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\centerline{MONT 107Q -- Thinking About Mathematics}
\centerline{Discussion -- An Old Babylonian Mathematical Table}
\centerline{February 8, 2017}
\bigskip
\noindent
A number of Old Babylonian tablets with information equivalent 
to the following table of base 60 numbers have been found.
This was clearly a standard part of the Babylonian mathematicians'
``calculation toolkit.''  To make things simpler for a first encounter
with ``the real thing,'' the convention for the cuneiform number symbols is:  in the 
base 60 digits, $<\ =\ 10$ and $\vee = 1$.  {\it Spaces} separate each
base 60 digit from the next one.  The first number on the row is
added just to help us identify the rows in the table.
$$
\matrix{
&1 &\vee\vee & \qquad\qquad & <<< \cr
&2 &\vee\vee\vee&              & << \cr
&3 &\vee\vee\vee\vee & & <\vee\vee\vee\vee\vee\cr
&4 &\vee\vee\vee\vee\vee & & <\vee\vee\cr
&5 &\vee\vee\vee\vee\vee\vee & & < \cr
&6 &\vee\vee\vee\vee\vee\vee\vee\vee & & \vee\vee\vee\vee\vee\vee\vee\quad <<< \cr
&7 &\vee\vee\vee\vee\vee\vee\vee\vee\vee & & \vee\vee\vee\vee\vee\vee\quad <<<< \cr
&8 &< & & \vee\vee\vee\vee\vee\vee\cr
&9 &<\vee\vee & & \vee\vee\vee\vee\vee\cr
&10 &<\vee\vee\vee\vee\vee & & \vee\vee\vee\vee \cr
&11 &<\vee\vee\vee\vee\vee\vee & & \vee\vee\vee\quad <<<<\vee\vee\vee\vee\vee \cr
&12 &<\vee\vee\vee\vee\vee\vee\vee\vee & & \vee\vee\vee\quad << \cr
&13 &<< & & \vee\vee\vee \cr
&14 &<<\vee\vee\vee\vee & & \vee\vee\quad <<<\cr
&15 &<<\vee\vee\vee\vee\vee & & \vee\vee\quad <<\vee\vee\vee\vee\cr
&16 & <<\vee\vee\vee\vee\vee\vee\vee & & \vee\vee\quad <\vee\vee\vee\quad <<\cr
&17 &<<< & & \vee\vee \cr
&18 &<<<\vee\vee & & \vee\quad <<<<<\vee\vee\quad <<<\cr
   & \vdots & \cr
}$$
\bigskip
\noindent
Thus for instance, the 6th row would translate to 
base 60 numbers expressed like this in our notation.  
$$ (8)_{60} \qquad{\rm and}\qquad (7 ; 30)_{60} $$
(note the space between the $\vee\vee\vee\vee\vee\vee\vee$ and the $<<<$ on 
the right on this row).  The equivalent base-10 numbers are
$$8 \qquad{\rm and}\qquad 7 + {30\over 60} = 7.5$$
\bigskip
\noindent
{\it Figuring out the table}
\bigskip
\item{A)}  Translate all the table entries
into base 10 numbers like this.
\item{B)} Then figure out what the table is and how the numbers
in the second column relate to the numbers in the first.  Also, is there any
ambiguity involved?  {\it Hint:}  A good way to approach this is to interpret the numbers
in the left column as whole numbers.  However, the ones on the 
right are best understood as base-60 {\it fractions}, which you will convert to decimal
numbers.  (Where does the {\it sexagesimal point} or the decimal point go?)
\item{C)}  Babylonian mathematicians would have used a table like this to 
compute things like the base-60 form of fractions like $5/32$.  How could this table
(and perhaps information from another table) be used for that?
\bye
