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Topic: Egyptrian mathematics: an over view
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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Re: Egyptian mathematics: an over view
Posted: Feb 15, 2007 4:50 PM
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Dear Forum members,

Reponding to a friend's comment on the Akhmim Papyrus, that Kevin Brown and I jointly analyzed about 12 years ago, the old data is summarized by:

http://www.ecst.csuchico.edu/~atman/Misc/horus-eye.html

A few updates need to be made related to Silger's 2002 publication of Fibonacci's 1202 AD Liber Abaci, which points out that a first and a second subtraction were allowed, thereby explaining several hard to evaluate data points in the n/17 table, as had been suggested by Brown and myself.

The Akhmim Papyrus was written around 500 AD. It is a Coptic text that used two forms of the Hultsch-Bruins method, a first and second subtraction. The first method: 2/p - 1/A = (2A -p)/AP was cited be Fibonacci per his methods 4, 5 and 6, as listed at the end of his seventh chapter, as aptly translated by Sigler. Note that Fibonacci's methods 4 and 5 stressed unit fraction first partitions, the style followed by Ahmes, while a medieval method 6 allowed vulgar fractions, such as 3/8 raised to a multiple of 6 or 18/48, to be a first partition.

The second Hultsch-Bruins method appearing in the Akhmim Papyrus is a modified form. The modification was read by
Sylvester in 1891 as only an early version of the greedy algorithm. The Coptic version, and the data provided by Fibonacci, clearly show a second subtaction, as Leonardo (Fibonacci) lists as his 7th method, of 7, was used for
4/17 and several other n/17 table entries (i.e 8/17 and 13/17), and that n-subtractions as a greedy algorithm would require, were not mentioned by Fibonacci, nor by the Coptic scribe.

As a summary, beginning with Wilbur Knorr's 1982 Historia Mathematica data, the following analysis is pertinent,
and therefore the data has been updated to be consistent with the RMP and the Liber Abaci.

n/17 Akhmim P. Value 1650 BC Egyptian Frac
---- ----------------- ---------------------------------
2/17 12' 51' 68' ; 2/17 - 1/12 =(4 + 3)/(12*17)
3/17 12' 17' 51' 68'; 2/17 + 1/17
4/17 12' 15' 17' 68' 85'; 4/17-1/12 =(3+12+16)/(12*17)
= 68' + 17' + 16/(12*17), or
4/51-1/15 = 9/(15*51) = 85'

5/17 4' 34' 68'; 5/17 - 1/4 = (2 + 1)/(4*17)
6/17 3' 51'; 6/17 - 1/3 = 51', trivial
7/17 3' 17' 51'; 6/17 + 1/17
8/17 3' 15' 17' 85'; 8/17 - 1/3 = (3+4)/51,
4/51-1/15=(51+9)/(51*15)
= 15' + 85'

9/17 1/2 34'; 9/17 - 1/2 = 34', trivial
10/17 1/2 17' 34'; 9/17 + 1/17
11/17 1/2 12' 34' 51' 68'; 10/17 + 2/17
12/17 1/2 12' 17' 34' 51' 68'; 11/17 + 1/17
13/17 1/2 4' 68'; 13/17-1/2 = 9/34,
9/34-1/4 =2/(4*34)= 1/68

14/17 1/2 4' 17' 68'; 13/17 + 1/17
15/17 1/2 3' 34' 51'; 9/17 + 6/17
16/17 1/2 3' 17' 34' 51'; 15/17 + 1/17

Legend: 4' = 1/4 following the Greek style of writing
Egyptian fractions.

Best Regards to all,

Milo Gardner



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