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\centerline{MATH 110 - 02 -- Algebra Through History}
\centerline{Information for Final Exam}
\centerline{December 11, 2019}
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\noindent
{\it General Information}
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The final exam will be given in our regular classroom at 
{\it 11:00am on Wednesday, December 18}, as announced by
the Registrar.
It will cover the historical material we have studied
since the start of the semester.  

This exam will be comprised of about $60\%$ on short answer questions 
related to the history we have discussed, and $40\%$ on
a roughly 2-page essay on an assigned topic (see below).
If you are well-prepared, I'm expecting the exam will take about
1.5 hours to complete, but you will have the full 2.5 hour
exam period to work on it if you need more time.
{\it Note:}  I will {\it not} be asking any questions where
you need to solve mathematical problems, although you will
see that some of the historical questions will ask for details
about the mathematical contributions of the people we studied.

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\noindent
{\it Mathematical/Historical Topics} 
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\item{$\bullet$} From our study of Old Babylonian mathematics: 
\itemitem{a.}  Know the approximate historical period represented by the 
Old Babylonian mathematical texts.  
\itemitem{b.}  Know the base-60 number system and number symbols and be
able to read ``the real thing'' and convert to base-10 form (given information about where the 
fractional part of the number starts).
\itemitem{c.}  Know and understand what the tablets YBC 6967 and YBC 7289 contain
\itemitem{d.}  The role of addition, multiplication, reciprocal, $n^3 + n$, ... tables 
in Babylonian arithmetic
\itemitem{e.}  Know the different interpretations of the Babylonian ``quadratic algebra''
problem texts like YBC 6967 given by Otto Neugebauer and more recent historians like Jens Hoyrup
and Eleanor Robson.  
\item{$\bullet$}  From our study of the Greek algebra and number theory:
\itemitem{a.}  Know the approximate historical periods of Euclid and Diophantus.
\itemitem{b.}  The meaning of incommensurability of magnitudes, the discovery of incommensurable 
magnitudes like $\sqrt{2}$, know the way that the diagonal of a square 
was proved to be incommensurable with the side of the square (see class materials for 
September 27 for the proof from ``Book X, Proposition 117'' in Euclid).  Know the way the
Greeks dealt with the fact that these incommensurable magnitudes exist.
\itemitem{b.}  Euclid's ``geometric algebra'' from Book II of the {\it Elements}; 
Know the proof of Proposition 5 (the geometric form of the 
identity $ab + ((a+b)/2 - b)^2 = ((a+b)/2)^2$.)  See pages 7 and 8 of the class materials
for September 25.  
\itemitem{c.}  Diophantus' {\it Arithmetica}.  In particular, know
the algebraic symbolism Diophantus introduced and be able to ``decode''
the proof/example for Proposition 4 from Book I (translate his notation into
modern algebra, carry out his steps and solve the problem).     
\itemitem{d.}  How know Proposition 8 from Book II of the {\it Arithmetica}
led to ``Fermat's Last Theorem'' and when by whom that was finally established.
\itemitem{e.}  Know the time frame for Pappus of Alexandria, his mathematical
contributions, and in general terms what his ideas about {\it analysis} and {\it synthesis}
in mathematics were.  
\item{$\bullet$}  From our study of medieval Islamic mathematics:
\itemitem{a.}  Know when and where Al-Khwarizmi was active, what his main contributions to algebra 
were, how he presented solutions of different types of quadratic equations, and 
in particular how he said to solve the ``squares equal to roots and numbers'' case.
\itemitem{b.} Know the other Islamic mathematicians we discussed (Thabit Ibn Qurra, Umar
al-Khayyami $=$ Omar Khayyam), and their contributions to algebra that we discussed
\itemitem{c.} Know (in general terms) how the Greek mathematical works translated
into Arabic and the works of the Islamic mathematicians were transmitted to western Europe
after about 1100 CE.  Know the roles of Gerbert of Aurillac, Gerard of Cremona, Robert of Chester, 
and Leonardo Pisano (also known as Fibonacci).
\item{$\bullet$}  Renaissance algebra:
\itemitem{a.}  Tartaglia, Ferrari, Cardano, and the solution of cubic and quartic equations.
Know the period these mathematicians lived and the story of how the solution of these
equations became known.
Know Tartaglia's formula for solving cubics of the ``cubes and roots equals numbers'' form, i.e.
$x^3 + px = q$.  Recall that this is:
$$x = \root 3 \of{{q\over 2} + \sqrt{\left({q\over 2}\right)^2 + \left({p\over 3}\right)^3}}
+ \root 3 \of{{q\over 2} - \sqrt{\left({q\over 2}\right)^2 + \left({p\over 3}\right)^3}}$$
\itemitem{b.}  Fran\c cois Vi\`ete and {\it The Analytical Art}.  Know the timeframe, the
major characteristics of what he did--for instance, how did his work go beyond what
Diophantos had done before?  Also know 
Vi\`ete's point of view about Al-Khwarizmi and the Islamic algebra he had inherited,
and how he based his work on ideas from Pappus of Alexandria (see above).  Know 
Vi\`ete's Homogeneity Principle and how that factored into his ideas about algebra.
\itemitem{c.}  Descartes, {\it La G\'eometrie} and the development of ``analytic geometry.''
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{\it Essay}
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The essay question will be the following:
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Although this has mainly been a course about the historical development
of algebra, when you look back at what we have learned, you should see that
one consistent theme throughout has been the ways algebra
and geometry have been connected with each other, and the changing ways mathematicians have
thought about that relationship.  Discuss the relationship 
between algebra and geometry in each of the following periods, or mathematical works.
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\item{(1)}  The Old Babylonian period and the problem texts such as YBC 6967.  And is there
a difference of opinion about how to interpret what is algebra and what is geometry there?
\item{(2)}  The presentation of elementary mathematics in Euclid's {\it Elements}, 
especially in Book II.
\item{(3)}  The {\it Arithmetica} of Diophantos.
\item{(4)}  The {\it Hisab al-jabr w'al-muqabala} of Al-Khwarizmi.
\item{(5)}  {\it La G\'eometrie} of Ren\'e Descartes.  
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[Note: This next part is asking for an opinion; no single ``right answer'':]  In teaching 
algebra today, which way of dealing with algebra and geometry do you think would
help students most to learn the subject? And would knowing some of this history
also help?
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{\it Miscellaneous Groundrules}
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No use of cell phones, pagers, I-pods, or any other electronic devices
beyond a calculator will be allowed during the exam -- turn them off
and stow them in your backpack.
\bye
