# Egyptian number theory -

Forums:

Dear Forum members,

Classical Greeks like Plato studied in Egyptian
schools for a very good reason. Egyptian number
theory and its use of rational numbers to exactly
measure volume and other units developed into a
form of finite arithmetic - that many oddly label
as Egyptian fractions was created from a different
foundation.

The fact of history reported in the Akhmim Wooden
Tablet and elsewhere named this 2,000 BCE form of
numeration Remainder Arithmetic. Egyptian fractions
formed only one phase of writing a vulgar fraction,
as noted by:

n/p or n/pq = Q + R

To assist the division of any rational number special
substitutions were used, such as a hekat unity = 64/64
that allowed 1/nth of a volume unit - the hekat - to be
stated as a two-part number, in the form:

(64/64)/n = Q/64 + (5*R/n)* 1/320

with (5*R/n) being the Egyptian fraction component -
easily found whenever needed, hence this ancient
method had been created in a generalized form.

There were problems with the system. In the beginning
it may have only been used for bread or beer measuremnents -
where n was allowed allowed to reach 64. However, very soon
smaller units were needed, the next one being a hin, or
1/10th of a hekat (as reported by Gillings in the RMP,
problem #31, as I recall, a system that Gillings did not
understand in the deepest terms). Again, 1/320 ( a unit
called ro - actually it should be seen as a common divisor)
was factored from the remainder term reducing the size of
the vulgar fraction, that was to be converted to an Egyptian
fraction series.

Concerning medical prescriptions, as reported around 1,700 BCE
to 1,000 BCE smaller units were needed. Here the numerator
had to be increased to allow larger divisors than 640, the
limit of the hin system. Tannja Pemererening, a recently
minted PhD wrote her thesis on this subkject, publishing the
thesis in 2005, following up her 2002 master's thesis (which
is available on the web) so the world of Egyptian math and
its facilitating weights and measures systems are now
coming to light - in very new ways.

Any comments on these long overlooked facts?

Best Regards,

Milo Gardner

### Re: Egyptian number theory -

Due to my relative inability to follow a calculation without
an extremely detailed presentation thereof, I wasn't entirely sure
what you were talking about everywhere. If you can write a
succinct entry in the encyclopedia that makes all the computations
and their usefulness clear, I would be interested in reading it.

I noticed that "egyptian fractions" are defined on PlanetMath,
at http://planetmath.org/encyclopedia/UnitFraction.html

### Re: Egyptian number theory -

Three Encyclopedia posts : remainder arithmetic,
remainder arithmetic vs Egyptian fractions and
Egyptian fraction, Hultsh-Bruins opens a little of
the use of aliquot parts known to Greeks, as
taught by Egyptians.

Please excuse the roughness of the Encylopedia
entries. They will be worked on and hopefully
improved over the next couple of weeks.

Milo