Egyptian number theory -

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# Egyptian number theory -

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

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## 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