\documentclass[a4paper,titlepage]{article}
\title{Complex Numbers}
\author{Emily Lawson 1418516}
\date{\today}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{tikz}
\usepackage[margin=0.5in]{geometry}
\usepackage{array}
\begin{document}
\maketitle
\tableofcontents
\pagenumbering{roman}
\emph{*NOTE : This student has a disability affecting written expressions. Please ignore minor errors of spelling, grammar or punctuation.*}
\section{Introduction}\label{intro}
Some equations are impossible to solve using the set of reals numbers, for example $x^2 + 4 = 0$. The only way to solve this equation is by using a number that is not real. By rearranging this equation, we get $x^2 = - 4$. A special non-real number called $i$ can be used here to solve this equation. $i$ has the property that $\textit{i}^2 = -1$, so the solution to the equation above is $x = 2i$.
Below is a function written in Sage that when run will produce the graph in Figure \ref{graph}, the graph shows the positions of three different complex numbers.
\begin{verbatim}
def cpoint(x,y,color,d=0.5):
p = point((x,y),color=color)
t = text('(
return p + t
c = cpoint(6,4,'red')+cpoint(2,2.5,'blue')+cpoint(4,1,'purple')
c += text('Real axis',(7,-1),color='black',horizontal_alignment='right')
c += text('Imaginary axis',(-1,5),color='black',rotation='vertical',vertical_alignment='top')
c.show(xmin=-1, xmax=7, ymin=-1, ymax=5)
\end{verbatim}
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=12cm]{sagegraph}
\end{center}
\label{graph}
\caption{$2 + \frac{5}{2}i$, $4 + i$ and $6 + 4i$ shown on here graphically}
\end{figure}
\cite{graph}
\section{Definition of Complex Numbers}\label{def}
\subsection{Real and imaginary parts of a complex number}
A complex number is an expression $a + bi$, where a and b are both real numbers, and $i$ is as defined above in section \ref{intro} where $i^2 = - 1$. For a complex number $z = a + bi$, this is written in standard form. The \textit{real part} of $z$ is denoted as $Re(z)$ and the \textit{imaginary part} of $z$ is denoted as Im(z). Therefore, when $z = a + bi$,
$$
Re(z) = a, Im(z) = b
$$
We can use Sage to find the imaginary and real parts of a complex number, below is some code showing this with the example $z = 4 - 7i$.
\begin{verbatim}
z = 4 - 7 * i # this defines z as a complex number
z
real_part(z) # finds the real part of z
4
imag_part(z) # finds the imaginary part of z
-7
\end{verbatim}
\section{Conjugation and Exponentiation of Complex Numbers}
\subsection{Conjugation of complex numbers}
Multiplication of complex numbers is very similar to multiplication of real numbers. In order to be able to divide a complex number by another you must first find the \textit{conjugate} of the complex number you are dividing by.
\newline
\newline
The conjugate of a complex number $z$ is denoted as $\overline{z}$, the real part of $z$ and $\overline{z}$ is equal but the imaginary part of $\overline{z}$ has the opposite sign to that of $z$. For the complex number $z = a + bi$, where $a$ and $b$ are both real numbers,
$$
\overline{z} = \overline{a + bi} = a - bi
$$
Sage can be used to find the conjugate of a complex number, below is some code demonstrating this with the example $z = 4 - 7i$,
\begin{verbatim}
z = 4 - 7 * i # this defines z as a complex number
conjugate(z) # finds the complex conjugate of z
7*I + 4
\end{verbatim}
\textbf{Division of Complex Numbers}
\\
Given that $w$ and $z$ are both complex numbers, $w = a + bi$ and $z = c + di$, by using conjugation of complex numbers we can now find the result of $\frac{w}{z}$. In order to carry this out, we can use the identity below for the quotient of two complex numbers.
$$
\frac{w}{z} = \frac{w\overline{z}}{z\overline{z}}
$$
For the complex numbers $w$ and $z$ as defined earlier, provided $a$, $b$, $c$ and $d$ are all real non-zero numbers, we can find the quotient of $w$ and $z$:
\begin{align}
\frac{a + bi}{c + di} &= \frac{(a + bi)\overline{(c + di)}}{(c + di)\overline{(c + di)}} \\
&= \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) - (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i
\end{align}
Sage can be used to calculate the quotient of two complex numbers. For the examples $z = 3 - 7i$ and $w = 6 + 2i$, below is some Sage code which produces a further complex number to $\frac{z}{w}$,
\begin{verbatim}
z = 3 - 7 * i
w = 6 + 2 * i
z/w
-6/5*I + 1/10
\end{verbatim}
Here is some Sage code which calculates $\frac{z\overline{w}}{w\overline{z}}$. The two solutions to the equations are the same, proving the law of using conjugates in the division on two complex numbers, so $\frac{z}{w} = \frac{z\overline{w}}{w\overline{w}}$.
\begin{verbatim}
z = 3 - 7 * i
w = 6 + 2 * i
(z * conjugate(w)) / (w * conjugate(w))
-6/5*I + 1/10
\end{verbatim}
\subsection{Exponentiation of complex numbers}
For a complex number $z$ and a real number $n$, $z^n$ is defined to be as follows,
$$
z^n =
\begin{Bmatrix}
z...z & n > 0 & \text{where z is multiplied by itself n times}\\
1 & n = 0 \\
\frac{1}{z...z} & n < 0 & \text{where z is multiplied by itself n times and z $\neq$ 0}\\
\end{Bmatrix}
$$
Using the above definition, another known properties of complex multiplication and division, it can be shown that integer exponentiation of complex numbers and integer exponentiation of real numbers satisfy the same properties. From this we can assume the following statements, where $z$ and $w$ are complex numbers, and $m$ and $n$ are real numbers,
\begin{align*}
z^m z^n &= z^{m + n}, \\
\frac{z^m}{z^n} &= z^{m - n}, \\
(z^m)^n &= z^{mn}, \\
(wz)^n &= w^nz^n, \\
\left(\frac{w}{z}\right)^n &= \frac{w^n}{z^n}.
\end{align*}
Information from lecture notes \cite{exponentiation}
\section{Different Forms of Complex Numbers}
There are number of ways in which to represent a complex number, so far in Section \ref{intro} and Section \ref{def} we have been using the standard form, where a complex number $z$ is represented as $a + bi$. It is possible to represent a complex number graphically using an Argand Diagram, this will be shown in Section\ref{Argand}. Another way is in the exponential form, where $z = re^i\theta$ where $r$ is the modulus of $z$ and $\theta$ is the argument of $z$, this will be demonstrated in Section \ref{exp}. The final way of showing a complex number that I am going to talk about is by representing it in the polar form, this puts the complex number in terms of $\sin$ and $\cos$, this will be described in Section \ref{polar}.
\subsection{Argand Diagrams}\label{Argand}
Let $z$ be a complex number where $z = 2 + 3i$. Below is an Argand Diagram showing $2 + 3i$.
\begin{figure}[!htbp]
\begin{center}
\begin{tikzpicture}[scale=0.8]
\begin{scope}[thick,font=\scriptsize]
\draw [->] (-4,0) -- (4,0) node [above left] {$\Re\{z\}$};
\draw [->] (0,-4) -- (0,4) node [below right] {$\Im\{z\}$};
\foreach \n in {-3,...,-1,1,2,...,3}{
\draw (\n,-3pt) -- (\n,3pt) node [above] {$\n$};
\draw (-3pt,\n) -- (3pt,\n) node [right] {$\n i$};}
\draw [thick, color=red] (0,0) -- (2,3);
\draw [color=blue, fill=blue] (2,3) circle(0.05);
\node [color=black] at (3,3) {$ 2+3i$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{\textit{\textbf{2+3i}} plotted on an Argand Diagram}
\label{Argand Diagram}
\end{figure}
The length of the line representing $z$ is the modulus, this can be worked out by the following Pythagoras equation. For $z = a + bi$, where $a$ and $b$ are real numbers,
$$
\mid{z}\mid = \mid{a + bi}\mid = \sqrt{a^2 + b^2}
$$
Below is some Sage code that calculates the modulus of the complex number shown in Figure \ref{Argand Diagram},
\begin{verbatim}
z = 2 + 3 * i
abs(z) # finds the modulus of z and prints this first
arg(z) # finds the argument of z, prints second
sqrt(13)
arctan(3/2)
\end{verbatim}
The argument is the angle between the positive x-axis and the line representing the complex number itself. It must comply to the following inequality,
$$
-\pi < arg(z) \leq \pi
$$
The argument can never be greater than $\pi$, so if the line produced by drawing a complex number is in the positive Imaginary axis, the argument will be positive. If the line produced is in the negative Imaginary axis, then the argument will be negative. If the line produced lies on the Real axis where $z$ is a negative real number, the argument will be $-\pi$.
Information from lecture notes \cite{Argand}
\subsection{Exponential form}\label{exp}
For any complex number $z$ and any real number $\theta$, the following definition holds true,
$$
z = re^{i\theta} \ \text{and also} \ re^{i(\theta + 2n\pi)} = re^{i\theta}
$$
In the above definition, $r$ is the modulus and $\theta$ is the argument of the complex number $z$, the modulus and argument are both explained in Section \ref{Argand}.
\subsection{Polar form}\label{polar}
For any complex number $z$, by knowing the modulus and the argument, it is possible to show $z$ as follows,
$$
z = r(cos(\theta) + i(sin(\theta))
$$
\cite{polar}
\subsection{Comparing forms of complex numbers}
Table \ref{t1} pictured below shows multiple different ways of showing each complex number.
\begin{table}[h ]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\bf{Standard Form} & \bf{Exponential Form} & \bf{Polar Form} \\[5pt]
\hline
$a + bi$ & $re^{i\theta}$ & $r(cos(\theta)$ + $isin(\theta))$ \\[5pt]
\hline
$3 + 3\sqrt{3}$ & $6e^{\frac{\pi}{3}i}$ & $6(cos(\frac{\pi}{3}) + isin(\frac{\pi}{3}))$ \\[5pt]
\hline
$-5$ & $-5$ & $-5(cos(2\pi) + isin(2\pi))$ \\[5pt]
\hline
$7i$ & $7e^{\frac{\pi}{2}i}$ & $7(cos\left(\frac{\pi}{2}\right) + isin\left(\frac{\pi}{2}\right)$ \\[5pt]
\hline
\end{tabular}
\end{center}
\caption{Table showing different forms of representing complex numbers}
\label{t1}
\end{table}
\section{Conclusion}
In this document I have been exploring many different aspects of complex numbers. I have looked at how to plot graphs in LaTex and in Sage, showing the Sage code in the document. If I were to further continue this project I would investigate differential equations using complex numbers, and look at how to solve them using Sage.
\bibliographystyle{plain}
\bibliography{Bibliography}
\end{document}
