|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Message
|
|
|
| ``Re: Socrates's math is difficult to follow''
by milogardner on 2008-10-28 13:03:23 |
|
| The sad issue of historians only reading the practical side of Egyptian mathematics is well documented, per your conclusion " IF it's true that they had a theoretical basis for the number, then I would certainly want to read something about this. As far as I knew (and as far as history books on the subject report) their notions of calculation and number came to them from Egypt, where the notions were developed on an empirical basis and handled case by case."
To read details of the theoretical side of Egyptian mathematics from and Akhmim Wooden Tablet's proof that 64 exactly partitioned 64/64 by 3, 7, 10, 11 and 13 into binary quotients and scaled remainders was published by Hana Vymazalova in 2002. Hana V. was a masters student at Charles U., Prague, before gaining her PhD in Egyptology in 2006. As a master's project she gained a fresh copy of the AWT from Cairo's main museum, and corrected two of George Daressy's 1906 errors (related to the n = 11 and n = 13 cases).
The ancient AWT scribe proved that each quotient and scaled remainder answer was correctly written by multiplying the answer by the initial divisor. In all five AWT cases 64/64 was returned, meaning that a hekat unity had been defined in that manner. Hana V. should be congratulated for publishing the hekat unit fact, with the n = 3 example, written as
1/4 + 1/16 + 1/64 + (1 + 2/3)*1/320 multiplied by 3 such that
(16 + 4 + 1)/64 + 5/3*(1/320)* 3 = 63/64 + 1/64 = 64/64
and likewise for n = 7, 10, 11 and 13 cases.
To fully enter the practical Egyptian fraction world of Ahmes a second fact needs to be reported.
Ahmes showed that n = 70 could only divide a hekat unity when greater than one hekat was the subject. Ahmes chose 100 hekat, writing
(6400/64)/70 = 91/64 + (30/70)*(5/5)*(1/64) =
(64 + 16 + 8 + 2 + 1)/64 + (150/70)*(1/320) =
1 + 1/4 + 1/8 + 1/32 + 1/64 + (2 + 1/7)*(1/320)
a problem that Robins-Shute muddled the intermediate steps in 1987 in their RMP book published by the British Museum.
More over, Ahmes in RMP 81 wrote out 29 examples of the quotient and scaled remainder statements were re-written into 1/10 units, by using
10/n hin,
as also translated by Tanja Pemmerening in 2002 for the 1/64 dja
64/n dja
as Ahmes himself wrote out for the 1/320 case per
320/n ro
Note that 10/n hin, 64/n dja and 320/n ro define the prsctical side of Egyptian weights and measures, and the quotient and scaled remainder level has not been reported by historians until 2002.
Note also that a theoretical side of Ahmes' 2/n table was suspected by F.Hultsch in 1895 by an aliquot part method. Bruins independently confirmed Hultsch's view of the 2/n table, a topic that Robins-Shute tried to refute in 1987.
It is clear that Ahmes 2/n table was created as the EMLR was created, by selecting multiples, multipliers of vuigar fractions as 5/5 scaled hekat unity divisions. A theoretical analysis of Ahmes 2/n table by optimal multiples is found on:
http://rmprectotable.blogspot.com
and the related EMLR's 26 conversions by non-optimal conversions by:
http://emlr.blogspot.com
Best Regards,
Milo Gardner
|
| | [ reply | up | top ] | |
|
|
|
|
|
|
|
