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Title: THE EGYPTIAN MATHEMATICAL LEATHER ROLL, ATTESTED SHORT TERM AND LONG TERM. By: MILO GARDNER <[email protected]> SACRAMENTO, CALIFORNIA Date: Feb. 16, 2002 1. ABSTRACT The Egyptian Mathematical Leather Roll (EMLR), housed at the British Museum, contains 26 unique Egyptian fraction series. In summary, five attested methods converted 1/p and 1/pq to the 26 series. A breakdown includes three classes of identities (18 series), a class of remainders (three series), and a class of algebraic identities (five-eight series). The paper discusses clues to Middle Kingdom scribal school teaching methods that are hidden in the EMLR. Here is a mystery worthy of Conan Doyle and Sherlock Holmes. The EMLR indicates the student scribe was introduced to Egyptian fraction methodologies that were an early form of abstract mathematics, as needed to work RMP problems. 2. INTRODUCTION A complete EMLR translation is included as Appendix I, written in reverse order, as the EMLR student actually wrote. Note that Appendix I contains no questions or applications, only answers. Horus-Eye numbers (1/2, 1/4, 1/8, 1/16, 1/32, 1/64) stand out along with 1/3, 1/5, 1/7, 1/9, 1/10, 1/11, 1/13 and two ways of stating 1/15th, amidst other 1/p and 1/(p x q) conversions, to exact Egyptian fraction series. The EMLR series gave exact answers with no remainder, showing that the round-off problems of the older infinite series were attempted to be resolved by Middle Kingdom scribes. This innovation arguably introduced an improved perspective of rational numbers. This paper departs from the usual history of Egyptian mathematics, where a system of multiplication was connected to a base 2 decimal fraction, duplation, infinite series numeration (Robins-Shute, MacTutor). This is a critical point, since using the Old Kingdom duplation methodology to explain Middle Kingdom texts confuses this writer, and hopefully the reader. This paper, therefore, stresses a flip side of Horus-Eye math, opening a long lost door into the development of exact Egyptian mathematics, known as Egyptian fractions. Appendix II discusses the Horus-Eye fraction method in one older sense, awkwardly computing the units of weights and measures, even when the easier Egyptian fractions are hidden within the calculation. The new Middle Kingdom hieratic system apparently was an improvement that endured in the Western Tradition for over 3,500 years, from 2,000 BC to 1585 AD, ending with the formalization of base 10 decimals (Ore). One obvious reading of the EMLR, as a test paper, infers that all Egyptian fraction methods eliminated rounding-off practices, except when associated with Old Kingdom weights and measures units. Similar round-off practices existed in the base 60 Babylon numeration system (van der Waerden). Middle Kingdom Egyptians learned to eliminate round-off within a rational number system that used exact unit fraction series, thereby greatly improving mathematical accuracy related to our pre-base ten decimal system. Analysis of the EMLR unit fraction series introduces five different methods, as the student test paper is read. It
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This note was uploaded on 02/21/2011 for the course AIBS 9982 taught by Professor Dr.ngchongchin during the Fall '10 term at Shandong University.

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