Elena's review of the history of zero was very interesting,
and important to ponder in several respects. Zero has long
been reported in USA math textbooks, and elsewhere, to have
exclusively been the invention of the Vedic Math tradition,
and diffused to Europe (Germany around 1200 AD) by the means
of Islamics.
Elena's HM post leaves out several important foundations of
the Western Traditon zero, before zero became a place
holder in our modern base 10 decimal system, formalized
by Stevins in 1585, by two books, one for business and
one for science.
I will not ponder the earliest uses of zero as a positional
number place holder in non-Western Tradition cultures,
such as Mesoamericans, at this time. What I will show is
over 3,000 years of use of zero in the Babylonian and Egyptian
tradition, prior to Vedic uses. That is, the earliest uses of
zero should not be ignored as number theory, as is so often
the case in USA textbooks. Accurate reporting of the past
should be a standard on which our state math frameworks
can and should structure their math pedagogies, K -12.
To summarize Babylon's use of zero, Otto Neugebauer may
not have been first to see its use. I will not cover the
use of zero before 1500 BC in this region of the world,
at this time. What I will show is the Babylon and
Egyptian scribes shared many ideas, like zero, well
before the use of Egyptian demotic script formalized
a large body of information after 1500 BC.
What I will show is the Egyptian side of zero, before
1500 BC, first within the oldest hieroglyphic script,
and second within hieratic and demotic scripts.
As most math historians are aware, hieroglyphic script
included a binary fraction numeration system that
dominated Egypt's oldest period, the pyramid builders,
before 2600 BC. At that time zero was fully used in
several ways, one mathematical and one spiritually.
The mathematical system created the Horus-Eye round
off system where the first 6-terms of the series:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
with,
0 = 1 - 1/2 - 1/4 - 1/8 - 1/16 - 1/32 - 1/64 - ... - 1/2n - ...
were used as both business and scientific notations.
When numbers smaller than 1/64 were desired, ro units
were introduced.
On the weights and measures side of Egyptian culture,
that lasted at least through the Classical Greek and
Hellene period, and into European history, 1 was a standard
of perfection, say of contents of wheat in bread. 0 was a
standard of impurity, say of the contents of wheat in bread.
On the spiritual side of Egyptian culture, 1 was infinity,
god, the ultimate; while zero was the smallest known idea.
What is also interesting is the number theory side of Egyptian
mathematics found in hieratic script, that fully emerged
in the Middle Kingdom, as the Moscow Papyruus of 2,000 BC
and other documents detail. Exactness of rational numbers
and proportions were stated in unit fraction series
using several methods, eliminating the need for round-off
(as a separate but equal numeration system, as our modern
base 10 decimal system has not duplicated).
I will not re-state the use of hieratic fractions at this
time, other than to say, Egypt knew very well the limits
of zero, stated in both infinite (Horus-Eye) and finite
(hieratic) series, as the an early use of zero and several
of its number theory properties inspired.
To cite one Egyptian reference, zero in the RMP is clearly
reported by Robins-Shute 'The Rhind Mathematical Papyrus',
British Museum Publications Ltd, and re-published by Dover
(1987), covers basic points, mentioned above.
I again thank Elena for sparking a most important subject,
beginning a review of the deeper historical roots of the
Western Tradition's first use and applications of zero
as a mathematical number.
Regards to all,
Milo Gardner