This section is aimed at students in upper secondary education in the Danish
school system, some objects will be simplified some details will be omitted.
Real Numbers \(\mathbb{R}\)
Some of the first numbers discovered were the "counting" numbers, or the
non-negative whole numbers, denoted by \(\mathbb{N}\) and with these we can add
numbers and we can multiply numbers without issues.
At some point in history, people wanted to be able to denote negative numbers
also so we added the negative whole numbers yielding all the whole numbers,
denoted by \(\mathbb{Z}\supset\mathbb{N}\) and this allowed us to do the
opposite of addition and subtract numbers without running into issues.
As you may have guessed, at some point it became necessary to be able to do the
opposite thing of multiplication, division, and the rational numbers were born,
denoted by \(\mathbb{Q}\supset\mathbb{Z}\supset\mathbb{N}\). These numbers can be
represented as two whole numbers divided by each other or as decimal numbers with
repeating digits, i.e. at some point the same sequence of digits will
repeat infinitely.
Now this is a powerful set and you can pretty much do anything with these numbers,
but being notorious sticklers for details, mathematicians required more precision,
in fact infinite precision, and the real numbers emerged, denoted by
\(\mathbb{R}\supset\mathbb{Q}\supset\mathbb{Z}\supset\mathbb{N}\). These are
represented by decimal numbers with any number of non-repeating digits, also infinitely many,
and some examples of real numbers that have infinitely many non-repeating digits are \(\pi\),
Euler's number and the golden ratio, impractically precise numbers with extremely
beautiful and useful properties.
Complex Numbers \(\mathbb{C}\)
Now many of you probably thought the story ended there, but it did in fact not. New
problems emerged, one being that some polynomials didn't have any roots, e.g.
\(p(x)=x^2+1\). If it did have a root it would have the property that \(x^2=-1\)
which is not true for any real number. These issues caused mathematicians to
consider the complex numbers, denoted by
\(\mathbb{C}\supset\mathbb{R}\supset\mathbb{Q}\supset\mathbb{Z}\supset\mathbb{N}\).
Cartesian Form
These numbers can be represented in the so called "Cartesian" form \(a+ib\), where
\(a,b\in\mathbb{R}\). We say that a is the "real part" and b is the "imaginary part"
which is recognizable by that weird "i" which is called the "imaginary unit" which
is the solution to our previous polynomial since it has the property that \(i^2=-1\).
This can be represented visually as a point in 2D space with a as the "x-coordinate"
and b as the "y-coordinate" which in effect means that we have two coordinate axes
where one of them is "real" and the other is "imaginary". Note that these are just
names and not properties, imaginary numbers have multiple properties that suggest
they are just as real as any other number.
Multiplication
These numbers have every property of real numbers, so we can do the exact same
operations on these as the real numbers, the only difference is that we have the
imaginary unit which squares to -1. This means that we can multiply complex numbers
and get a new complex numbers, which is a benefit over vectors since the canonical
"multiplication" of vectors yields a scalar, i.e. changes the object type. So lets
demonstrate how you multiply two complex numbers by using the normal rules of
multiplication:
$$(a+ib)(c+id)=ac+aid+ibc+ibid=ac+iad+ibc+i^2bd=ac-bd+i(ad+bc)$$
Unit Circle
Lets pull back and restrict ourselves to the unit circle, i.e. we will consider
complex numbers of the form \(f(x)=\cos(x)+i\sin(x)\) where x is the arc length from
the real axis. This in fact is a function of arc length which allows us to
differentiate it,
$$f'(x)=-\sin(x)+i\cos(x)=i^2\sin(x)+i\cos(x)=i(\cos(x)+i\sin(x)=if(x)$$
This means that f has the property that \(f'=cf\) where \(c=i\) and there is only one
function that has this property, namely the natural exponential function
\(f(x)=ce^{ix}\) but \(c=f(0)=cos(0)+i\sin(0)=1\) which means that
$$e^{ix}=\cos(x)+i\sin(x)$$
which in turn yields arguably the most beautiful formula in all of math
$$e^{i\pi}=\cos(\pi)+i\sin(\pi)=-1+i\cdot0\implies e^{i\pi}+1=0$$
This is not the only immediately interesting consequence of that result, another is that
if we take two numbers on the unit circle and multiply them we get
\begin{align}
e^{i\theta}e^{i\varphi}&=(\cos(\theta)+i\sin(\theta))(cos(\varphi)+i\sin(\varphi))\\
e^{i(\theta+\varphi)}&=\cos(\theta)\cos(\varphi)-\sin(\theta)\sin(\varphi)
+i(\cos(\theta)\sin(\varphi)+\sin(\theta)\cos(\varphi))\\
&=\cos(\theta+\varphi)+i\sin(\theta+\varphi)
\end{align}
which means that
$$\cos(\theta+\varphi)=\cos(\theta)\cos(\varphi)-\sin(\theta)\sin(\varphi)$$
and
$$\sin(\theta+\varphi)=\cos(\theta)\sin(\varphi)+\sin(\theta)\cos(\varphi)$$
Polar Form
Any complex number, \(z=a+ib\), can then be written as a unit number
\(e^{i\theta}\) multiplied by its magnitude \(|z|=r=\sqrt{a^2+b^2}\), i.e.
\(z=|z|e^{i\theta}=re^{i\theta}\) and this is called the "polar form".
Complex Conjugate and Division
For every complex number \(z=a+ib\) there is another complex number called it's
conjugate \(\bar{z}=a-ib\) which is just \(z\) reflected by the real axis. We can use the
conjugate to define complex division in the following way. We consider the product
\(z\bar{z}=(a+ib)(a-ib)=a^2-(ib)^2=a^2+b^2=|z|^2\) which means that
$$z\frac{\bar{z}}{|z|^2}=1\implies z^{-1}=\frac{\bar{z}}{|z|^2}$$
This means that in order to "divide" by the complex number \(z\) we need to multiply by
\(\frac{\bar{z}}{|z|^2}\)