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%TCIDATA{<META NAME="Title" CONTENT="Complex Numbers">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:09:24 1997}
%TCIDATA{Language=American English}
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\input{tcilatex}
\begin{document}


\section{Complex Numbers\label{Complex numbers}}

Complex numbers are numbers of the form $a+b\sqrt{-1}$ where $a$ and $b$ are
real numbers. The arithmetic of the complex numbers is obtained by observing
all of the standard rules of arithmetic of real numbers together with the
identity $\left( \sqrt{-1}\right) ^{2}=-1$.

The default notation for the square root of $-1$ is $i$. If you prefer to
use $j$ you can change the default as follows.

\begin{quote}
$\blacktriangleright $ To modify the system to recognize $j$ for $\sqrt{-1}$
\end{quote}

\begin{itemize}
\item  Choose \textsf{Maple + Settings}

\item  Click the \textsf{Definition Options} tab.

\item  Under \textsf{Imaginary Unit}, check \textsf{Change from i to j }
\end{itemize}

If you carry out this procedure, \textsl{Scientific Notebook} will recognize
and return $j$ rather than $i$ for the square root of minus one. The letter $%
i$ is used throughout this document.

\strut

\subsection{\ Basic Operations}

For addition, subtraction, multiplication, and division write the
expressions in standard form and apply \textsf{Evaluate}. \textsl{Scientific
Notebook} will return the result in the form $a+bi$ or $a+ib$, with $a$ and $%
b$ real numbers.

\strut

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\sqrt{-5}=\,i\sqrt{5}$\vspace{6pt}

$\left( 1+i\right) \left( 3-2i\right) =\,5+i\vspace{6pt}$

$\dfrac{i}{1+i}=\,\dfrac{1}{2}+\dfrac{1}{2}i\vspace{6pt}$

$\left( 23-5i\right) +\left( 1+16i\right) =\allowbreak 24+11i\medskip $

$\left( 2.5+3i\right) -\left( 3.59+16i\right) =\allowbreak
-1.\,09-13.0i\medskip $

$\left( 2.5+3i\right) \left( 3.59+16i\right) =\allowbreak
-39.\,025+50.\,77i\medskip $

$\left( 23-5i\right) /\left( 1+16i\right) =\allowbreak -\frac{57}{257}-\frac{%
373}{257}i\medskip $

$\frac{2.5+3i}{3.59+16i}=\allowbreak .\,21189-.\,10871i\medskip $

$\left( 1+16i\right) \left( \left( 2.5+3i\right) -\left( 3.59+16i\right)
\right) =\allowbreak 206.\,91-30.\,44i$
\end{quotation}

\strut

\begin{quotation}
\emph{Note\quad }For some of the expressions with complex numbers, the
keyboard shortcut \textsc{ctrl } \textsc{+} \textsc{e} for evaluation is not
as effective as the menu item \textsf{Evaluate}.
\end{quotation}

\strut

\subsection{\ Powers and Radicals 
\index{Powers@Powers}\label{Powers}}

To raise complex numbers to powers, use common notation for powers and apply 
\textsf{Evaluate}. 
\index{Exponents@Exponents}\label{Exponents} 
\index{Radicals@Radicals}\label{Radicals}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$i^{2}=\allowbreak -1%
\vspace{4pt}$

$\left( 3+2i\right) ^{4}=\allowbreak -119+120i\vspace{4pt}$

$\left( 3+2i\right) ^{-4}=\allowbreak -\frac{119}{28561}-\frac{120}{28561}i%
\vspace{4pt}$

$\left( \dfrac{2}{5}-\dfrac{3}{4}i\right) ^{5}=\allowbreak \frac{113221}{%
400000}+\frac{43737}{128000}i\vspace{4pt}$

$\left( 0.4-.75i\right) ^{5}=\allowbreak .\,28305+.\,3417i\vspace{4pt}$

$\sqrt{2.34-i}=\allowbreak 1.\,5628-.\,31994i\vspace{4pt}$

$\left( 2.5+0.5i\right) ^{\frac{4}{5}}=\allowbreak 2.\,088+.\,3325i\vspace{%
4pt}$

$\left( a+bi\right) ^{-1}=\allowbreak \frac{a}{a^{2}+b^{2}}-i\frac{b}{%
a^{2}+b^{2}}\vspace{4pt}$

$\left( 0.16-3i\right) ^{-1}=\allowbreak 1.\,7727\times 10^{-2}+.\,33239i%
\vspace{4pt}$

$\left( 8i\right) ^{\frac{1}{3}}=\allowbreak \frac{1}{2}\sqrt[3]{8}\sqrt{3}+%
\frac{1}{2}i\sqrt[3]{8}\vspace{4pt}$

$\sqrt{2+3i}=\allowbreak \frac{1}{2}\sqrt{\left( 4+2\sqrt{13}\right) }+\frac{%
1}{2}i\sqrt{\left( -4+2\sqrt{13}\right) }$

$\sqrt[3]{i}=\allowbreak \frac{1}{2}\sqrt{3}+\frac{1}{2}i$

$\sqrt{1+i}=\frac{1}{2}\sqrt{\left( 2+2\sqrt{2}\right) }+\frac{1}{2}i\sqrt{%
\left( -2+2\sqrt{2}\right) }$ \medskip
\end{quotation}

\strut

Note that \textsf{Evaluate} returns a different answer for $\left( \frac{2}{5%
}-\frac{3}{4}i\right) ^{5}$ and $\left( 0.4-.75i\right) ^{5}$. The fraction
displayed for $\left( \frac{2}{5}-\frac{3}{4}i\right) ^{5}$ is the exact
answer, and the number displayed for $\left( 0.4-.75i\right) ^{5}$ is the
best $5$-digit approximation to the exact answer.

\strut

For some roots you can obtain the answer in simpler form by applying \textsf{%
Simplify} after (or in place of) \textsf{Evaluate.}

\strut

$\blacktriangleright $ \textsf{Evaluate}

\begin{quotation}
$\left( 8i\right) ^{\frac{1}{3}}=\allowbreak \frac{1}{2}\sqrt[3]{8}\sqrt{3}+%
\frac{1}{2}i\sqrt[3]{8}$
\end{quotation}

\strut

$\blacktriangleright $ \textsf{Simplify}

\begin{quotation}
$\left( 8i\right) ^{\frac{1}{3}}=\allowbreak \sqrt{3}+i$
\end{quotation}

\strut

$\blacktriangleright $ \textsf{Evaluate Numerically}\label{Enter radical}

\begin{quotation}
$\left( 8i\right) ^{\frac{1}{3}}=\allowbreak 1.\,7321+1.0i$
\end{quotation}

\strut

\strut

\begin{quotation}
\emph{Tip}\textbf{\quad }To enter the $3$ in $\sqrt[3]{1+i}$, do one of the
following.
\end{quotation}

\begin{itemize}
\begin{itemize}
\item  
\begin{quotation}
Click \FRAME{itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{radical.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.2992in;height 0.2992in;depth
0.0692in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1.0179";cropright "1.0179";cropbottom "0";filename
'RADICAL.WMF';file-properties "XNPEU";}}; place the insertion point inside
the radical sign, press the \textsc{tab} key, and enter the $3$ in the small
input box that appears;
\end{quotation}
\end{itemize}
\end{itemize}

\begin{quotation}
\strut

- or -
\end{quotation}

\begin{itemize}
\begin{itemize}
\item  
\begin{quotation}
\strut Select the radical sign; click \FRAME{itbpF}{0.3001in}{0.3001in}{%
0.0701in}{}{}{property.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
0.3001in;height 0.3001in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Property.wmf';file-properties "XNPEU";}} or
choose \textsf{Edit + Properties}; and enter the $3$ in the small input box
that appears.
\end{quotation}
\end{itemize}
\end{itemize}

\strut

\label{Im}You can find the real and imaginary parts of a complex number with
the functions $\func{Re}$ and $\func{Im}$. When you enter these functions in
mathematics mode, they will automatically turn gray.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\func{Re}\left( \dfrac{a+bi}{c+di}\right) =\dfrac{ac}{c^{2}+d^{2}}+\dfrac{bd%
}{c^{2}+d^{2}}\vspace{6pt}=\allowbreak \dfrac{ac+bd}{c^{2}+d^{2}}$

$\func{Im}\left( \dfrac{a+bi}{c+di}\right) =\dfrac{bc}{c^{2}+d^{2}}-\dfrac{ad%
}{c^{2}+d^{2}}\vspace{6pt}=\allowbreak \dfrac{bc-ad}{c^{2}+d^{2}}$

$\dfrac{a+bi}{c+di}=\dfrac{ac+bd}{c^{2}+d^{2}}+i\left( \dfrac{bc-ad}{%
c^{2}+d^{2}}\right) \vspace{6pt}$
\end{quotation}

$\strut $

\begin{quotation}
$\func{Re}\left( \dfrac{3.6+6i}{5-3.25i}\right) =-4.2179\times 10^{-2}$%
\vspace{6pt}

$\func{Im}\left( \dfrac{3.6+6i}{5-3.25i}\right) =1.1726$
\end{quotation}

\strut

\subsection{\ Absolute Value\label{Absolute value}}

The \emph{absolute value } 
\index{Absolute value@Absolute value}of a complex number $z$, the distance
of $z$ from zero, is denoted $\left| \,z\right| $.

\begin{quote}
$\blacktriangleright $ To put vertical bars around an expression
\end{quote}

\begin{enumerate}
\item[1.]  Select the expression with the mouse.

\item[2.]  Click 
\FRAME{itbpF}{0.2992in}{0.2992in}{0.0692in}{}{}{fences.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.2992in;height 0.2992in;depth
0.0692in;original-width 24.0625pt;original-height 22.5625pt;cropleft
"0";croptop "1.0179";cropright "1.0179";cropbottom "0";filename
'FENCES.WMF';file-properties "XNPEU";}} or choose \textsf{Insert + Brackets}.

\item[3.]  Select the vertical bar.\medskip

\item[4.]  Choose \textsf{OK}.
\end{enumerate}

\strut

\begin{quote}
$\blacktriangleright $ To take the absolute value of a complex number
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in an expression enclosed between
vertical bars.

\item[2.]  Choose \textsf{Evaluate}.
\end{enumerate}

\strut

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left| a+bi\right| =\allowbreak \sqrt{\left( a^{2}+b^{2}\right) }\medskip $

$\left| 2+3i\right| =\,\sqrt{13}\medskip $

$\left| \sqrt{1+2i}\right| =\allowbreak \sqrt[4]{5}\medskip $

$\left| 2.5-16.3i\right| =\allowbreak 16.\,491\medskip $

$\left| e^{i\pi }\right| =\,1$
\end{quotation}

\strut

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\left| 2+3i\right| =\allowbreak 3.\,6056\medskip $

$\left| \sqrt{1+2i}\right| =\allowbreak 1.\,4953$
\end{quotation}

\strut

\subsection{\ Complex Conjugate \label{Complex conjugate}}

The complex conjugate of a complex number $a+ib$ is the complex number $a-ib$%
. To find the complex conjugate 
\index{Complex conjugate} of $a+ib$, evaluate the expression $\left(
a+ib\right) ^{*}$.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( a+ib\right) ^{*}=a-ib$ \qquad \qquad \qquad \qquad \qquad \qquad $%
\qquad \,\,$

$\left( 5-14i\right) ^{*}=5+14i$

$\left( 
\dfrac{3.6+6i}{5-3.25i}\right) ^{*}=-4.2179\times 10^{-2}-1.1726i$\qquad

$\left( a+ib\right) \left( a+ib\right) ^{*}=a^{2}+b^{2}\vspace{6pt}$
\end{quotation}

\strut \strut

\subsection{\ Numerical Approximations \label{Numerical approximations}}

\textsl{Scientific Notebook} returns an exact, or symbolic, answer whenever
appropriate and otherwise returns a numerical approximation. You can force a 
\index{Numerical results@Numerical results}\label{Numerical results}%
numerical result to any evaluation either by choosing\textsf{\ Evaluate\
Numerically} 
\index{Evaluate numerically@Evaluate numerically}\label{Evaluate numerically}%
from the \textsf{Maple}\ menu or by starting with numbers in decimal
notation. You obtain numerical approximations in response to any operation
when you enter numbers with decimals.

You can change the number of digits displayed in these approximations by
changing the setting in the \textsf{%
\hyperref{\textsf{Maple + Settings}}{}{}{DM2-3.tex#Maple settings}} dialog.
You can also change the number of digits used in computations and the
threshold for scientific notation in that dialog. Numerical approximations
are discussed in greater detail in the section \hyperref{Numerical
Approximations}{}{}{DM2-3.tex#Numerical approximations} under Real Numbers.

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Numerical Approximations}{}{}{DM2-3.tex#Numerical
approximations}

\item  \hyperref{The Display of Numerical Results}{}{}{DM2-3.tex#Maple
settings}
\end{itemize}

\rule{0.67in}{0.01in}

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"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Absolute value}{}{}{DMIndex.tex#Absolute}, \hyperref{Complex
conjugate}{}{}{DMIndex.tex#Complex conjugate}, \hyperref{Complex numbers}{}{%
}{DMIndex.tex#Complex numbers}, \hyperref{Imaginary part}{}{}{%
DMIndex.tex#Imaginary}, \hyperref{Imaginary unit}{}{}{DMIndex.tex#Imaginary}%
, \hyperref{Numerical approximations}{}{}{DMIndex.tex#Numerical
approximations}, \hyperref{Powers}{}{}{DMIndex.tex#Powers}, \hyperref{%
Radicals}{}{}{DMIndex.tex#Radicals}, \hyperref{Real numbers}{}{}{%
DMIndex.tex#Real numbers}}

\end{document}
