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.

The 2/n table's 51 series were created from optimized red auxiliary numbers, least common multiples (LCMs). Scholars had confused proposed 2/n table construction methods by assuming additive limitations. Suggestions included five classes of methods that may have been used. Ahmes was reported creating 2/p (where p is a prime number) by possibly using two methods to convert 2/p (where p is a prime number), and three methods to convert 2/pq with composite denominators. Actually only one method had been used by Ahmes, an optimized red auxiliary LCM method.

The Egyptian Mathematical Leather Roll (EMLR), circa 1,900 BCE, reports the creation of non-optimal Egyptian fraction series likely selecting non-optimized red auxiliary numbers LCMs. The EMLR was likely a student scribe's introduction to the optimized 2/n table's LCM method. Ahmes explicitly practiced the selection of optimized LCMs in three problems, RMP 21 - 23.

Today, it is understood that red auxiliary numbers were used to create non-optimal EMLR and optimized 2/n table series. The EMLR 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), written in red numbers, 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 non-additive method that was suggested to parse denominators of the first partition into aliquot parts. H-B was only a modern analytical method. It was unknown to Ahmes. The modern method allows 2/p to be stated as optimal unit fraction series very near Ahmes' actual thinking.

In 2002, a single LCM method was reported that converts 2/101 by a multiple 6, creating 12/606 or (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606 as Ahmes' RMP shorthand reports. The Egyptian Mathematical Leather Roll also used a multiple of 6 to convert 1/4, 1/7, 1/9, 1/10 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 different 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 were published 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 LCM method created nearly optimal unit fraction series. Scribal students they were introduction to 2/n table multiples by working Egyptian Mathematical Leather Roll non-optimal multiples, and three rmpo problems, RMP 21-23.

[edit] References

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