )}$