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\centerline{MONT 108N -- Mathematics Through Time}
\centerline{Problem Set 3 -- Babylonian Mathematics}
\centerline{{\bf due:}  Friday, October 8}
\bigskip
\noindent
I.  A Old Babylonian tablet from about 1700 B.C.E (now 
held by the Louvre in Paris) has the following
problem:  "Find how long it will take a certain sum
of money to double itself at compound annual interest
of $20\%$."  This means that you will have $1.2 \times $
the original amount after 1 year, $(1.2)^2 \times $ the
original amount after 2 years, $(1.2)^3 \times $ the
original amount after 3 years, and so on.  The question
is: How many years will be needed until you have twice
the original amount? (Fractional parts of years 
are also allowed.)
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\item{A)}  The Babylonian method of solution (written
with base 10 numbers and in modern language, of course) was
this:  First compute the powers to see that 
$(1.2)^3 = 1.728$ and $(1.2)^4 = 2.0736$.
So the doubling will happen between the 3rd and 4th year.
To find the doubling time, find the point on the straight
line through $(3,(1.2)^3) = (3,1.728)$ and $(4,(1.2)^4)=(4,2.0736)$ 
with $y = 2$.  The $x$-coordinate of that point 
is the doubling time.  Carry out the calculations to find this time.
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\item{B)}  The Babylonian tablet gives the answer by this
method as the base 60 number 
$$(3.47:13:20)_{60}$$
(with fractional part).  
Is this correct (does it agree with with you did in part A)?
\item{C)}  Is this method exact or an approximation?  Explain. 
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\item{D)}  Solve the  problem {\it exactly} by modern methods and 
compare with the Babylonian answer.  How different are they?
{\it Hint:}  logarithms.
\bigskip
\noindent
II.  Old Babylonian tablets containing values of $n^3 + n^2$ for 
$n = 1, ..., 30$ has been found.
\medskip
\item{A)}  Make such a table for $n = 1, \ldots, 10$.  
\medskip
\item{B)}  Use it to solve the cubic equation $x^3 + 2x^2 = 3136$.
(Note:  The trick is to multiply both sides of the equation by 
an appropriate number first, then consult the table!)  {\it Answer:  $x = 14$,
but full details of how this is derived must be shown for credit.}
\medskip
\item{C)}  A tablet of about 1800 B.C.E. from Susa in present-day Iran
asks for a solution of the system of equations
$$\eqalign{xyz + xy &= 7/6\cr
            y &= 2x/3 \cr
            z &= 12x\cr}$$
Use the last two equations to eliminate $y,z$ and get an equation
in $x$ alone.  Then use your table from part A) to find the solution.
\bye
