I ABSTRACT

Student scribes from 2,000 BCE to 1,500 BCE, the Egyptian Middle Kingdom studied non-optimal arithmetic (per the EMLR) before learning optimized, but not optimal, arithmetic methods (per the Kahun and AHmes papyri).

Middle Kingdom arithmetic fragments must be stripped away to fairly translate omitted scribal initial and intermediate calculations into unified modern arithmetic statements. Ahmes' corrected initial calculation expose rational numbers written in modern-looking addition, subtraction, multiplication and division operations.

II Summary

Recently decoded scribal vulgar fraction data expose LCM multipliers, and aliquot parts of divisors. written within optimized red auxiliary number 2/n tables, and an economic context. RMP 36 provides this clearest set of example calculations. Scribal vulgar fraction (rational number) need to be stripped away to see the ancient proto-number theory. For example, Ahmes' arithmetic notation confused 20th century scholars. Scholars suggested that scribal division consisted of 'single false position', a medieval root finding method that was unknown to Middle Kingdom scribes. Updated Middle Kingdom rational numbers and arithmetic operations expose modern looking arithmetic recorded in the following ways:

A. addition: (n/p + m/q) = (nq + mp)/pq 1. common: (1/p + 1/q) = (p + q)/pq 2. frequent: (1/p + 1/np) = (n + 1)/np

B. subtraction: (n/p - m/q) = (nq - mp)/pq, 51 times, as a secondary 2/nth table method.

C. multiplication: (n/p x m/q) = (nm/pq), and (n/p x (m/m) = nm/mp, with mp aliquot parts = nm, writing 51 exact unit fractions series in 2/n table. A second Old Kingdom duplation method was used to prove answers' correct value, by working the problem backwards (as RMP 1-28 detail). The second method followed a doubling pattern (duplation), while the first and second methods looked and acted as modern arithmetic definitions.

D. division: (nm/pq)/(m/q) = (nm/pq) x (q/m) = n/p, with division by m/q, an inverse q/m multiplication operation.

III CONCLUSIONS

Updated 21st century analysis of Ahmes' 87 problems strip away Middle Kingdom Egyptian fraction notations thereby exposing ancient rational numbers written with ancient arithmetic operations used within an economic context. Scholars in the 20th century transliterated hieratic numerical data within an additive notation. To fully decode and translate Ahmes' rational number arithmetic into modern arithmetic statements a wider point of view was needed. One successful view has stripped and replaced Ahmes' unit fraction notation with ancient and modern rational number data that includes Ahmes' initial statements.

Weights and measures units used in the RMP and the Middle Kingdom contained several classes of errors and poor translations. Serious translation errors have been corrected by recognizing theoretical and practical properties of ancient Egyptian fraction mathematics. Updated theoretical weights and measures, arithmetic, algebra and geometry 'initial calculations' assist scholars to fairly report scribal methods by double checking the accuracy of 'practical' unit sizes, and scribal final answers. Interdisciplinary teams are being established in 2009 to correct 20th century translation errors related to the Ebers Papyrus, a medical text, and other hieratic to better report Middle Kingdom hieratic texts, and ancient scribes recorded the information.

Considering the 3,650 year life of the Egyptian fraction system, few theoretical changes were introduced from 2,000 BCE to 800 AD and 1454 AD, though notational changes were implemented by Greeks, Arabs and others. Arab scribes, for example, replaced ciphered hieratic and Greek numerals with base 10 numerals while also substituting a subtraction method for Ahmes' 2/n table created multiplication and aliquot part method. Fibonacci reported a medieval version of the Egyptian system in the Liber Abaci. With the fall of the Byzantine Empire in 1454 AD, the Liber Abaci fell out of general use, as did the Egyptian fraction system. By 1600 AD, Stevin's 1585 AD decimal system, written within an Arab algorithm, erased the majority of the theoretical features of Egyptian fraction arithmetic, algebra, geometry, and its weights and measures systems. Today, few reminders of the greatness of Egyptian fraction mathematics, like aliquot parts, remain.

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