I ABSTRACT

After 2050 BCE Egyptian math students learned Middle Kingdom unit fraction arithmetic written in ciphered hieratic script. Beginning students studied non-optimal methods in the Egyptian Mathematical Leather Roll (EMLR). Advanced students learned concise, but not optimal, 2/n table methods in the Kahun Papyrus and Ahmes Papyrus, the later also known as the Rhind Mathematical Papyrus (RMP).

Ahmes' arithmetic style, for example, omitted initial and intermediate calculations. Replaced in the 21st century, Ahmes' complete rational number statements expose unified modern arithmetic statements. That is, fragmented Middle Kingdom arithmetic statements have been stripped away to replace omitted scribal initial and intermediate calculations written in modern 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.

Bibliography

1

A.B. Chace, Bull, L, Manning, H.P., Archibald, R.C., The Rhind Mathematical Papyrus, Mathematical Association of Amnerica, Vol I, 1927. NCTM reprints available.

2

Milo Gardner, An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati, MD Publications Pvt Ltd, 2006.

3

Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books, 1992.

4

Oystein Ore, Number Theory and its History, McGraw-Hill Books, 1948, Dover reprints available.

5

T.E. Peet, Arithmetic in the Middle Kingdom, Journal Egyptian Archeology, 1923.

6

Tanja Pommerening, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005.

7

Gay Robins, and Charles Shute Rhind Mathematical Papyrus, British Museum Press, Dover reprint, 1987.

8

L.E. Sigler, Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation, Springer, 2002.

9

Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.