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\markright{MATH 110 -- Mathematics Through Time, Midterm Exam Solutions}

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\noindent
I.  Short answer.  Answer any 6 of the following.  
\vskip 10pt
\begin{enumerate}
\item[A)] (2.5)  What base 10 number is represented by these Egyptian
hieroglyphs? 
\vskip 10pt
\leftline{Answer: $372$}
\vskip 10pt
\item[B)] (2.5)  What base 10 number is represented in the
Babylonian way by these symbols?
\vskip 10pt
\leftline{Answer: $25 \times 60 + 47 = 1547$}
\vskip 10pt
\item[C)] (2.5)  Approximately {\it when} was the Middle Kingdom period in 
Egyptian history?
\vskip 10pt
\leftline{Answer: Approximately 2000 to 1600 BCE}
\vskip 10pt
\item[D)] (2.5)  The name for the writing system used in question B is
\vskip 10pt
\leftline{Answer: cuneiform}
\vskip 10pt
\item[E)] (2.5)  What are the two most important surviving records of ancient Egyptian 
mathematics?
\vskip 10pt
\leftline{Answer: The Rhind and Moscow mathematical papyri}
\vskip 10pt
\item[F)] (2.5)  Most of the geographical area of the Old Babylonian 
kingdoms is contained in what present-day country?
\vskip 10pt
\leftline{Answer: Iraq}
\vskip 10pt
\item[G)] (2.5)  Approximately when did Thales live and what part of the Greek 
world did he come from?  
\vskip 10pt
\leftline{Answer: Approximately 624BCE - 547BCE; Miletus in Asia Minor (present-day Turkey)}  
\vskip 10pt
\item[H)] (2.5)  Who founded the Academy in Athens whose entryway was inscribed ``let no one unversed in 
geometry enter here?''
\vskip 10pt
\leftline{Answer: Plato}
\end{enumerate}
\vfill\eject
\noindent
II.  Compute ``in the Egyptian way"
\begin{enumerate}
\item[A)] (15) $37 \times 136$
\item[] {\it Solution:}  Begin with successive doublings until you see the next multiplier of 136 would be larger than 37:
\begin{eqnarray*}
1 \times 136 &=& 136 -\\
2 \times 136 &=& 272\\
4 \times 136 &=& 544 -\\
8 \times 136 &=& 1088\\
16 \times 136 &=& 2176\\
32 \times 136 &=& 4352 -\\
\end{eqnarray*}
($2 \times 32 = 64$ so we don't need to go farther).  
Then $37 = 32 + 4 + 1$, so $37 \times 136 = 4352 + 544 + 136 = 5032$.

\item[B)] (15) $52 \div 18$  (``compute with $18$ until you find $52$'')
\item[] {\it Solution:}  Proceed as in part A to begin:
\begin{eqnarray*}
1 \times 18 &=& 18\\ 
2 \times 18 &=& 36
\end{eqnarray*}
(Then $4 \times 18 = 72 > 52$, so we don't need that.)  Then $52 - 36 = 16$.  So now we start
in with fractions of 18.  The simplest way to do this to recall that the Egyptians did
also allow $\frac{2}{3} = \overline{\overline{3}}$, then successively halve:
\begin{eqnarray*}
 \overline{\overline{3}} \times 18 &=& 12\\
\overline{3} \times 18 &=& 6\\
\overline{6} \times 18 &=& 3\\
\overline{18} \times 18 &=& 1.
\end{eqnarray*}
Then $16 = 12 + 3 + 1$, so $16 \div 18 = \overline{\overline{3}} + \overline{6} + \overline{18}$ and 
$$52 \div 18 = 2 +  \overline{\overline{3}} + \overline{6} + \overline{18}.$$
\end{enumerate}
\vskip 10pt
\noindent
III. (20) An Old Babylonian problem text asks for the side of a square
if the area of the square minus the side is the base
60 number $14,30;0$.  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.''
\vskip 10pt
In modern language, the problem is to solve for $x$ if $x^2 - x - 870 = 0$.  
Find $x$ by solving this equation, then explain how the Babylonian method of solution
is essentially the same as using the quadratic formula for the
equation $x^2 - x - 870 = 0$.  
\vskip 10pt 
\noindent
{\it Solution:}  If we apply the quadratic formula to solve the equation, we get 
$$x = \frac{1\pm \sqrt{1 + 4 \cdot 870}}{2} = 30, -29.$$
The negative root is not relevant for the geometry, so we take $x = 30$, which comes
with the $+$ sign.  Note that result from the quadratic formula can be rearranged as follows
$$30 = \frac{1}{2} + \sqrt{\left(\frac{1}{2}\right)^2 + 870}$$
In modern algebraic notation this gives the same steps as described on the Babylonian
tablet:  Under the radical, we start by squaring $\frac{1}{2}$, add $870$, take the 
square root, then add $\frac{1}{2}$. 
\vskip 10pt
\noindent
IV.  (10) What is Proposition 1 in Euclid's {\it Elements}?  Describe the 
associated construction.
\vskip 10pt 
\noindent
{\it Solution:}  The statment is:  To construct an equilateral 
triangle with any given line segment as one side.  The construction is as follows.
Say the line segment is $AB$.  Using Postulate 3, construct one circle with 
center at $A$ and radius $AB$, then a second circle with center at $B$ and radius $AB$.
Let $C$ be one of the intersections of those circles.  Then $\Delta ABC$ is equilateral
(since $AB = AC = BC$).   
\vskip 10pt
\noindent
IV.  Essay.  (25) ``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 $x = 60/x + 7$.
\vskip 10pt
\noindent
{\it Model Answer}:  The main historian we discussed who would agree with this 
statement is Otto Neugebauer.  His view was that even when a Babylonian problem
could be interpreted using geometric ideas, the Babylonians were probably thinking 
in numerical and algebraic terms.  His evidence for saying this was, first, that many of 
the surviving texts used extensive tables of numerical data (think of the $60/n$
table that we looked at in Discussion 1), and second, that in some cases (as in the 
problem from question III above), the questions the Babylonians asked often combined lengths
and areas in ways that would not really make sense in geometric terms.  Neugebauer's
explanation for the YBC 6967 problem was that the step-by-step solution given
could be explained by using the algebraic identity
$$\left(\frac{a + b}{2}\right)^2 - \left(\frac{a - b}{2}\right)^2 = ab$$
with $a = x$ and $b = \frac{60}{x}$.  Given that $a - b = 7$, and knowing $ab = 60$, 
you can use the identity to solve for $a$ and $b$ and that is one way to read what
the solution says.   

The main historian we discussed who would disagree with this is Jens H\o yrup, although
Eleanor Robson would also agree with this point of view.
H\o yrup's main ideas about YBC 6967 were that algebraic interpretations, although
perhaps mathematically equivalent to what the Babylonians said, are probably 
conceptually anachronistic in the sense that the Babylonians just did not have
the formalism and the algebraic tools to have approached the problem that way.  
Moreover, even when we might tend to interpret a Babylonian text 
in algebraic terms, the actual language they used indicates that they
were more likely thinking geometrically.  For instance, H\o yrup gives a way to 
understand the YBC 6967 solution as a series of ``cut and paste'' geometric operations 
on rectangles.  



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