Ahmes

Ahmes (c. 1680 BC-c. 1620 BC) (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. A surviving work of Ahmes is part of the Rhind Mathematical Papyrus now located in the British Museum (Newman, 1956). Ahmes states that he copied the papyrus from a now-lost Middle Kingdom original, dating around 1650 BC. The work is entitled "Directions for Knowing All Dark Things" and is a collection of problems in arithmetic, algebra, geometry, weights and measures, business and recreational diversions. The 51 member 2/nth table and the following 84 problems were presented with solutions. But the scribe only offered brief notes for his often hard-to-read steps. However, bringing in additional documents like the Akhmim Wooden Tablet, Egyptian Mathematical Leather Roll, Reisner Papyrus and the Moscow Mathematical Papyrus, a broader view of Ahmes's math is being found. For example, the 2/nth table may have generally converted 1/p, 1/pq, 2/p, 2/pq and higher vulgar fractions into exact Egyptian fractions series. One or more, standardized conversion methods were employed. A generalized multiple method, as discussed in the Liber Abaci may have allowed a singular conversion method to be used. If additional conversion methods were employed by Ahmes, the Egyptian Mathematical Leather Roll methods may have been known by Ahmes.

On a broader level, considering the RMP and its parent documents, the Egyptian fraction notation was developed around 2000 BCE, most likely as a method to replace the Old Kingdom's binary fraction Horus-Eye notation, and its awkward round-off system. Finally, considering the connections provided by all the Middle Kingdom texts, the why's and how's of the mathematics that Ahmes drew upon is finally coming into focus. Ahmes' methods, as taught to him, and followed by later scribes always wrote vulgar fractions in exact ways, never rounding off when rational numbers were involved.

Yet there were times when Ahmes did make use of approximations. For example, Ahmes states without proof that a circular field with a diameter of 9 units is equal in area to a square with sides of 8 units (Beckmann, 1971). In modern notation, Ahmes' method implies the following approximation:

: $pi\; left\; (\; frac\{9\}\{2\}\; ight\; )\; ^\{2\}\; approx\; 8^\{2\}$

This method leads to a value of pi approximately equal to 3.16049, which comes within 0.6% of the true value of pi. This approximation for pi as an irrational number reached well beyond the rational number domain of Egyptian mathematics. It was used to compute the volume of a hekat, and its many sub-units, including the hin, ro, and dja as recorded in the RMP, Akhmim Wooden Tablet.

External links

* [http://rmprectotable.blogspot.com/ Breaking the RMP 2/n Table Code]