
Re: RMP # 48
Posted:
Oct 3, 2007 7:53 AM


Lee,
While citing the infinite to finite connections provided by scaled Egyptian fraction remainders, summarized by:
"... MMP 10 stressed a proposed link to Eudoxus and Archimedes without stating the 1/4th geometric (infinite) series Egyptian fraction details,
4A/3 = A + A/4 + A/16 + A/64 + ... + A/4n + ...
which Archimedes converted to a finite Egyptian fraction series,
4A/3 = A + A/4 + A/12
as one slice needed to exactly sum the area, much as the MMP did 1,500 years earlier.
...
Beauty is in the eye of beholder  and Peet saw no nonadditive beauty in Egyptian fractions, as Ahmes and the Middle Kingdom scribes saw (beauty) when they (he) solved the HorusEye problem exactly writing
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...
as an exact Egyptian fraction series,
by using any rational number as an exact remainder, such as, the remainder 1/64 as 5/320 into ro units, per "
needs to be simplified to an additive statement
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 5 ro
That is, Ahmes divided 100 hekat, stated as (6400/64), using a nonadditive method so that a division by 70 could take place in HorusEye fractions, as a quotient: 91/64, and Egyptian fractions, as a remainder: 30/(64*70), scaled to ro units: 150/70 *(1/320) , or
(6400/64)/70 = (64 + 16 + 8 + 2 + 1)/64 + (2 + 1/7)*1/320
= 1 + 1/4 + 1/8 + 1/32 + 1/64 + (2 + 1/7)ro
as RobinsShute listed in their 1987 RMP book as the proper answer without directly explaining Ahmes' arithmetical steps, as noted above.
RMP 81 includes 29 additional hekat partition problems of this type. Hence Ahmes thought and wrote in this manner, points that Gillings did not mention in 1972 in "Mathematics in the Time of the Pharaohs". All 29 RMP 81 problems were first written into ro units using a nonadditive method that is well defined and proven in the Akhmim Wooden Tablet (written 350 years prior to the RMP). More importantly a hekat unity value of (64/64), validated by Vymazalova in 2002, allowed Ahmes to use divisors n that ranged from 1/64 to 64, with ro and hin (1/10 of a hekat) units.
The two part HorusEye and Egyptian fraction context (points that Gillings, Peet and others had not reported as Ahmes clearly wrote), were easily written into additive hin units by a method that should be obvious to all: 10/n hin, with n being the divisor.
Best Regards.
Milo Gardner

