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\assigntitle{17}{The Polar Form of Complex Numbers}

Now that you have a foundation in the basics of complex numbers, this
week, as promised, we're going to explore the extremely fascinating
\emph{polar representation} of complex numbers.

But first, a digression\dots

\section{The number line}
\label{sec:number-line}

``The number line?'' I hear you exclaim. ``That's, like, SO first
grade.  Why are we learning about the number line?''  Well, bear with
me for a minute\dots

\topic{the number line}
As you no doubt learned in first grade (with successive refinements to
include negative numbers, and then fractions, and then real numbers),
we can think of the real numbers as inhabiting a one-dimensional space
called the \term{number line}.  \pref{fig:numberline-integers}
illustrates the set of integers $\Z$ located at discrete points on a
number line; the real numbers, of course, inhabit the entirety of the
line.

\diagrampst{numberline-integers}{The number line}

\topic{arithmetic as geometry}
So, why is this useful or interesting?  The key point is that
\emph{simple arithmetic operations on the real numbers have simple
  geometric interpretations on the number line}.

\topic{addition is translation}
For example, as illustrated in \pref{fig:numberline-addition},
addition corresponds to \emph{translation}.  If you start with some
number $n$, adding four to it corresponds to moving four units right
along the number line.  Likewise, subtracting four (that is, adding
negative four) corresponds to moving four units left.

\diagram{numberline-addition}{Real addition corresponds to
  translation on the number line}

\topic{multiplication is scaling/reflection}
Multiplication by a positive number corresponds to \emph{scaling};
multiplication by a negative number also corresponds to
\emph{reflection}. 

\begin{problem}
  Draw pictures (something like \pref{fig:numberline-addition}) that
  illustrate the scaling and reflection properties of multiplication
  on the number line.  You can submit these pictures in one of two
  ways:
  \begin{enumerate}
  \item If you draw them using some sort of computer program, you can
    email them along with your assignment.
  \item If you draw them on paper, you can mail them to me. (In this
    case, just mail them by the due date for your assignment; it
    doesn't matter when they reach me.)
  \end{enumerate}

\end{problem}

What about complex numbers?  Can we come up with a spatial model for
the complex numbers, where complex arithmetic has a nice geometric
interpretation? 

\section{The complex plane}
\label{sec:complex-plane}

\topic{the complex plane}
Just as we can picture the integers or real numbers as being located
on a one-dimensional structure, the \emph{number line} or \emph{real
  number line}, we can picture the complex numbers in the
two-dimensional \term{complex plane}, as shown in
\pref{fig:complex-plane}. 

\diagram{complex-plane}{The complex plane}

\topic{real and imaginary axes} The real numbers are located along the
horizontal axis (marked \textbf{\textsf{R}} in the picture
above\footnote{It really should be marked $\R$ instead of
  \textbf{\textsf{R}}, but\dots well, it's a long story.}), and the
imaginary numbers are located along the vertical axis (marked
\textbf{\textsf{I}}).  So instead of the $x$-axis or $y$-axis we talk
about the \term{real axis} and the \term{imaginary axis}. Zero, which
is of course both a real number and an imaginary number, is at the
origin, the intersection of the real and imaginary axes.  So, for
example, the complex number $3 + 2i$ is located three units right and
two units up from the origin, exactly where $(3,2)$ would be located
if we were talking about the Cartesian plane. (In some sense, we
\emph{are} talking about the Cartesian plane.)

\topic{complex arithmetic as geometry?}
So, this \emph{seems} rather natural, but it would be really nice if
complex arithmetic operations had geometric interpretations in the
complex plane, just as real arithmetic has geometric interpretations
on the number line.

Well, for addition, it turns out that this isn't too hard:

\begin{problem}
  What is a geometric interpretation of complex addition?  In other
  words, if you start at some point in the complex plane and add $a +
  bi$, where do you end up?
\end{problem}

However, multiplication is not as obvious!

\begin{problem}
  Get out a piece of graph paper and plot each of the following points
  in the compex plane (you may want to use a different graph for each
  subproblem, so you can see what's going on).
  \begin{subproblems}
    \item $i$, $i^2$, $i^3$, $i^4$
    \item $3 + 2i$, $-2+1$, and $(3+2i)(-2+i)$
    \item $4i$, $-1/2$, $(-1/2) \cdot 4i$
  \end{subproblems}
\end{problem}

\begin{problem}
  Before you read on, write down some of your observations or guesses
  about what multiplication does geometrically in the complex plane.
\end{problem}

To really see what is going on with multiplication, we'll have to talk
about\dots

\section{Complex numbers in polar form}
\label{sec:complex-polar}

\topic{complex numbers using polar coordinates}
Now that we are thinking of complex numbers as occupying points on a
plane, an idea naturally suggests itself: what would happen if we
thought of complex numbers in terms of polar coordinates, instead of
Cartesian coordinates?  

\diagram{complex-polar}{Representing complex numbers in polar coordinates}

\begin{problem} \label{prob:z-polar-as-rect}
  Consider \pref{fig:complex-polar}.  It shows some complex number $z$
  in the complex plane, with its distance $r$ and angle $\theta$ from
  the origin labeled.  Write $z$ in Cartesian ($a + bi$) form, in
  terms of $r$ and $\theta$.
\end{problem}

\topic{a Really Amazing Fact}
And now for a Really Amazing Fact: it turns out that
\begin{equation}
  \label{eq:complex-exp-def}
  e^{i\theta} = \cos \theta + i \sin \theta.
\end{equation}

\topic{huh?}
Now, you might very well ask whether I am pulling your leg.  What does
it even \emph{mean} to raise $e$ to the power of an \emph{imaginary}
number!?  Unfortunately, there isn't room on this assignment to
explain---and even if there were, every way I know how to explain it
requires calculus!  But you can take my word for it that if you think
about what raising $e$ to an imaginary power \emph{could possibly}
mean, there is only one possible definition that works out correctly
with everything else we already know about $e$, exponentiation,
imaginary numbers, sine, cosine, and so on---and that definition is
\pref{eq:complex-exp-def}!  Hopefully you can see how surprising and
beautiful this equation is.  Why should cosine and sine show up if you
raise $e$ to an imaginary power?  Before seeing this equation, you
would have no reason to suspect that $e$, cosine, and sine even have
anything at all to do with one another.

\begin{problem}
  \topic{a beautiful equation}
  Substitute $\theta = \pi$ into \pref{eq:complex-exp-def}, and
  simplify using your knowledge of sine and cosine.  Now add one to
  both sides, and write down the equation you get. This is called
  \term{Euler's identity}.  Why do you think this is considered one of
  the most beautiful equations in all mathematics?
\end{problem}

\begin{problem} \label{prob:polar-form}
  \topic{the polar form of a complex number}
  Use \pref{eq:complex-exp-def} to rewrite your answer to
  \pref{prob:z-polar-as-rect} in terms of $r$, $\theta$, $i$, and $e$.
\end{problem}

Your answer to \pref{prob:polar-form} is the \term{polar form} of a
complex number.  Although it's difficult to add two complex numbers in
polar form (it would easier to just convert them to Cartesian form
first), the polar form makes multiplication very easy!

\begin{problem}
  \topic{multiplication of polar forms}
  Using the polar form you found in \pref{prob:polar-form}, show that
  if one complex number is $r_1$ units away from the origin at angle
  $\theta_1$, and another complex number is $r_2$ units away from the
  origin at angle $\theta_2$, then their product is $r_1 r_2$ units
  away from the origin at an angle $\theta_1 + \theta_2$.
\end{problem}

\topic{complex multiplication is rotation!}
This, then, is the answer: in the complex plane, multiplication
corresponds not just to \emph{scaling} (like it did on the number
line) but also to \emph{rotation}!  For example, multiplying by $i$
corresponds to a counterclockwise rotation of $\pi/2$ ($90^\circ$) in
the complex plane.  And multiplying by $2e^{i\pi/4} = \sqrt{2} +
i\sqrt{2}$ corresponds to a rotation by $\pi/4$ and scaling by
$2$---that is, any complex number, when multiplied by $\sqrt{2} +
i\sqrt{2}$, will end up rotated by an angle of $\pi/4$, and twice as
far away from the origin.

\begin{problem}
  Plot $1+i$, $(1+i)\left(\sqrt{2} + i\sqrt{2}\right)$,
  $(1+i)\left(\sqrt{2} + i\sqrt{2}\right)^2$, and $(1+i)\left(\sqrt{2}
    + i\sqrt{2}\right)^3$ on your graph paper.  What happens?
\end{problem}

\section{Beyond 2D?}
\label{sec:beyond}

\topic{quaternions!}
You might well ask whether there are any sorts of numbers which can be
envisioned as inhabiting some sort of space with a dimension higher
than two.  And the answer is\dots yes!  There are four-dimensional
numbers called \term{quaternions}. The quaternions still have an
imaginary number $i$ with $i^2 = -1$; but they have two additional
imaginary numbers, called $j$ and $k$.  $j^2$ and $k^2$ are both equal
to $-1$, just like $i^2$; additionally, $ijk = -1$.  Just as
complex numbers can be written in the form $a + bi$, where $a$ and $b$
are real numbers, quaternions can be written in the form $a + bi + cj
+ dk$, where $a$, $b$, $c$, and $d$ are all real numbers.

Interestingly, just as we lose a nice property when generalizing from
the real numbers to the complex numbers (the fact that the real
numbers are in a certain \emph{order}, so we can talk about real
numbers being less than or greater than other real numbers), we lose
another property when generalizing to the quaternions: quaternion
multiplication is not \emph{communitive}, that is, if $x$ and $y$ are
quaternions, it is not necessarily true that $xy = yx$ (this is true
for complex numbers).  Quaternions can represent rotations in three
dimensions, so they are sometimes used in computer graphics, as well
as in other applications.  

There are also eight-dimensional \term{octonions} (octonion
multiplication is not even \emph{associative}), and
sixteen-dimensional \term{sedenions}, but by that point so many nice
properties have been lost that it's not even clear whether these
things should be called ``numbers'' anymore!


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