Introduction to Complex Numbers

Definitions

Complex numbers are an extension of the real numbers. A general complex number $z\in\mathbb{C}$ (where $\mathbb{C}$ denotes the set of all complex numbers} can be written in the form \begin{align} z = x + iy \qquad x,y\in\mathbb{R} . \end{align} The symbol $i$ is the so-called imaginary unit and is defined by the property $i^2 := -1$. The two real components $x$ and $y$ are referred to as the real part $\Re{z}$ and the imaginary part $\Im{z}$ of the complex number respectively.^[1]

There are different conventions about if the imaginary unit is written in front of or behind the imaginary part. I personally prefer to write in front because then it becomes immediately clear that an imaginary expression is following.

Why Study Complex Numbers?

The field of real numbers is not closed algebraically, i.e. there are polynomials which are defined solely in terms of real numbers but do not have any real solution.^[2] A simple example is the equation \begin{align} x^2 = -4 . \end{align} There is no real number that yields -4 when being multiplied by itself - square roots of negative numbers are not defined within the real numbers. However, for $x\in\mathbb{C}$ this equation can be solved easily by using the expression $-1 = i^2$: \begin{align} \Leftrightarrow \quad x^2 = (-1)(4) = 4 i^2 \quad \Rightarrow \quad x_\pm = \pm 2i \end{align}

Geometric Representation: Complex Plane

In contrast to real numbers, the complex ones have two independent components: The real part $x$ and the imaginary part $y$. While the real numbers can be represented geometrically as a line, this does not work for the complex numbers anymore. Here a two-dimensional plane is required in order to represent all possible combinations of values for $x$ and $y$. It is very intuitive to represent these components $x$ and $y$ in analogy to cartesian coordinates along a rectangular grid:

Such a two-dimensional representation of the complex numbers is referred to as an Argand diagram.^[3]

Calculation Rules

Complex Conjugate

The complex conjugate $\overline{z}$ of a complex number $z= x + iy$ is obtained by switching the sign of the imaginary part:^[4] \begin{align} \overline{z} := x - i y \end{align} Using the notation of complex conjugates is really helpful. For instance one can compute both the real and the imaginary part of a complex number by using its complex conjugate: \begin{align} \Re{z} &= \frac{z + \overline{z}}{2} = \frac{x+iy + x-iy}{2} = x \\ \Im{z} &= \frac{z - \overline{z}}{2i} = \frac{x+iy - \left( x-iy \right) }{2i} = y \end{align} Furthermore, if one defines the absolute value of a complex number in accordance to the Euclidean norm, one can make use the complex conjugate as well: \begin{align} \left| z \right| := \sqrt{ x^2 + y^2 } = \sqrt{ \left( x+iy \right) \left( x-iy \right)} = \sqrt{ z \overline{z} } \end{align}

Arithmetic Operations

If one treats the symbol $i$ like an unknown variable (with the additional property $i^2=-1$), then one can just adopt the calculation rules from real numbers. Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$ be two complex numbers. Then one can easily perform the following operations:^[5]

• addition: \begin{align} z_1 + z_2 = x_1 + x_2 + i \left( y_1 + y_2 \right) \end{align}
• subtraction: \begin{align} z_1 - z_2 = x_1 - x_2 + i \left( y_1 - y_2 \right) \end{align}
• multiplication: \begin{align} z_1 z_2 &= \left(x_1 + i y_1 \right) \left( x_2 + i y_2 \right) \nonumber\\ &= x_1 x_2 + i^2 y_1 y_2 + i \left( x_1 y_2 + y_1 x_2 \right) \nonumber\\ &= x_1 x_2 - y_1 y_2 + i \left( x_1 y_2 + y_1 x_2 \right) \end{align}
• division: \begin{align} \frac{z_1}{z_2} = \frac{x_1 + i y_1}{x_2 + i y_2} \end{align} This one is a bit tricky. It is not at all obvious what a complex denominator actually means. One therefore wants to turn it into something real. This can easily be done by expanding the fraction with the complex conjugate of the denominator: \begin{align} \frac{z_1}{z_2} &= \frac{\left( x_1 + i y_1 \right) \left( x_2 - i y_2 \right) }{ \left( x_2 + i y_2 \right) \left( x_2 - i y_2 \right) } \nonumber\\ &= \frac{x_1 x_2 + y_1 y_2}{ x_2^2 + y_2^2} + i \frac{ x_1 y_2 - y_1 x_2 }{ x_2^2 + y_2^2} \end{align}