\docname{sess6.2}
\doctitle{Complex Arithmetic Examples}

\mylhead
\myrhead

\title{\doctitle}
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In the following we let $z=2+3i$ and $w=4+5i$.

\section{Real and imaginary parts}
\[\Repart(z)=2, \quad \Repart(w)=4, \quad \Impart(z)=3, \quad \Impart(w)=5.\]
Note: the imaginary part does not include $i$.

\section{Addition and Subtraction}

\begin{align*}
z+w&=(2+3i)+(4+5i)=6+8i\\
z-w&=(2+3i)-(4+5i)=-2-2i.
\end{align*}

\subsection{Multiplication}
\[z\cdot w=(2+3i)(4+5i)=8-15+i(10+12)=-7+22i.\]

\section{Complex Conjugate and Magnitude}

\begin{align*}
\overline{z}&=\overline{2+3i}=2-3i\\
|z|&=\sqrt{4+9}=\sqrt{13}\\
z+\overline{z}&=2+3i+2-3i=4=2\, \Repart(z)\\
z\cdot\overline{z}&=(2+3i)(2-3i)=4+9=13=|z|^2
\end{align*}

\section{Division} 
Multiply numerator and denominator by the complex conjugate of the
denominator:

\[\frac wz=\frac{4+5i}{2+3i}=\frac{4+5i}{2+3i}\cdot\frac{2-3i}{2-3i}=\frac{8+15+i(-12+10)}{13}=\frac{23}{13}-\frac{2}{13}i.\]
