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Large Cuneiform Numbers
For computation, the Mesopotamians used what is
usually referred to as a 'sexagesimal' (i.e., base-60) system.
Technically, this is a slightly inaccurate designation as they used
only combinations of two symbols bundled together for writing numbers
up to 60. For writing numbers greater than 60, they just
repeated the symbols in different columns, just as we do, except that
where for us a '1' in the 'tens' column means 10, for the Babylonians
a
in the 'sixties' column meant 60. Each column increased the value of
the number by a factor of 60, and the Babylonians wrote their numbers
with the largest values to the left, just as we do. Here are some examples of cuneiform numbers,
their transliterations and values in our notation.
There are a few differences between the way we
write our numbers and the way the Babylonians did. First, they
had no special way to mark an empty column. We would write a
zero to mark the place, they would often leave a space, but not
always. For example, it is not always clear if
should mean '2' or '61', or even '3601'. In practice, empty
columns don't arise that often in a base-60 system and so this was not
such a problem as you may think. Later on, in the Neo-Babylonian
and Seleucid times, when astronomers needed to do lots of many-place
sexagesimal computations, they did introduce an empty-column marker. One of the great advantages of a place-value
system is that you can use the same symbols to make ever larger
numbers. There is no limit to how large a number you can write
down. Another advantage is that you can continue
writing numbers in places to the right of the units column in order to
denote fractions. All that distinguishes the number 1234 from
the number 1.234 is the use of a decimal point (or comma in Europe) to
mark where the units come. Computations with fractions are just the same as
computations with whole numbers. The Babylonians used the same
idea, except that they did not bother with a decimal point - that
absolute size of a number was 'determined by inspection.'
For example, the number
could mean 160000, as noted above, but it could also be 1/81, the
reciprocal of 81, which is why it was widely used. In the early days of deciphering Mesopotamian
mathematics, people were puzzled as to why they would go to the
trouble of writing a 160000-times multiplication table. The last
sexagesimal number given in the table above,
,
also has a more useful meaning than 305470. |

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