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1. Context

As a reminder, these are the common number sets:

\[ \begin{array}{rcl} \mathbb{N} & = & \{ 0, 1, 2, 3, 4, \dots \} \\ \mathbb{Z} & = & \mathbb{N} \; \bigcup \; \{ -1, -2, -3, -4, \dots \} \\ \mathbb{Q} & = & \mathbb{Z} \; \bigcup \; \{ \frac{1}{2}, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{3}, \frac{2}{3}, -\frac{2}{3}, \dots \} \\ \mathbb{R} & = & \mathbb{Q} \; \bigcup \; \{ \sqrt{2}, \sqrt{3}, \pi, \dots \} \\ \end{array}\]

1.1. History of Numbers

First there was set set of natural numbers \(\mathbb{N}\). Addition never posed a problem, but subtraction \(a - b\) was only possible if \(a \geq b\).

To make subtraction \(a - b\) possible where \(a < b\), such as \(3 - 5\), negative numbers were invented (\(\mathbb{Z}\)). All was good, until someone noticed that division wasn’t always possible: while \(8 / 2\) was perfectly defined, what is \(8 / 3\) equal to?

Rational numbers solve this division problem by introducing rational numbers (\(\mathbb{Q}\)). Then someone came along pointing out that \(\sqrt{2}\) could not be written as a fraction. According to legend, this person was prompty thrown out to sea for his efforts.

The real numbers (\(\mathbb{R}\)) were then defined so as to make it possible to take the root of any number. Any positive number, that is.

1.2. Imaginary Numbers

Mathematicians have a tendency to be quite the rebels. Whenever someone tells them something is impossible, they promptly set out to axiomatically state that the answer exists. In our case, while everyone tells you \(\sqrt{-1}\) is not possible, some mathematician stated the opposite: the square root of \(-1\) does exist and it has a name: \(i\), short for imaginary unit.

2. Definition

The imaginary unit \(i\) is defined as

\[ i = \sqrt{-1}\]

A complex number has the form

\[ a + b \cdot i\]

where \(a, b \in \mathbb{R}\). The set of complex numbers is called \(\mathbb{C}\).

When working with complex numbers, you simply have to treat \(i\) as if it were some unknown value of which you know that \(i^2 = -1\).

3. Interpretation

While real numbers form a totally ordered set, which means that for any two real numbers \(x\) and \(y\), either \(x < y\), \(x > y\) or \(x = y\) must be true. This has as consequence that real numbers can be laid out on a line: take a line, pick a point to represent \(0\), and all other numbers must then be placed according to the rule "if \(x < y\) then \(x\) must be placed to the left of \(y\)".

Complex numbers do not fit on a single line as they are inherently 2 dimensional. Instead, we speak of the complex plane.

The complex number \(a + b \cdot i\)'s position can be found as follows: interpret \(a\) and \(b\) as the \(x\) and \(y\) coordinates. The result looks like this:

complex plane

4. Operations on Complex Numbers

Most standard operations are defined on complex numbers:

\[ (a_1 + b_1 \cdot i) + (a_2 + b_2 \cdot i) = (a_1 + a_2) + (b_1 + b_2) \cdot i\]
\[ (a_1 + b_1 \cdot i) - (a_2 + b_2 \cdot i) = (a_1 - a_2) + (b_1 - b_2) \cdot i\]

Multiplication consists of applying distributivity and simplifying \(i^2\) to \(-1\):

\[ (a_1 + b_1 i) \cdot (a_2 + b_2i) = a_1a_2 + a_1b_2i+a_2b_1i+b_1b_2i^2 = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) \cdot i\]

The absolute value of a complex number is its distance from \(0\):

\[ |a + bi| = \sqrt{a^2 + b^2}\]

5. Visual Interpretation of Multiplication

Multiplication of complex numbers coincides with rotation in 2D. Say you have a point \((x, y)\) that you want to rotate by an angle \(\theta\) around the origin, you can find it using

\[ x' + y' i = (x + y i) \cdot (\cos(\theta) + \sin(\theta) i) = (x \cos(\theta) - y \sin(\theta)) + (x \sin(\theta)+y \cos(\theta)) i\]

which yields the same result as when using rotation matrices:

\[ \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \cdot \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x \cos(\theta) - y \sin(\theta) \\ x \sin(\theta)+y \cos(\theta) \end{bmatrix}\]