% Math 370 hw5 Spring 2014
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\title{History of Mathematics \\
Homework 5
}
\author{Ethan Bolker}

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The first question requires work before Tuesday. It's all due
on Thursday, March 13 .

\begin{enumerate}

\item Come prepared to discuss possible term paper topics.

Your paper should follow the model we've used in the course: present
some mathematics in the form appropriate to the period or author
you're discussing, along with a modern version and connections to your
own education. Think small -- one theorem, one novelty, in context,

 I'd like it
  best if you chose a topic that you know you want to know more
  about, but I am willing to make some suggestions. Here are a few.

\begin{itemize}
\item Biography. There are good ones on
  Srinivasa Ramanujan, David Hilbert, Richard Courant, Julia Robinson,
  Leonhard Euler, Paul Erd\H{o}s, \ldots .

  More at \url{http://www.ericweisstein.com/encyclopedias/books/MathematicsBiography.html}

  Note: You have to read a whole book, then choose and and write about
  some actual mathematics in the way I suggest above. You can't just
  cut some stuff from web pages. 

  \item Follow up on some topic  we've touched on in class
    that you would like to know more about. The text suggests lots of
    references. 

\item Write about some mathematical contribution from a nonwestern
  culture -- China, India, Babylon, Africa, \ldots

\item Write about the history of some mathematics that connects to its
  use in other sciences -- perhaps calculus and Newtonian mechanics,
  matrix algebra and quantum mechanics or Riemannian geometry and
  relativity theory. 

   \item Check out papers from previous essay
     competitions at \url{http://historyofmathematics.org/} --
     the web site for the MAA's special interest group on the History
     of Mathematics. Any paper you write that's good enough to submit
     for the 2014 contest will earn you an automatic A in this course.
(The 2013 contest closes March 19, so it's too late for that one.)

\end{itemize}



\item Do Exercise 4.38 in the text.

\item What is the etymology of the word ``cyclotomic''?

\item The Euclidean Algorithm (which we will study in class Tuesday).

\begin{enumerate}

\item Find a solution in integers to the equation
%
\begin{equation*}
1001x + 512y = 1 .
\end{equation*}

\item Where in Euclid will you find the Euclidean algorithm?
 Did Euclid invent/discover it?

\item  Show that after every two steps in the Euclidean algorithm, the
  remainder is reduced by at   least one half. 
Conclude that the Euclidean algorithm terminates in at most
$2\log_2(b)$ steps.
In particular, show that the number of steps is at most seven
times the number of digits in $b$.

\item Experiment with the Euclidean algorithm applied to find the
  greatest common divisor of consecutive Fibonacci numbers. Discover
  something cool.

\end{enumerate}


\end{enumerate}

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Here is the \LaTeX{} source for this document. You can cut it from the
pdf and use it to start your answers. I used the \verb!\jobname! macro
for the source file name, so you can call your file by any name you like.

\verbatiminput{\jobname}

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