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\begin{document}

\section{Homework 1 - due Monday}

\begin{enumerate}

\item Given distinct points $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$, suppose that $P = (x,y)$ is any
point on a line through $P_1$ and $P_2$. Show that $x$ and $y$ satisfy the equation
$$\frac{y_2-y_1}{x_2-x_1}=\frac{y-y_1}{x-x_1}$$
if $x_2 \neq x_1$. Explain why this equation is the equation of a straight
line. What happens if $x_2 = x_1$?

Parallel lines, not surprisingly, turn out to be lines with the same slope.
Show that distinct lines $y = ax+c$ and $y = a_1x+c_1$ have a common point
unless they have the same slope $a = a_1$. Deduce  that the parallel to a line $\ell$ is the unique line
through $P$ with the same slope as $\ell$.

\item Using distance it's fairly straightforward to derive the equation of a circle once you know that a circle is a set of all points equidistant from a given point. What is the set of points equidistant from two points? How do you prove that?

\item Distances are useful not in just proving that some are equal, but that some are unequal! Prove triangle inequality.\footnote{Choose a convenient triangle!}

\item Hilbert needed an axiom that told him how many times a circle and a line (or circle and a circle) can intersect. Explain why you don't need any such axiom. 


\item Let's briefly venture into the world of complex numbers.
Consider the numbers $n = 4 + 6i, j = 2 + i, a = 4 + i$. 
\begin{itemize}
\item Calculate the results when each of $n, j$, and $a$ is multiplied
by $i$.
\item  Calculate the results when each of $n, j$, and $a$ is multiplied
by $i$ twice.
\item  Calculate the results when each of $n, j$, and $a$ is multiplied
by $i$ thrice, um. I mean three times.
\end{itemize}

How about graphing these as well? \\

 Do the same, but now multiply and graph the results of multiplication by $-i$. 


\item You may want geogebra for this\footnote{Click on View/Spreadsheet. In the cell A1 enter 1+i. Click on cell B1 and type =A1*(1+i). Click on A2 and type =B2. Then copy A2 down and B1 down. There may be an easier way....}: Consider the complex number $z =1 + i$. Plot and label
each of these on the same complex plane.
\begin{itemize}
\item $z$
\item $z^2$
\item $z^3$
\item $z^4$
\item $z^5$
\end{itemize}

Theresa stands at the origin $(0, 0)$ and stares at the powers
of $z = 1+ i$ as they are built. What happens to these numbers from Theresa's persepctive?


\item  Consider the complex number $z = \frac{3}{5} + \frac{4}{5} i$. Plot and label
each of these on the same complex plane.
\begin{itemize}
\item $z$
\item $z^2$
\item $z^3$
\item $z^4$
\item $z^5$
\end{itemize}


Connie stands at the origin $(0, 0)$ and stares at the powers
of $z = \frac{3}{5} + \frac{4}{5} i$ as they are built. Describe what happens as accurately
as you can. How does it compare to what happens
with the powers of $1 + i$? Why is it different?

\item You observed what happened to powers of $1+i$. What happens to a point $z$ in the complex plane if it is multiplied by $1+i$? As it is multiplied by $1+i$ repeatedly? 

\item 
You observed what happened to powers of$z = \frac{3}{5} + \frac{4}{5} i$. What happens to a point $z$ in the complex plane if it is multiplied by $z = \frac{3}{5} + \frac{4}{5} i$? As it is multiplied by $z = \frac{3}{5} + \frac{4}{5} i$ repeatedly? 


\end{enumerate}

\pagebreak
\section{Monday}

Today we will describe all the isometries of the plane. 

\end{document}
