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\leftline{MONT 104Q -- Mathematical Journeys}
\leftline{Solutions for Midterm Exam, November 2, 2015}
\vskip 10pt
\noindent
I.  
\begin{enumerate}
\item[(A)]  (10) G. H. Hardy included two theorems and proofs from Euclid's {\it Elements} as
prime examples of ``real, serious, beautiful mathematics'' in his book 
{\it A Mathematician's Apology}.  Give the statements of those two theorems.  
{\it You do not need to supply the proofs for these}.
\vskip 10pt
\item[]  First theorem:  {\it Solution:} There are infinitely many prime integers.
\item[] Second theorem:  {\it Solution:} The number $\sqrt{2}$ is irrational (that is, it cannot
be written in the form $\sqrt{2} = \frac{a}{b}$ where $a,b$ are integers).  
\item[(B)]  (5)  Hardy says ``the mathematics which has permanent aesthetic value ... may
continue to cause intense emotional satisfaction to thousands of people after thousands of 
years.''   About how much time (to within 100 years) separates the time of Euclid from Hardy's
own time? 
\item[] {\it Solution:}  It was about 2200 years, give or take.  (Euclid lived about
300 BCE and Hardy was alive 1877 - 1947 CE, so taking 1900 CE as a date for Hardy, that's
about 2200 years.)
\end{enumerate} 
\vskip 10pt
\noindent
II.  
\begin{enumerate}
\item[(A)] (10)  State the 5 Common Notions (Axioms) and 5 Postulates at the start of 
Book I of the {\it Elements} of Euclid.  
\item[] {\it Solution:}  The Common Notions (This is just one possible way to state all of these.  I'll accept alternates meaning the same, and they don't need to be in this order)
\begin{enumerate}
\item[(1)]  Things that are equal to the same thing are equal to one another.
\item[(2)]  If equals are added to equals the wholes are equal.
\item[(3)]  If equals are subtracted from equals, the remainders are equal.
\item[(4)]  Things that coincide with one another are equal to one another.
\item[(5)]  The whole is greater than the part. 
\end{enumerate}
The Postulates:  (This is just one possible way to state all of these.  I'll accept alternates meaning the same.)
\begin{enumerate}
\item[(1)]  (It is possible) to draw a straight line from any one point to any other point.
\item[(2)]  (It is possible) to extend a line indefinitely in both directions.
\item[(3)]  (It is possible) to construct a circle with any given point as center and radius 
equal to any given line segment.
\item[(4)]  All right angles are equal.
\item[(5)]  If a line falling on two other lines makes angles on one side totaling less
than two right angles, then the two lines, when extended indefinitely, meet on the
side on which the two angles totaling less than two right angles are located.
\end{enumerate}
\vfill\eject
\item[(B)]  (5)  What is the purpose of the statements from part A in Euclid's logical
scheme?  How are the Common Notions different from the Postulates?
\item[] {\it Solution:}  The purpose of these statements is to give a starting point
for reasoning about geometry.  These statements are understood as unproved starting
assumptions and the basic facts of plane geometry are deduced from these.  The Common
Notions are different from the Postulates in that they (the Common Notions) are more
general -- they are principles of quantitative reasoning in general.  The Postulates
are more specfically statements about geometry. 
\end{enumerate}
\vskip 10pt
\noindent
III.  (10) Proposition 1 in Book I of Euclid's {\it Elements} is a construction for 
an equilateral triangle with side equal to any given line segment AB.  Give the construction
and the proof that the construction is valid, with justifications for each of the steps in 
the proof based on your answers to question II.  
\vskip 10pt
{\it Solution:}  The construction:  Given: the segment $AB$.
\begin{enumerate}
\item[(1)]  With center $A$ and radius $AB$, form one circle (Postulate 3).
\item[(2)]  With center $B$ and radius $AB$, form a second circle (Postulate 3).
\item[(3)]  Let $C$ be one of the intersections of the two circles and connect
$AC$ and $BC$ with line segments (Postulate 1).
\end{enumerate}
\noindent
Proof:  The claim is that $\Delta ABC$ is equilateral.  First, $AB = AC$ since they are
both radii of the first circle.  Similarly, $AB = BC$ because they are both radii
of the second circle.  Hence $AC = BC$ as well by Common Notion 1.  Therefore $\Delta ABC$
is equilateral.
\vskip 10pt
\noindent
IV.  Proposition 47 in Book I of the {\it Elements} is a famous statement from geometry
illustrated by the figure above.  Use the labeling here in your answers to all parts.
\begin{figure}
\begin{center}
\includegraphics[width=.5\linewidth]{EuclidI47.jpg}
\caption{Figure for Proposition 47, Book I}
\end{center}
\end{figure}
\begin{enumerate}
\item[(A)]  (5)  Give the statement {\it in Euclid's form} and the usual name of this result. 
\item[] {\it Solution:}  Euclid's statement is:  Let $\Delta ABC$ be a right triangle with
right angle at $A$.  Then the square on the side $BC$ opposite the 
right angle (the hypotenuse) is equal in area to the sum of the squares on the sides
$AB$ and $AC$.  This is the {\it Pythagorean Theorem}, but stated in terms of areas.   
\item[(B)] (5)  How is the dotted line $AM$ in the figure constructed?
\item[] {\it Solution:}  It's constructed to pass through $A$ and be {\it parallel} to 
the line containing $BD$.  The construction for that is given in a previous proposition
(Proposition 31, to be exact).
\item[(C)] (5)  In the first part of the proof, Euclid shows that $\Delta GBF$ has the same
area as what other triangle in the figure?  Why does that follow?
\item[] {\it Solution:}  $\Delta GBF$ has the same area as $\Delta CBF$. This follows
because those two triangles have the same base and are in the same parallels (Proposition 37).
Euclid establishes that $CG$ and $FB$ lie on parallel lines by considering the alternate
interior angles for the transversal line containing $AB$.
\item[(D)] (5)  The second part of the proof consists of showing that $\Delta FBC$ and 
$\Delta ABD$ are congruent.  How does that follow?  (Show that is true using one of the 
triangle congruence results proved before in Book I.)  
\item[] {\it Solution:}  
We have $FB = AB$ since they are two sides of the same square.  Similarly $BC = BD$ since
they are two sides of the same square.  Finally, $\angle FBC = \angle FBA + \angle ABC =
\angle ABC + \angle CBD = \angle ABD$, where the middle equality uses the facts
that $\angle FBA$ and $\angle CBD$ are both right angles and Postulate 4. Then
$\Delta FBC$ and $\Delta ABD$ are congruent by the SAS congruence criterion (Proposition 4).
\item[(E)] (5)  How does Euclid conclude that $ABFG$ and $BLMD$ have the same area?  And
how does he conclude the proof?
\item[] {\it Solution:}  First he shows that $\Delta ABD$ and $\Delta BDM$ have
the same area using Proposition 37 again.  Then Common Notion 1 says $\Delta GBF$ and
$\Delta BDM$ have the same area.  But $\Delta GBF$ has half the area of the square
$ABFG$ and $\Delta BDM$ has half the area of the rectangle $BLMD$.  So $ABFG$ and 
$BLMD$ also have the same area (Common Notion 2).  Euclid concludes the proof by 
saying that a similar argument shows the area of the other square $ACKH$ is 
equal to the area of the rectangle $LMEC$.  Then adding we get that the area of the
square $BDEC$ is equal to the sum of the areas of the squares $ABFG$ and $ACKH$.  
\end{enumerate}
\vskip 10pt
\noindent
IV.  Essay.  (35) George G. Joseph, the author of an interesting book about the history of mathematics called {\it The Crest of the Peacock}, offered 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 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?   What is Joseph getting at? 
Then, based on what he said in {\it A Mathematician’s Apology}, how would G.H. Hardy respond
to Joseph?  Finally, which side of this debate do you come down on personally? Should all the mathematics we learn and do have practical usefulness or significance?
\vskip 10pt
\noindent
{\it Sample Response:}  The last sentence in the quote from Joseph's book is referring to 
the fact that in texts like Euclid's {\it Elements}, much more attention is paid to 
the abstract geometric relationships of triangles, parallelograms, squares, etc. 
and the logical sequence of the postulates and propositions rather than to any possible
uses of geometry to solve real-world problems.  Euclid in particular to care more about
how statements are proved (``how they arrived at a conclusion'') than in potential 
applications.  Moreover, Joseph is saying that this focus on proof and on the logical
structure of mathematical deduction held Greek mathematics back because it divorced
mathematics from the applications that would have lead to other new ideas and 
discoveries.  [{\it Comment:}  For Joseph, and for many modern mathematicians 
(but not Hardy!), a lot of really interesting mathematics comes from the practical 
problems and questions from the sciences and other areas of inquiry.]   

Based on the ideas he expressed in {\it A Mathematician's Apology}, we can safely assume
that Hardy would have {\it disagreed vehemently} with Joseph.  For Hardy, 
any ``practical significance'' of results is definitely of secondary importance.  He 
is more interested in the logic of proofs and the sense of beauty and unexpectedness he 
finds in ``serious, real'' mathematics like the propositions from Euclid that he quotes as 
his prime examples (from question I before).  Joseph would say those deal with
``abstract, idealized constructs,'' while Hardy would say in effect that that is 
why they are interesting to him.  Indeed, Hardy finds most of applied
mathematics ugly and boring.  While he acknowledges that mathematics can be useful, 
it is not that side of the subject that appeals to him.  

[Any personal opinion is OK for the last part!]     
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