Rhind Mathematical Papyrus 2/n table

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Rhind Mathematical Papyrus 2/n table

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.

The 2/n table's 51 series were created from (red auxiliary) least common multiples. Scholars had confused the contents of the 2/n table by suggesting five methods may have been used. Ahmes was reported creating 2/"p" (where "p" is a prime number) by using two methods to convert 2/"p" (where "p" is a prime number), and three methods to convert 2/"pq" with composite denominators. The Egyptian Mathematical Leather Roll had not been clearly reported as a student scribe's introduction to an advanced 2/n table method.

Today, it is clear that one multiple method was used to create EMLR and 2/n table series. The EMLR had used 8 multiples (2, 3, 4, 5, 6, 7, 10 and 25) converting 22 rational numbers to non-optimized unit fraction series. The RMP used 14 multiples (2, 3, 4, 6, 8, 12, 20, 24, 30, 36, 40, 56, 60 and 70) converting 51 rational numbers to nearly optimal unit fraction series.

The scholarly record on decoding the 2/n table table has been mixed. 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 stated as optimal unit fraction series. In 2002, a single method converted 2/101 by the multiple 6, creating 12/606 or (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606. The Egyptian Mathematical Leather Roll used multiple of 6 to convert 1/4, 1/7, 1/9, 1/00 and 1/15. In addition, Eye of Horus 1/2, 1/4, 1/8, 1/16 numbers were converted in the EMLR by multiples reporting non-optimal Egyptian fraction series. As practice, 1/8 was converted by three multiples.

In 2002, three scholarly 2/"pq" conversion methods reported a 'red auxiliary' least common multiple method. The multiple method had been used in the 250 year older Egyptian Mathematical Leather Roll. The method converted 2/19 and 2/95 by a multiple of 12. Multiples 30 and 70 converted 2/35, and 2/91, respectively. Beyond the 2/"n" table, Egyptian fractions reported in Egyptian weights and measures, i.e. hin, dja, ro and hekat sub-units, followed the red auxiliary multiple method Egyptian fraction remainders. A longer time view of 2/n table methods includes four of seven conversion methods reported in the 1202 AD Liber Abaci, facts noticed in 2006.

In summary, Ahmes, a KP scribe, and likely all other scribes converted 2/n entries to nearly optimal Egyptian fraction series by a single least common multiple method. The red auxiliary method created nearly optimal unit fraction series. As scribal students they were introduction to 2/n table multiples by working Egyptian Mathematical Leather Roll non-optimal multiples.

References

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