Rhind Mathematical Papyrus 2/n table

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The Rhind Mathematical Papyrus contains, among other mathematical contents, a table of Egyptian fractions created from 2/n. The text reports 51 rational numbers converted to short and concise unit fraction series. The document was written in 1650 BCE by Ahmes. Aspects of the document may have been copied from an unknown 1850 BCE text. Another ancient Egyptian papyrus containing a similar table of Egyptian fractions, the Kahun Papyrus, written around 1850 BCE is about the age of one unknown source for the Rhind papyrus. The Kahun 2/n table differs in minor respects to the Rhind Papyrus' 2/n table.

Scholars had suggested, over the years, five methods that Ahmes may have used to create 2/p (where p is a prime number) expansions. Two methods use the form 2/p (where p is a prime number), and three methods use the form 2/pq with composite denominators. The Egyptian Mathematical Leather Roll had been seen as a student scribe's introduction to 2/n table methods since the 1930's. In 2002 a multiple method aptly connects all 2/n table and all EMLR Egyptian fraction series.

Prior to 2002 the Hultsch-Bruins method, named for F. Hultsch (1895) and E.M. Bruins (1945), was the best known method that parsed the denominator of the first partition into aliquot parts. H-B allowed 2/p to be converted to optimal unit fraction series though in fragmented ways. In 2002, a single method converted 2/101 by the multiple 6, or an identity 1 = (1/2 + 1/3 + 1/6), creating 12/606 or (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606. The EMLR also used a multiple of 6 to convert 1/101 by reporting 6/606 = (3 + 2+ 1) /606 = 1/202 + 1/303 + 1/606. In addition 21 other 1/n unit fractions, mostly Eye of Horus numbers, were converted in the EMLR by multiples 2, 3, 4, 5, 6, 7 and 25 to create non-optimal Egyptian fraction series.

In 2002, three scholarly 2/pq conversion methods began to be reported as a single multiple method. The method was also reported in the Egyptian Mathematical Leather Roll creating non-optimal Egyptian fraction series. The method converted 2/95 by factoring 1/5 times 2/19. Hints of the H-B method may have been used to Ahmes to convert 2/19 to an Egyptian fraction series, with 1/5 being multiplied to create the optimal Egyptian fraction series. Another method may have converted 2/35, and 2/91. Yet 2/35 and 2/91 can be created by multiples. Alternatives 2/n table construction methods continue to be suggested in the professional literature. Given that the 2/n table system of Egyptian fractions used in Egyptian weights and measures, via hin and ro hekat sub-units, reported in RMP 81, offers a wider view of Egyptian arithmetic and the 2/n table problem. One longer time view includes the consideration of seven 1202 AD Liber Abaci rational number conversion methods. Four of the seven Liber Abaci methods may have been known by Ahmes. A once stale additive 2/n table debate is heating up, and may be solved by bringing together a wide range of Egyptian fraction texts than were considered prior to 2002.

In summary, Ahmes and a KP scribe likely converted 2/n by a single multiple method, listing their efforts in tables. The method created short and concise unit fraction series, as well and non-optimal EMLR series. Scribal shorthand omissions had confused scholars for over 120 years. Scholars, as a group had been unable to solve either the 2/n table problem or the EMLR problem until 2002. Scholars had relied on intuition and personalized mathematical senses to suggest incomplete details of Ahmes' and the EMLR's construction method(s). Today, debates focus on the 2/n table, the EMLR's 1/p, and 1202 AD conversion method(s). A final resolution to the 2/ n table problem may be near.

[edit] References

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