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\centerline{MONT 104Q -- Mathematical Journeys}
\centerline{Information for Midterm Exam}
\centerline{October 14, 2015}
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\noindent
{\it General Information}
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The midterm exam announced in the course syllabus 
will be given in class on Monday, November 2.  The whole
class period will be devoted to the exam.
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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 and historical material we have studied
since the start of the semester (i.e. starting from our discussion of Hardy's
{\it Mathematician's Apology}), through the end of our discussion of
the {\it Elements} of Euclid.  
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\noindent
This means in particular:
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\item{1.}  Know the approximate historical periods of the following figures:  
Pythagoras, Plato, Euclid, G.H. Hardy, where they came from (if known), and 
where they were active.
\item{2.}  Know the statements and proofs of the two theorems Hardy gives as 
supreme examples of ``real mathematics'' (from the first group discussion day).  
\item{3.}  Know the 5 Common Notions (Axioms) and 5 Postulates that appear at the beginning
of Book I of Euclid's {\it Elements}.
\item{4.}  Know the statements and the proofs of Propositions 1 and 5 of Book I of the {\it Elements}.  Be able to identify what Axioms, Postulates, and/or previously proved results 
are used in the proofs of these. 
\item{5.}  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 ... .''
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\noindent
{\it Format}
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Approximately 65\% of the exam will consist of questions related to the topics above.
The remaining approximately 35\% will be a short essay (target length: 1-2 handwritten 
pages).  The essay will be on one of the two topics below (I will choose the one for 
the exam).  To prepare well, you will want to think (and probably write) out 
practice essays on both 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.  (I may not be able to resist correcting things, but any
corrections I make as I read will not feature in the grading.)
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{\it The Essay Question}
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\noindent
The essay question will be on {\it one of the following topics} (yes, these exact questions):
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\item{1)} 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?   
\item{2)}  Peter Rudman, the author of a book about the extent to which
the Pythagorean theorem was anticipated by work in the Old Babylonian period
(about 2000 - 1600 BCE) wrote this: ``High school mathematics
education today, with its emphasis on creating high scores on standardized
tests, 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 
are the ``derivations'' that Rudman is referring to?  Why do you think he says
that that ``is where mathematics is learned?''  How does what he
is saying connect with things we read in Hardy's {\it Mathematician's Apology} and that
we have done so far in this course?  Does
his description of high school mathematics match your experience?    
\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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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\noindent
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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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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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 $x = 60/x + 7$.
