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\centerline{MONT 104Q -- Mathematical Journeys}
\centerline{Information for Final Exam}
\centerline{December 7, 2015}
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\noindent
{\it General Information}
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The final exam announced in the course syllabus 
will be given at 11:30am on Friday, December 18 in our regular classroom, Swords
209.  You will have the entire 2.5 hour exam period to work on the exam if you 
want or need that much time.  However, if you are well-prepared and work steadily, 
I do not expect that it will take any longer than 1.5 hours to complete.  
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\noindent
{\it Miscellaneous Groundrules}
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No use of cell phones, pagers, I-pods, I-pads,  or any other electronic devices
will be allowed during the exam.  There will not be any numerical computations
requiring use of a calculator.  You will be asked to turn all devices off
and stow them in your backpack.
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\noindent
{\it What Will Be Covered}
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The exam will cover the mathematical material we have studied
since the start of the semester, but {\it not} the CHQ common readings (except for
one option on the essay portion of the final -- see below). 
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\item{$\bullet$} Our discussion of Hardy's {\it A Mathematician's Apology}, Hardy's
ideas about pure and applied mathematics, his sense of mathematical beauty (esthetics), 
his thoughts about his own life and why he did what he did
\item{$\bullet$} Book I of the {\it Elements} of Euclid
\item{$\bullet$} Our proof of the ``power theorem'' (which said that
if $m$ is a prime number and $1 \le r \le m - 1$, then $r^{m - 1} R m = 1$).  Incidentally,
the ``official name'' of this result is {\it Fermat's ``Little Theorem''}.  (It's called
that to distinguish it from Fermat's ``Last Theorem'' which was an unsolved problem until 
about 1994, when Andrew Wiles and Richard Taylor completed a proof).
\item{$\bullet$} Imre Lakatos' ideas about the development of mathematics through proofs
and refutations and how he essentially rules out the idea of absolute certainty derived
from proofs of mathematical statements.  (You're free to disagree with that, 
of course; many mathematicians would also disagree.  But you should understand that that
{\it is} Lakatos' point of view, at least about the current frontiers of mathematics, the
things mathematical researchers are actually working on at any given time.)
\item{$\bullet$} The relation $V - E + F = 2$ for convex polyhedra as developed
through our reading of the first sections of {\it Proofs and Refutations}.
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\noindent
This means in particular that I might ask you about
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\item{1.}  The statements {\it and proofs} of the two theorems Hardy gives as 
supreme examples of ``real mathematics'' (from the first group discussion day).  (A word 
to the wise:  We did not
do the proofs on the midterm; you should expect to see one of the proofs on the final.)
\item{2.}  Know the statements and the proofs of Propositions 27, 28, 29 and 32 of Book I of the 
{\it Elements}.  Be able to identify what Axioms (Common Notions), Postulates, and/or previously proved results are used in the proofs of these, how Propositions 27 and 29 are related, and 
know especially why Proposition 29 is such an important step in the development of Book I.  
\item{3.}  Know the statement and the proof of Proposition 47 in Book I of the {\it Elements}.
For this one, you needn't memorize the numbers of the previous Propositions that are 
used in the proof (because there are quite a few).  It will suffice just to say ``by a 
previous proposition, we know ... .''  (Note:  This appeared on the midterm exam too, but
it's very important and a high point of what we have done!)
\item{4.}  Know the pattern that we found in the remainders $r^{m-1} R m$ for $m$ a prime number and
$1 \le r \le m-1$.  Also know the proof we developed that this always holds.
\item{5.}  Know the relation $V - E + F = 2$ for convex (and other) polyhedra and how
we sketched a proof in that case.  (Recall, this was an unsatisfyingly incomplete 
``monster-barring'' exercise for Lakatos, but it is an interesting proof even if it is not
``the whole story'' about when $V - E + F = 2$ holds for polyhedra.)
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\noindent
{\it Format}
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Approximately 65\% of the exam will consist of questions related to the mathematical results
and theorems above.
The remaining approximately 35\% will be an essay (target length: about 2 
handwritten pages), on one of the topics below.  To prepare well, you 
will want to think (and probably write) out practice essays on the topics.  
In the evaluation of the exam essay, I will be looking at the content and 
organization of what you are saying, but {\it not} at mechanical issues of 
grammar, punctuation, spelling, etc. 
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\noindent
{\it The Essay Question}
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\noindent
For the essay question, you will have a {\it choice} of either option A or option B below.  
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\item{$\bullet$} Option A:  Write on {\it one of the following topics} (yes, one of these exact 
questions).  I will choose which of these two appears on the exam.
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\item{1)} Oliver Heaviside, 1850-1925, an English engineer, applied mathematician, 
and physicist, once wrote the following about the role of Euclid in mathematical education in 
his time in England:  ``As to the need of improvement there can be no question whilst the reign 
of Euclid continues.  My own idea of a useful course is to begin with arithmetic, 
and then not Euclid but algebra.  Next, not Euclid, but practical geometry, solid 
as well as plane; not demonstration, but to make acquaintance. Then not Euclid, 
but elementary vectors, conjoined with algebra, and applied to geometry ... Elementary 
calculus should go on simultaneously ... . Euclid might be an extra course for 
learned men, like Homer.  But Euclid for children is barbarous.''  On the other hand, 
about 5 years ago, Peter Rudman, a contemporary physicist, wrote this: ``High school 
mathematics education today, ... , 
all too often neglects the derivations where mathematics is learned and emphasizes 
memorizing the equations that provide quick solutions in the standardized tests but 
that are then rapidly forgotten ... .''  What aspects of mathematics does each of 
these authors seem to value most highly and think students should learn?  How does 
what each of them says relate to the ideas of G.H.Hardy in {\it A Mathematician's 
Apology}?  Why might Heaviside say that teaching Euclid to children is ``barbarous?''  
Was your high school mathematics more or less like what Heaviside is recommending?  
Was your experience like that Rudman describes?  Do you think that emphasizing proofs 
more would have made mathematics more interesting for more people?  Or is that too much 
to hope for?
\item{2)}  Alex Bellos, in his book {\it Here's Looking at Euclid} (catchy title, no?)
says, refering to Euclid's proof of Proposition 47 in Book I of the {\it Elements}, ``the 
thrill of math[ematic]s is the moment of instant revelation, from proofs 
such as this, when suddenly everything makes sense. It is immensely satisfying, 
an almost physical pleasure. The Indian mathematician Bhaskara (1114-1185, CE) was so taken by 
this proof that underneath a picture of it in his twelfth-century math[ematic]s 
book {\it Lilavati}, he wrote no explanation, just the word `Behold!' ''  If a 
proof is a journey, can a journey take place instantaneously?  How
does this relate to some of what Hardy says about the esthetics of mathematics
in {\it A Mathematician's Apology}?  Can a picture supply a proof of a mathematical
theorem by itself, with no explanation?  Is there room for this sort of sudden
insight
in Lakatos's conception of the development of mathematical understanding through
proofs and refutations?  Have you ever had this sort of ``aha moment'' working to 
find or understand a proof?  If so, what had you done before that moment?  Was
it really instantaneous?  And is that flash of insight the only way understanding
of mathematics comes to people?
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\item{$\bullet$} Option B:  Write an essay about the poem {\it Ithaka} by C. Cavafy posted
on our course homepage (I'll provide the text of the poem for you to refer to on the final 
if you choose this option).  In particular, this poem clearly draws on themes from 
the {\it Odyssey}, but does it just retell parts of Homer's story, or does it
end up making something quite different of them?  
In particular, is the {\it return} of Odysseus the main
point here?  Why doesn't Cavafy mention Telemachus or Penelope? 
Finally, how do you think what Cavafy is saying here relates to 
the CHQ theme (especially the ``how then shall we live?'' part)?
\bye


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\noindent
{\it Practice Questions}
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The following are some practice questions indicating the types
of mathematical questions I might ask on the midterm exam.
Recall that these will account for about 60\% of the total
points.  The remainder of the points will be distributed
between short answer questions dealing
with some of the historical background, and a short essay
as described on the information sheet.  There are answers
for these questions posted on the course home page, but I think it will be
best to work through complete solutions and then check your
work -- don't just look at the answer and say ``yeah, that
makes sense.''  You need to be able to do this for yourself
with questions you are seeing for the first time.
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\noindent
{\it Disclaimer}
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These practice problems show some of the range of topics that
will be included and the approximate level of difficulty
of the exam questions.  The actual exam questions may look
different and combine topics we have discussed in different
ways.
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\noindent
{\it Sample Questions}
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\noindent
I.
\item{A)}   Solve the equation $x^2 - 7x - 60 = 0$ by factoring the
quadratic.
\item{B)}  Solve the equation $x^2 - 7x - 60 = 0$ by means of
the quadratic formula and show that the answers you obtain are the
same as in part A.
\item{C)}  Explain how the Old Babylonian problem text YBC 6967 
can be understood as solving this quadratic equation.
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II.
\item{A)}  Express in base 10:  $(101101)_2$.
\item{B)}  Express in base 10:  $(10,23,24)_{60}$.
\item{C)}  Express in base 60:  $(2691)_{10}$.
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III.  The following two questions are typical of the geometry
problems from the Moscow mathematical papyrus.  Solve them
using modern methods.
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\item{A)}  The area of a rectangle is $48$ and the width is
$3/4$ the length.  What are the dimensions?
\item{B)}  One leg of a right triangle is $5/2$ times
the length of the other.  The area is 20.  What are the dimensions?
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IV.  Compute ``the Egyptian way:"
\item{A)}  $56 \times 125$
\item{B)}  $134 \div 24$ 
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V.  A Babylonian problem asks for the side of a square
if the area of the square, minus the side is the base
60 number $(14,30)_{60}$.  The tablet says to do this to
solve the problem (all numbers in base 60, of course!):
``Take half of 1, which is $0;30$, square that
to get $0;15$, add the $14,30$ to get
$14,30;15$.  The last number is the square of
$29;30$.  Now add the $0;30$ to get $30$, which
is the side of the square.''
\item{A)}  Solve this problem using modern methods.
\item{B)}  Is 30 the correct answer?
\item{C)}  Explain how the Babylonian method of solution
is essentially the same as using the quadratic formula(!)
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VI.  An integer $n$ is {\it base-60 regular}
if $n = 2^a 3^b 5^c$ for some $a,b,c \ge 0$ (that is, if
$n$ is divisible only by the primes occurring in the factorization
of $60$).
\item{A)}  What are all the $n \le 50$ that are base-60 regular?
\item{B)}  Many Babylonian tables giving $n, 60/n$ are known
and all of them contain only base-60 regular $n$'s.  Why do you suppose that is
that true?
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VII.  I might also ask you to state and prove Proposition 1 or Proposition 5
from Book I of the {\it Elements} of Euclid.
\vfill\eject
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{\it Answers}
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I.
\item{A)}  $x^2 - 7x - 60 = (x - 12)(x + 5)$, so $x = 12$ or $x = -5$.
\item{B)}  By the quadratic formula
$$x = {7 \pm \sqrt{49 + 280}\over 2} = {7 \pm \sqrt{289}\over 2} = 
{7 \pm 17\over 2} = 12, -5.$$
\item{C)}  The YBC 6967 problem is:  Find a number $x$ such that $x$ exceeds
$60/x$ by $7$.  This is equivalent to $x = 60/x + 7$, or $x^2 - 7x = 60 = 0$.
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II.  A)  $45$  B) $37404$  C)  $(44:51)_{60}$
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III.
\item{A)}  Call the length $\ell$.  Then $w = 3\ell/4$ and $A = 48 = \ell w = {3\ell^2\over 4}$.
Solving for $\ell$,  $\ell = 8$ and $w = 6$.
\item{B)}  Call the length of one leg $\ell$.  Then the other is $5\ell/2$ and
the area is $20 = {1\over 2}\ell{5\ell\over 2}$.  So $\ell^2 = 80/5 = 16$ and $\ell = 4$.
The dimensions are $4, 10$.
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IV.
\item{A)}  By repeated doubling, we get $125, 250, 500, 1000, 2000, 4000$ (we can stop
there since the next double would be $64 \times 125$, but $64 > 56$.  Then since
$56 = 32 + 16 + 8$,
$$56 \times 125 = 4000 + 2000 + 1000 = 7000$$.
\item{B)}  By repeated doubling, we get 
$$\eqalign{
  1 \times 24 &= 24\cr 
  2 \times 24 &= 48\cr
  4 \times 24 &= 96\cr}$$  
The next double is larger than $134$ so we stop there.   
Then starting with $2/3 \times 24 = 16$, we
get
$$\eqalign{
2/3 \times 24 &= 16\cr
  1/3 \times 24 &= 8\cr
  1/6 \times 24 &= 4\cr
  1/12 \times 24 &= 2\cr
  1/24 \times 24 &= 1\cr}$$
Since $134 = 96 + 24 + 8 + 4 + 2$, we have $134 \div 24 = 5 + 1/3 + 1/6 + 1/12$.
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V.
\item{A)}  Say the length of the side is $x$.  Then the condition given says
$x^2 - x = 870$, or $x^2 - x - 870 = 0$.  By the quadratic formula we get
$$x = {1 \pm \sqrt{1 + 3480}\over 2} = 30, -28$$
Only the positive solution is relevant for the geometry, so $x = 30$.
\item{B)}  Yes!
\item{C)}  The Babylonian ``directions'' applied to any equation of the
form $x^2 - px = q$ would say to compute
$$x = \sqrt{\left({p\over 2}\right)^2 + q} + {p\over 2} $$
This is equivalent to the solution given by the quadratic formula(!)
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VI.
\item{A)}  $2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50$
\item{B)}  Since those are the $n$ for which $1/n$ has a {\it finite} base
60 expansion! (See Discussion 1 for one of those tables.)

\bye
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\noindent
I.  Compute ``the Egyptian way"
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\item{A)} (10) $51 \times 187$
\item{B)} (10) $1332 \div 47$ 
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II.  
\item{A)} (10) Express the base 60 number $(2,31;40)_60$ in base 10.
\item{B)} (10) Express the base 2 number $(111000111)_2$ in base 60.
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\noindent
III.  A Babylonian problem text asks for the side of a square
if the area of the square minus the side is the base
60 number $(14:30)_{60}$.  The tablet says to do this to 
solve the problem (all numbers in base 60, of course!):
``Take half of 1, which is $0.30$, square that
to get $0.15$, add the $14:30$ to get 
$14:30.15$.  The last number is the square of
$29.30$.  Now add the $0.30$ to get $30$, which
is the side of the square.''
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\item{A)} (15) Solve this problem using modern methods.
\vglue 4.0truein
\item{B)}  (5)  Explain how the Babylonian method of solution
is essentially the same as using the quadratic formula for an
equation of the form $x^2 - px = q$.  
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\noindent
IV.  Short answer:
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\item{A)} (2.5)  What base 10 number is represented by these Egyptian
hieroglyphs? 
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\item{B)} (2.5)  What base 10 number is represented in the
Babylonian way by these symbols?
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\item{C)} (2.5)  Approximately {\it when} was the Old Babylonian period in 
Mesopotamian history?
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\item{D)} (2.5)  The name for the writing system used in part B is
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\item{E)} (2.5)  What are the surviving records of Egyptian mathematics?
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\item{F)} (2.5)  Most of the geographical area of the Old Babylonian 
kingdoms is contained in what present-day country?
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\item{G)} (2.5)  What was special about the Egyptian way of 
dealing with fractions?   
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\item{H)} (2.5)  Which Old Babylonian tablet apparently contains
a table of Pythagorean triples?
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\noindent
V.  Essay.  (20) ``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 interpretations your historians would give 
for the YBC 6967 problem of solving the equation $x = 60/x + 7$.
