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Topic: One way to study 3,600 years of Egyptian fraction mathematics
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Milo Gardner

Posts: 1,105
Registered: 12/3/04
One way to study 3,600 years of Egyptian fraction mathematics
Posted: Dec 29, 2008 6:35 PM
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To study Egyptian fraction mathematics, a notation that was abandoned around 1600AD, when replaced by the modern decimal system, decoding doors need to be identified and entered with care.

The first door to open is Fibonacci's 1202 AD Liber Abaci. The Liber Abaci was Europe's arithmetic book for 250 years. Fibonacci had gathered Arab sources written from 800 AD to 1200 AD to write his book. There are many 800 AD to 1200 AD texts that are available to amplify the contents of the Liber Abaci, especially its first 124 pages of the 500 page book. The first 124 pages of Sigler's 2002 translation converts rational numbers to concise unit fraction series by seven rules. Five of the rules appear to date to the time of Ahmes, 1650 BCE. Check out this door, and see what you find. Those that read Arabic offer a wonderful resource to this project.

Once gaining an outline of Fibonaci's Egyptian fraction arithmetic, skip over the Greek texts, since few are available. We can come back to this topic later.

Let's jump to the Egyptian mathematical texts written in the 2,000 BCE to 1500 BCE era. There is a relatively large supply of hieratic texts.

A simple rule needs to be followed. To read hieratic numbers, they must be parsed from their ciphered hieratic words. This step takes practice. For example, ro meant 1/320 of a hekat, nothing more, nothing less.

Given numbers in a problem context, such as Ahmes' 2/n table, or RMP 40 and its arithmetic progression, the ancient problem can be investigated to a deeper numerical level. Excessive abstractions, often seen in reading pyramid and other texts, can be minimized, by mathematicians, and by linguists.

Note that standard 'dictionary' meanings of the words that introduce problem should be set aside (for the moment). Let the numbers speak for themselves. Then let several modified definitions of words be re-introduced at the end of the project.

That is, do not prejudge the numerical output that you may receive. It will not be rare to find a unique 'number' decoding path. Note that not one decoding path works to decode Ahmes' 2/n table members and the RMP's 84 problems.

That is, Ahmes' 2/n table and 84 problems were not all written in an additive context. The singular decoding path suggested in the 1920s, retarded research in serious ways. Today new groupings of hieratic mathematical problems need to be considered, each independently read, at first, by its own decoding path.

We do not know how many decoding paths will be needed to decode the available hieratic mathematical problems.

It is important to independently confirm at every step, be the problem Ahmes' 2/n table members, or RMP 40's arithmetic progression. Work in teams of mathematicians so that conflicting decoding paths can be resolved in your group, at a time of your group's choosing. Your group may come up with a better "Occam's Razor" solution than is published in Egyptology and math history journals.

Remember, Ahmes loved mathematical formulas. Modern code breakers are finding news classes of formulas outside of the additive context. Let's continue the work.

A third class of formula, decoded in 2005, shows that the Akhmim Wooden Tablet (AWT) numbers weights and measures units used in over 40 RMP examples. The 1923 additive translation of the AWT by Peet (1923) only read the 1/320 hekat aspect. This oversight is easily corrected today, with one version stating:

(64/64)/n = Q/64 + (5R/n)*ro

with Q = quotient, R = remainder and 1/320 replaced by the word ro, a form of exact quotient and remainder arithmetic formula that Ahmes used over and over again.

Once new formulas have been parsed, the 'dictionary' side of the problem context can be considered by professional linguists. Working together, mathematicians and linguists should discuss and agree upon the scope and the details contained in each problem.

The proposed process is likely a slow one. Its main elements have been tested and retested after 2000. Egyptian fraction mathematics used from 2,000 BCE to 1450 AD are being recovered following this method.

Best wishes to all,

Milo Gardner

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