RemainderArithmeticVsEgyptianFractions 2006-04-10 15:59:26 2009-10-04 12:47:53 Definition Egyptian mathematics arithmetic progressions Rhind Mathematical Papyrus Kahun Papyrus % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here INTRODUCTION: The 143 year old body of literature suggests decodings aspects of Egyptian fraction mathematics. The literature touches on aspects of 4,000 year old Egyptian fractions, one focus of this post. Modern research on 1650 BCE Egyptian mathematics began in 1879, 15 years after the Rhind Mathematical Papyrus(RMP), and the Egyptian Mathematical Leather Roll (EMLR) were deeded by the family of Henry Rhind to the British Museum. The 1879 debate began after a bootleg copy of RMP was taken from the British Museum and published in Germany. The RMP contents initiated a wide ranging debate. British, German, European, USA and Arab scholars have debated the RMP's themes beginning with its 2/n table. The arithmetic debate reached an additive acme in 1927 with Chace's view of the RMP. Scholars continued to discuss conflicting RMP themes well after after Chace's publication. Additional Egyptian fraction texts were brought into the debate. In 1927 scholars hoped that the EMLR would shed light on the RMP and its Egyptian fraction methods. The 26 line 1800 BCE EMLR was unrolled and additively read by British Museum scholars. An anticipated deeper understanding of Egyptian arithmetic was not reported. Several early 1930's German scholars suspected that the \PMlinkexternal{EMLR}{http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll}, at some point, would provide deeper insights. In 1933 the events of World War II virtually stopped Egyptian fraction research. Research started up again in 1945. Research slowly progressed over the next 50 years. Gillings published an excellent summary of the available Egyptian fraction texts in 1972 and indirectly mentioned the Akhmim Wooden Tablet. Gillings accepted the majority of the 1920's additive views of Peet, and Chace while adding several minor suggestions as aids to read the most popular texts. In 2002 three publications jump-started several scholarly views on Egyptian fraction arithmetic onto new arithmetic tracks. The first was an algebraic version of 22 EMLR conversions of rational numbers created by six multiples. The multiples may have been non-additive multiples. They hinted at a deeper arithmetic likely used by Ahmes. The EMLR multiple method was soon connected to the second publication, the 1202 AD Liber Abaci's and seven rational number conversion methods published by Sigler. The Liber Abaci had been read for years in fragmentary ways. Finally the full text was translated from Latin to English. The third publication was the 1900 BCE Akhmim Wooden Tablet. This text hinted at an abstract form of Egyptian remainder arithmetic. The paper was published by Vymazalova, a Charles U. graduate student. In 2007 a multiple method connected the first 2002 publication to the second 2002 publication. The newly reported link allowed EMLR and RMP Egyptian fraction data to be computed by an identical multiple method. It is clear by considering the entire scope of Egyptian fraction literature that six non-optimal EMLR multiples had been adapted by Ahmes, the RMP scribe, into a single optimal multiple method. Ahmes seemed to easily convert 51 2/n table rational numbers to optimized Egyptian fraction series by selecting an optimal multiple. Research continues to parse Ahmes' math specifics. All that is known for sure is that Ahmes used 'red auxiliary' numbers, an LCM method. Ahmes' selection of an optimal multiple may have also considered Akhmim Wooden Tablet and RMP remainder arithmetic. BACKGROUND: Using Webster's new collegiate dictionary, an Egyptian is defined by: 1. a native or inhabitant of Egypt; 2. the Afro-Asiatic language of the ancient Egyptian from the earliest time to the 3rd century A.D. By adding the word fraction to the word Egyptian, creates the phrase: Egyptian fraction. This narrative will show that Egyptians living before the 3 century A.D. wrote Egyptian fractions in ways that took over 115 years of debate to 'break the ancient scribal code' of Egyptian fraction arithmetic. The 115 year narrative's definition of an Egyptian fraction disallows the interjection of post-300 AD non-Egyptian fraction ideas and methods, such as represented by the modern greedy algorithm, and other modern decoding attempts that had hidden the ancient scribal methods from full view. For example, by removing the modern idea of algorithm, and its 800 AD birth, as a \PMlinkexternal{RMP}{http://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus} decoding possibility, the 2/n table and the EMLR methods began to be fairly decoded in other ways. That is, the possible greedy algorithm's use in the Liber Abaci (as noted by Sylvester in 1891 in the last of its Egyptian fraction methods) only included the use of a second subtraction step, and not an n-step algorithm. In other words, by placing algorithms, and other none scribal arithmetic suggestions (like false position) aside, the central outline of 2,000 BCE scribal arithmetic come into view. Three of the four scribal arithmetic operations look much like our own modern arithmetic operations. Of course, the oldest duplation multiplication operation was unique to Egyptian mathematics, a form of arithmetic that was disliked by Greeks, and Arabs. It should be noted, by the time of the Liber Abaci (1202 AD), Greek and Arab lattice multiplication came into dominance, thereby fully replacing scribal duplation methods. Yet, the oldest Egyptian arithmetic's use of addition, subtraction and division operations, looked the same in 1650 BCE, as they did in 1202 AD, as the three arithmetic operations look and perform today. Generally ancient scribes wrote rational numbers as exact unit fraction series in optimal ways. The scribal methods for converting rational numbers has been a murky subject, in several respects. Hence few modern scholars have ventured into the deeper aspects of all four of the ancient scribal arithmetic topics, as they relate to the 4,000 year time period, since their first appearance. In other words, this summary is intended to high-light a few of the murky aspects of this longer arithmetic subject reporting for the first time a unified definition of the oldest Egyptian fractions. Returning to a broader Egyptian fraction decoding topic, the first chapter stresses that the Egyptian Mathematical Leather roll and its conversion of 1/p and 1/pq unit fractions to Egyptian fractions was an ancient teaching tool for anyone wishing to become a scribe. The EMLR student raised its simple set of unit fractions to multiples of 2, 3, 4, 5, 7, and 25, as needed, and then parsed his/her denominators by multiples of the denominator. For example 1/3 was raised 2/2 = 2/6, allowing 1/3 + 1/3, a non-Egyptian fraction looking definition to be stated. Next 1/4, one of the binary numbers was raised to 4/4= 3/12, allowing (2 + 1)/12 to write 1/6 + 1/12. In total, the EMLR converted 26 lines of Egyptian fractions, several repeated 1/p or 1/pq unit fractions, converted by a different multiple of 2, 3, 4, 5, 7 and 25, and finding not-so-elegant Egyptian fraction series. The second decoding chapter begins with elegant two-term Egyptian fraction series for 2/pq vulgar fractions. Ahmes is reported as using an optimal \PMlinkexternal{RMP 2/n table}{http://en.wikipedia.org/wiki/RMP_2/n_table} method by using the same pattern, raising 2/pq to (p + 1). CONCLUSION: Egyptian and medieval Egyptian fractions methods report general conversions of rational numbers to optimal and elegant series. Ahmes, for example, wrote exact unit fraction series by a single multiple method. Over a longer period of time, seven conversions methods were summarized in the 1202 AD Liber Abaci. Four Liber Abaci methods had been used by earlier scribes writing 2/n tables. Scribal methods in 1202 AD and 1650 BCE had raised rational numbers to desired multiples to compute optional and elegant Egyptian fraction series. Early scribal methods had also partitioned a volume unit named the hekat to hin, dja, ro and other sub-units. Scribal weights and measures system scaled units and sub-units by exact Egyptian fraction remainders based on royal edits. Pharaohs were interested in controlling beer, bread, grain and other vital national inventories. Egyptian fraction volume controls were recorded in the Akhmim Wooden Tablet using a remainder arithmetic system. Remainder arithmetic units employed exact Egyptian fraction remainders. Hekat units included the hin, oipe, dja and ro. Hekat units partitioned vulgar fractions in weights and measures applications. Several classes of remainder arithmetic practices were cited in Egyptian fraction texts. One class was recorded in the Reisner Papyri and the RMP. The Reisner and RMP reported worker production rates measured in units of 10, production outputs of 10 workers days, likely in 10 hour days. Scholars often consider three math windows to parse Egyptian fraction texts. Modern and scribal algebra offers one Egyptian fraction window. Scribal 2/n tables and optimal multiple LCM methods offer a second window. Remainder arithmetic within weights and measures applications offers a third window. Considering the three math windows, scholars have hotly debated Egyptian fraction methods for over 100 years. Happily, the Egyptian fraction debate is cooling. Agreements are being reached concerning theoretical aspects of the oldest Egyptian fraction arithmetic, one being remainder arithmetic. \begin{thebibliography}{8} \bibitem{1} Georges Daressy, \emph{"Calculs Egyptiens du Moyan Empireâ?, Recueil de Travaux Relatifs De La Phioogie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, 62â72}, Paris, 1906. \bibitem{2} Milo Gardner, \emph{The Egyptian Mathematical Leather Roll Attested Short Term and Long Term, History of Mathematical Sciences}, Hindustan Book Company, 2002. \bibitem{3} Milo Gardner, \emph{An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati}, MD Publications Pvt Ltd, 2006. \bibitem{4}Richard Gillings, \emph{Mathematics in the Time of the Pharaohs}, Dover Books, 1992. \bibitem{5} T.E. Peet, \emph{Arithmetic in the Middle Kingdom}, Journal Egyptian Archeology, 1923. \bibitem{6} Tanja Pommerening, \emph{"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. \bibitem{7} L.E. Sigler, \emph{Fibonacci's Liber Abaci: Leonardo Pisano's Book of Calculation}, Springer, 2002. \bibitem{8} Hana Vymazalova, \emph{The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai}, Charles U Prague, 2002. \end{thebibliography}