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\centerline{MATH 110 -- Mathematics Through Time}
\centerline{Information for Midterm Exam}
\centerline{October 6, 2013}
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\noindent
{\it General Information}
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The first midterm exam will be given in class on Friday, October
25.
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{\it Miscellaneous Groundrules}
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No use of cell phones, pagers, I-pods, I-pads,  or any other electronic devices
beyond a calculator will be allowed during the exam.  If your only calculator
is on your phone, I can supply a few basic calculators for you to use.  You
will be asked to turn all other 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, through the start of our discussion of
the {\it Elements} of Euclid from class on October 7 and 9.  
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This means in particular:
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\item{1.}  Base 2,10,60 positional number systems.
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\item{2.} Egyptian mathematics
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\itemitem{a.}  Know the approximate historical period represented by
our primary Egyptian mathematical sources (the Rhind and Moscow mathematical
papyri).
\itemitem{b.}  Know the basic idea of their number system and
the number symbols for 1's, 10's, 100's, 1000's (see BJB)
\itemitem{c.}  ``Egyptian multiplication and division'' by repeated doubling
\itemitem{d.}  Egyptian computations with unit fractions via tables of
representations like the one from the Rhind papyrus on page ... of BJB).
\itemitem{e.}  Be prepared for problems like the ones on Problem Set 1.
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\item{3.}  Old Babylonian mathematics
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\itemitem{a.}  Know the geographical area occupied by the Old Babylonian civilization
and the approximate historical period represented by the
Old Babylonian mathematical texts.
\itemitem{b.}  Know their base-60 number system and number symbols and be
able to read numbers in their format
\itemitem{c.}  The role of addition, multiplication, ``reciprocal,'' tables
in Babylonian arithmetic
\itemitem{d.}  Know the different interpretations of the Babylonian 
problem texts like YBC 6967 given by Otto Neugebauer and Jens H\o yrup.
\itemitem{e.}  Know what is on the ``Plimpton 322'' tablet and the different ways that
Neugebauer and Eleanor Robson have explained what this tablet was.
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\item{4.}  Greek mathematics
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\itemitem{a.}  Know the approximate historical periods of the following figures:  Thales, Pythagoras,
Plato, Euclid, where they came from in the Greek world (if known), and where they were active.
\itemitem{b.}  Know the 5 Common Notions (Axioms) and 5 Postulates that appear at the beginning
of Book I of Euclid's {\it Elements}.
\itemitem{c.}  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.  
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{\it Format}
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Approximately 60\% of the exam will consist of two or three short mathematical
problems similar in format and content to the ones you have seen on the problem sets.
The remaining approximately 40\% will be distributed, in a way
to be determined, among a few short answer
questions and a short essay.  This 40\% will concentrate on aspects of
the history.
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{\it Essay}
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The essay question will be {\it one of the following}:
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\item{1)}  ``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 solving the equation $x = 60/x + 7$.
\item{2)}  How are the Egyptian and Old Babylonian ways of dealing
with fractions different from each other, and different from what
we do today?  Which of these three systems has the capability of dealing
with more fractions {\it in exact, finite terms} (that is, without
resorting to infinite repeating expressions)?  Explain.
\item{3)}  Peter Rudman, author of a book about the extent to which
the Pythagorean theorem was anticipated by the Old Babylonian problem 
texts states that ``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 ... .''   Is today's
approach really all that different, though, from the Egyptian and Old Babylonian
problem texts we have seen, where solutions of the problems posed are given
as series of ``do this, then do that'' steps for solving the problems?
In your opinion, what is the most effective way that mathematics can 
be taught and learned?  Does it matter what the specific goals of teaching
mathematics are?
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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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{\it Sample Questions}
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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.
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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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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.
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\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 $x = 60/x + 7$.
