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\centerline{MONT 106Q -- Mathematical Thinking}
\centerline{Problem Set 2 -- Egyptian Mathematics}
\centerline{{\bf due}: no later than 5:00pm, September 26, 2016}
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\noindent
I.  {\it Egyptian multplication and division.}
\item{A)} Multiply by successive doubling (that is, ``the Egyptian way''):  $434 \times 127$.
(Make it easier on yourself by a wise choice of the multiplier and multiplicand!)
\item{B)} Multiply by successive doubling:  $252 \times 59$.
\item{C)} Divide by the Egyptian method:  $167 \div 18$.
\item{D)} What {\it advantage(s)} does the Egyptian method of multiplication
have over the method commonly used now?  What shortcomings?
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\noindent
II.  {\it Egyptian fractions.} A part of the table of conversions of $2/n$ 
into unit fractions from the Rhind papyrus is given in 
the handout from class on Wednesday, September 21.
Each row gives an expansion
$${2\over n} = {1\over a} + {1\over b} + {1\over c} + {1\over d}$$
for one of the odd integers $3 \le n \le 101$.  
(if there is no entry for $c$ and/or $d$, then those terms 
are not present).
\item{A)}  Verify that the rows for $n = 13$ and $n = 29$ are
correct.
\item{B)}  Use the table to find
a decomposition of ${4\over 27}$ into unit fractions.  
\item{C)}  Show that for any positive integer $n$, 
$${2\over n} = {1\over n} + {1\over 2n} + {1\over 3n} + {1 \over 6n}.$$
Yet this expansion appears only once in the table from the Rhind
papyrus.  For which $n$ did the compiler of the table fall back
on this form?  Can you see why they might have preferred to avoid using this if
possible?
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\noindent
III.  {\it The {\bf seked} of a pyramid}.  The Egyptians measured the
steepness of a face of a pyramid by the ratio of the ``run''
(measured in palms) to the ``rise'' (measured in cubits).
There were $7$ palms in a cubit.  This ratio 
was called the {\it seked} of the pyramid, measured in
units of {\it palms per cubit}.  (See page 110 in {\it The Crest of the Peacock}.)
\item{A)}  Problem 56 of the Rhind papyrus asks the solver
to find the {\it seked} of a pyramid 250 royal cubits tall, 
with a square base 360 royal cubits on a side.  The answer
is given as $5 + {1\over 25}$ hands per royal cubit.  The computation
is given by Joseph, but not really explained.  Explain what is 
going on there with diagrams.  In particular, why does the calculation
start out by taking half of the $360$?
\item{B)}  The great pyramid at Giza (built by the Pharaoh Khufu, finished
about 2560 B.C.E., and the only one surviving of the ``seven wonders of 
the ancient world'') has a base 440 cubits on a side and
a height of 280 cubits.  What is its {\it seked}?
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\noindent
IV. {\it An Egyptian approximation to $\pi$.}  We take $\pi$
to be the ratio between the area of a circle and the square
of the radius:  $\pi = {A\over r^2}$.
\item{A)}  In the Rhind papyrus, the area of a circle with
diameter $d$ is given as 
$$A_{circ} = \left({8\over 9} d\right)^2.$$
(This is not exactly correct, but yields decent approximate results.)  
What is the implied approximation to the value of $\pi$ here?
\item{B)}  Here's one way this may have been derived. Take
a square of side $9$ units, divide the edges into 
three equal pieces, and cut off four corner triangles 
with side length 3 to make an octagon.  The octagon 
is pretty close in area to the circle inscribed in
the original square.  Explain why the octagon has area
$63 \doteq 8^2$, and then explain how the approximation
$A_{circ} = \left({8\over 9} d\right)^2$ would result.  
\bye
