Babylonia: 300 B.C.
 The Babylonians invented the
place value system. They had a sexigesimal number system, that
is, they based their counting on 60 as we base ours on ten. When
you count minutes in an hour or measure circles you arethinking
in sexigesimal.
Of course the Babylonians didn't use our numerals. They wrote
in cuneiform a writing system optimized for writing in damp clay tablets. They used two symbols that
represented all the numbers from 1 to 59. The wedge was used for
a one and the crescent equalled a 10. By grouping them together,
they created symbols for all 59 numbers.
5 crescents + 4 wedges = 54
Beginning at 60, we see a place value. The number 61 would be
written with one wedge to the left (1 sixty), and one to the right
(1 one).
The number 124 (2 sixties + 4 ones)
The number 1856 (30 sixties + 56 ones)
And here's the challenge that leads to the invention
of zero. How do you indicate that there's nothing in a particular
place? How would you show the number 3604? 3604 is 1 "60
squared" + 4 ones but nothing in the sixties column. Well,
scribes started leaving a blank space. But not all of them did
that and even when they did sometimes it was a pretty small space--it
was difficult to tell it was there. So one very bright scribe
put in a symbol that already existed as a separator in literature,
a sort of sideways, superscript, double wedge. Now it was easy
to distinguish whether you meant 3604 or 64:
top: 64 (1 sixty + 4 ones)
bottom: 3604 (1 sixty2 + 0 sixty + 4 ones)
Babylonian mathematicians used the separator
(effectively the first zero) in the middle position only.
The person doing the calculations knew what order of magnitude
he was working with and didn't add any separators at the end
of his notations. However, the astronomers
started using the zero placeholder in at the end and at the
beginning of notations. This allowed them to note fractional
degrees and minutes of arc and made their computations more
accurate.
Despite the invention of zero as a placeholder,
the Babylonians never quite discovered zero as a number. On
an accounting tablet recording the distribution of grain there
is a notation at the end of a column of numbers that reads
"The grain is exhausted." Another example from the same era
is a description subtracting 20 from 20: "twenty minus twenty...you
see."
Although we have evidence of zero from tablets
in the Selucid era (4th to 1st C BC), it is possible that
the zero was invented before that time. Many of the Seleucid
era tablets are copies of much older documents. We'll never
know for certain, so we place the Babylonia zero around the
3rd century BC.
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