Date: Nov 9, 2012 2:14 PM Author: Milo Gardner Subject: Re: Why study Egyptian fraction math? Arithmetic progressions birthed arithmetic proportions in the Kahun Papyrus:

http://planetmath.org/encyclopedia/KahunPapyrusAndArithmeticProgressions.html

"The Kahun Papyyrus was discovered by Flinders Petrie in 1889. The fragmented hieratic text dates to 1825 BCE Egypt details mathematical and medical topics. Many of its fragments are kept at the University College London. Most fragments date to the reign of Amenemhat III. One of the fragments referred to as the Kahun Gynaecological Papyrus dealt with gynecological illnesses and conditions.

A second fragment began with a brief 2/n table. Middle Kingdom scribal converted rational numbers 2/n to exact and concise unit fraction series as the RMP scribe followed 200 years later. The KP 2/n table converted 2/3, ..., 2/21 to concise unit fraction series and implied n/p conversions in shorthand proofs as the RMP scribe also demonstrated.

A longer Rhind Mathematical Papyrus (RMP) 2/n table converted 50 rational numbers, 2/3, ..., 2/101 to concise unit fraction series. A longhand LCM m conversion method waa defined in RMP 36. The method scaled n/p by LCM m to mn/mp with the best divisors of mp recorded in red auxiliary numbers that summed tn mn thereby calculating concise unit fraction series.

A. Considering KP arithmetic and arithmetic progressions the raw data contained intellectual and finite operating components. Arithmetic progressions and geometric proportions were the highest forms of Egypotian mathematics. The KP scribe defined a 10-term arithmetic progression summed to 100, with a difference (d) of 5/6. The KP arithmetic progression was discussed in two RMP problems.

Scribal arithmetic progressions were also found in the RMP. Ahmes listed two columns of data (published by Gillings in 1972). Ahmes's thinking is shown in Gillings' column 11 by multiplying 5/12 times 9, a fact that was needed to find the largest term of the RMP progression. Ahmes then added 10 and wrote out the correct largest term of the arithmetic progression, and subtracted 5/6, nine times. Gillings found the remaining terms of the progressions by using the KP's method. To understand the KP method, readers must make arithmetic calculations as the Middle Kingdom scribes wrote down in their three problems, double and triple checking your work with several tools.

Gillings' 1972 analysis of both RMP versions of Middle Kingdom arithmetic progression failed to parse the method in a manner that was comparable, in every respect, to the KP method. For example, Gillings noticed similar problems in the RMP (RMP 40, 64). Gillings muddled three pages of his analysis on the topic. ... "