A History of Mathematics From Mesopotamia to Modernity

BabylonianMathematics 23

Fig. 4The basic cuneiform numbers from 1 to 60.

Fig. 5How larger cuneiform numbers are formed.

You can find the details of how the system works in various textbooks; in particular, there are
plenty of examples in Fauvel and Gray. (Notice that the sum which I gave above was one in which it
was not needed—why?) Again following a general convention, modern editors make things easier
for readers by inserting a semi-colon where they deduce the ‘decimal point’ must have come, and
inserting zeros as in ‘30, 0’ or ‘0; 30’. So ‘1, 20’ means 80, but ‘1; 20’ means 1+^2060 = 113. There
would be no distinction in a Babylonian text; both would appear as ‘1 20’.
To help themselves, the Babylonians, as we do, needed to learn their tables. They were, it would
seem, in a worse situation than us, since there were in principle 59 tables to learn, but they
probably used short cuts. A scribe ‘on site’ would quite possibly have carried tablets with the
important multiplication tables on them, as an engineer or accountant today will carry a pocket
calculator or palmtop; and in particular the vital table of ‘reciprocals’. This lists, for ‘nice’ numbers
x, the value of the reciprocal^1 x, and starts:

230

320

415

512

610

7, 30 8

87,30

96,40

Using this table it is possible to divide simply by multiplying by the reciprocal; dividing by 4 is
multiplying by 15 (and of course thinking about what the answer means in practical terms—what
size of number one should expect).