Appendix-complex_numbers.md

(complex_numbers)=

Appendix D: Complex numbers

Properties of complex numbers

Imaginary Numbers

The square root of a negative real number is referred to as an imaginary number. So $\sqrt{-2}$, $\sqrt{-5}$ and $\sqrt{-11}$ are all imaginary numbers. The unit of imaginary numbers is $\sqrt{-1}$ and is usually denoted by the letter $i$. To avoid confusion with the current $i$ in electronics, the letter $j$ is used. So

$$ i = j = \sqrt{-1}. $$ (eq1)

Complex numbers

A complex number is a number of the form

$$ z = x + j \cdot y. $$ (eq2)

where $x$ and $y$ are real numbers. The variables $x$ and $y$ are the real (Re$z$) and the imaginary (Im$z$) component of $z$ respectively. The complex number $z$ can also be written as:

$$ z = ({\rm Re} , z,, {\rm Im} , z) = (x,, y). $$ (eq3)

In fact a complex number has two coordinates and can be graphically displayed as a point in a plane, the complex plane. This rectangular or the Cartesian coordinates are plotted along the real and the imaginary axis (see {numref}figD1).

---
width: 400px
name: figD1
---
Complex plane

A complex number can also be represented in a polar notation:

$$ z = (|z|,\varphi), $$ (eq4)

where:

$$ |z| = r = \sqrt{({\rm Re} , z)^{2}+({\rm Im} , z)^{2}} $$ (eq5)

is the length of the vector (${\rm Re } , z$, ${\rm Im} , z$), also known as the modulus of $z$, and:

$$ \varphi = \arg z = \tan^{-1} \left( \dfrac{{\rm Im} , z}{{\rm Re} , z}\right ) \pm k \cdot \pi $$ (eq6)

is the angle this vector makes with the positive real axis, with $k$ as an integer, also known as the argument of $z$.

:::{note} The $\pm k \cdot \pi$ has to do with the fact that $\varphi$ is a continuous quantity, in principle with a range of $-\infty$ and $+\infty$, while the function $tan^{-1}$ is defined for the interval $-\pi/2$ to + $\pi/2$. :::

The inverse transformation from polar coordinates to Cartesian coordinates is given by:

$$ \begin{split} {\rm Re} , z &= |z|\cos \varphi \ {\rm Im} , z &= |z| \sin \varphi. \end{split} $$ (eq7)

A third way to express a complex number is to make use of the complex exponential function:

$$ z = |z|\exp^{j\varphi} = r\exp^{j\varphi}, $$ (eq8)

where:

$$ \exp^{j\varphi} = \cos \varphi +j\sin \varphi. $$ (eq9)

Using the above definitions, and relationships we can derive some rules.

For addition and subtraction:

$$ \begin{split} z & = z_{1} \pm z_{2}\Rightarrow \ {\rm Re} , z &= {\rm Re} , z_{1} + {\rm Re} , z_{2} \ {\rm Im} , z & = {\rm Im} , z_{1} + {\rm Im} , z_{2}; \end{split} $$ (eq10)

for multiplication:

$$ \begin{split} z & = z_{1} \cdot z_{2}\Rightarrow \ |z| & = |z_{1}| \cdot |z_{2}| \ \arg z & = \arg z_{1} + \arg z_{2}; \end{split} $$ (eq:D13)

for division:

$$ \begin{split} z & = \frac{z_{1}}{z_{2}} \rbrace \Rightarrow \ |z| & = \frac{|z_{1}|}{|z_{2}|} \ \arg z & = \arg z_{1} - \arg z_{2}. \end{split} $$ (eq:D14)