Complex Number

A complex number is a number with real and imaginary parts. They can be expressed in the following general form, where $a$ and $b$ are real numbers:

$$a+bi$$

Magenta colour-coding

All imaginary parts in expressions on this page will be highlighted a lovely shade of magenta, to make them easier to follow.

_{Yes, this took way too long to typeset.}

The set of all complex numbers is denoted by the symbol $\mathbb{C}$. Yes, with the funny font and all. What next, will the cool S become a maths symbol as well?

If $a$ is $0$, then the complex number can be expressed as just $bi$. Just like my chances of graduating with a first class degree, this number is considered purely imaginary.

A letter commonly used to represent a complex variable is the letter $z$. Why? It’s convention or something, I dunno.

Real and imaginary parts

Given some complex number:

$$z=a+bi$$

To get just $a$ or $b$ from a complex number, we use the $\mathrm{Re}$ and $\mathrm{Im}$ functions:

$$\begin{array}{r}\mathrm{Re}(z)=a\\ \mathrm{Im}(z)=b\end{array}$$

Modulus-argument form

An alternative form of expressing complex numbers, if you’re too cool for boring ol’ $a+bi$ form, is modulus-argument form (also known as polar form). This, unsurprisingly, is composed of a modulus and argument - but what the heck are they?

A good intuition for understanding modulus-argument form is thinking about plotting complex numbers on an Argand diagram. Using the general $a+bi$ form above is like plotting a point on a rectangular coordinate system. Nothing special there, we simply encode $a$ and $b$ as a horizontal and vertical position respectively. However, that’s not the only way to describe a point on a 2D plane - we could also borrow a leaf out of polar plotting’s book and describe a point as an angle and a distance from the origin.

Here’s an analogy: imagine giving directions to a lost stranger. Since you’re a huuuuge nerd, you decide there are two ways to tell them the directions to where they’re trying to get. You could either tell them to walk a certain distance north/south then a certain distance east/west, or you could simply point in a direction and tell them to walk a certain distance that way. Both ways can uniquely identify a location, and the latter is what polar plotting and modulus-argument form is like.

As hinted at by the diagram above, modulus-argument form takes the following algebraic shape:

$$r(\cos \theta +i\sin \theta )$$

It looks a little scary at first, but it just comes from right triangle trigonometry (SOHCAHTOA). If the hypotenuse has length $r$, then the two other sides of the triangle have lengths $r\cos \theta$ and $r\sin \theta$:

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Modulus

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Argument

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