MONT 108N – Mathematics Through Time Discussion 2 – An Old Babylonian Mathematical Table October 1, 2010

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,” I have set the table using our convention for the cuneiform number symbols: in the base 60 digits, < = 10 and ∨ = 1. 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.

1  ∨∨  <<<
2  ∨ ∨ ∨  <<
3  ∨ ∨ ∨∨  < ∨ ∨ ∨ ∨ ∨
4  ∨ ∨ ∨ ∨ ∨  < ∨∨
5  ∨ ∨ ∨ ∨ ∨∨  <
6 ∨ ∨ ∨ ∨ ∨ ∨ ∨∨  ∨ ∨ ∨ ∨ ∨ ∨ ∨ <<<
7 ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨  ∨ ∨ ∨ ∨ ∨∨ <<<<
8  <  ∨ ∨ ∨ ∨ ∨∨
9  < ∨∨  ∨ ∨ ∨ ∨ ∨
10  < ∨ ∨ ∨ ∨ ∨  ∨ ∨ ∨∨
11 < ∨ ∨ ∨ ∨ ∨∨  ∨ ∨ ∨ <<<< ∨ ∨ ∨ ∨ ∨
12 < ∨ ∨ ∨ ∨ ∨ ∨ ∨∨  ∨ ∨ ∨ <<
13  <<  ∨ ∨ ∨
14  << ∨ ∨ ∨∨  ∨∨ <<<
15 << ∨ ∨ ∨ ∨ ∨  ∨∨ << ∨ ∨ ∨∨
16 << ∨ ∨ ∨ ∨ ∨ ∨ ∨  ∨∨ < ∨ ∨ ∨ <<
17  <<<  ∨∨
18  <<< ∨∨  ∨ <<<<< ∨∨ <<<
⋮

Thus for instance, the 6th row would translate to base 60 numbers expressed like this in our notation.

(8)_{60} (7 : 30)_{60}

(note the space between the ∨ ∨ ∨ ∨ ∨ ∨ ∨ and the <<< on the right on this row).

Questions

A) Translate all the table entries into base 60 numbers like this. B) Then figure out what the table is and explain in a paragraph how the numbers in

the second column relate to the numbers in the first. Also, is there any ambiguity involved? 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 fractions. (Where does the sexagesimal point go?)

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?