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\begin{center}
\textbf{MATH 110-02 -- Algebra Through History} \\
\textbf{Sample Midterm Exam Questions}
\end{center}
\vskip 10pt
\noindent
{\it Disclaimer:}  The following questions indicate the rough length of the
coming midterm and the ``mix'' of different kinds of questions I might ask.
The actual exam might be set up somewhat differently and it might draw
on other topics from the review sheet.  
\vskip 10pt
\noindent
I.  Refer to the Old Babylonian tablet YBC 7289 in Fig 1 (at top of next page).
\vskip 10pt
\begin{enumerate}
\item[A)] The base-60 form of the number written directly over the diagonal of the 
square has just one base-60 digit to the left of the semicolon separating the whole number
part from the fractional part.  Give all the base-60 digits, separated by the semicolon and commas.
\item[B)] Express the whole number from part A in base 10.  (It's OK to leave this as a sum of fractions if you don't have a calculator.)
\end{enumerate}
\vskip 10pt
\noindent 
II. 
\vskip 10pt
\begin{enumerate}
\item[A)] What does it mean to say that that the diagonal of a square
and the side of that square are \emph{incommensurable}?
\item[B)]  Give the proof the ancient Greeks found for the fact in part A.  
\item[C)]  What consequences
did the discovery of incommensurable magnitudes have for the way the 
Greeks thought of (pure) mathematics?
\end{enumerate}
\vskip 10pt
\noindent
III.  Short answer.  Answer any four of the following.  If you answer
more than four, the best four will be used.
\vskip 10pt
\begin{enumerate}
\item[A)]  What is the main feature that makes the presentation of the number theoretic
problems in Diophantus' {\it Arithmetica} different from earlier mathematics and a forerunner
of what we do now.
\item[B)]  What is the ``algebraic form'' of the fact proved in Proposition II.5
in Euclid's {\it Elements}? (The statement: If a straight line is cut into equal and 
unequal pieces then the rectangle contained by the unequal pieces of the whole line,
plus the square on the difference between the equal and unequal pieces is equal 
to the square on half of the line.)
\item[C)] What geometric fact did Archimedes have inscribed on his tombstone? 
\item[D)] What are the approximate dates of the YBC 6967 and YBC 7289 tablets?  About
when did Euclid live?
\item[E)] What does the $M^o 20 \varsigma 1$ in Diophantus' Proposition I.4 mean?
\item[G)] What is ``Fermat's Last Theorem'' and what is its connection with Diophantos?
When and by whom was this finally solved?
\item[H)] How would the Old Babylonian scribes have approached a large calculation
such as multiplying $23,41;16$ by $17,28;30$?  (You don't need to do the calculation.)
Compare this with what we need to know to do multiplication in our base-10 positional number
system.
\end{enumerate}
\begin{figure}
\begin{center}
\includegraphics[width=.4\linewidth]{YBC7289c.jpg}
\end{center}
\caption{Drawing of the YBC 7289 tablet by Asger Aaboe.  I do not have his permission to use this, but
he died in 2007, so I think the copyright issue is effectively moot(!)}
\end{figure}
\vskip 10pt 
\noindent
IV.  Essay.  You have the choice of responding to {\it either prompt 1 or 2}.  State which
one you have chosen at the start of your essay.
\begin{enumerate}
\item[1)]  A certain older history of mathematics says, flatly, that  
``the distinguishing feature of Babylonian mathematics is
its algebraic character.''  Of the historians we have mentioned, who
would agree with this claim and who would disagree?  
Explain using the the interpretations your historians would give 
for the YBC 6967 problem of (what we would phrase as) solving the 
equation $x = 60/x + 7$.  
\item[2)]   George G. Joseph, the author of another book on the non-European roots of 
modern mathematics called {\it The Crest of the Peacock}, offers
this overall evaluation of the ultimate impact of Greek geometry:  ``There is no 
denying that the Greek approach to mathematics produced remarkable results, but 
it also {\it hampered} the subsequent development of the subject.  ...  
Great minds such as Pythagoras, Euclid, and Apollonius spent much of their time 
creating what were essentially abstract idealized constructs; how they 
arrived at a conclusion was in some way more important than any practical 
significance.'' First, what does the last sentence mean?  
Would this criticism seem to be apt for 
Diophantos' {\it Arithmetica} as well?  Is it necessary for all the mathematics 
we learn and do to have practical usefulness or significance?  
\end{enumerate}
\end{document}
