Akhmim wooden tablet

The Akhmim wooden tablet, is an ancient Egyptian artifact that has been dated to 2000 BC, near to the beginning of the Egyptian Middle Kingdom. It is currently housed in Cairo's Museum of Egyptian Antiquities. Its text was reported by Georges Émile Jules Daressy in 1901 and analyzed and published by Daressy in 1906. The first half of the tablet details five divisions of a hekat, partitioned from its unity (64/64), by 3, 7, 10, 11 and 13. The answers were written in Eye of Horus quotients, and Egyptian fraction remainders, scaled to a 1/320th factor named ro. The second half of the document proved the correctness of the five division answers by multiplying the two-part quotient and remainder answer by its respective divisor (3, 7, 10, 11 and 13). In 2002, Hana Vymazalova, gained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly returned to the initial (64/64) unity value by the scribe. The proof that all five AWT divisions had been exact was suspected by Daressy, but was not proven in 1906. Typographical errors in Daressy's copy of two problems, the division by 11 and 13 data, were corrected by Hana Vymazalova.

For example, the first problem divided a hekat unity (64/64) by 3, such that an intermediate quotient 21 was found by 64/3, with an intermediate remainder 1. A second intermediate remainder meant 1/3 of 1/64 of a hekat, or simply 1/192. The final remainder was further scaled to 1/320 units of a hekat, namely 1 + 2/3. The final remainder was an Egyptian fraction (series) followed by the scaling factor ro, or (1 + 2/3)ro.

In order words, the intermediate quotient restated 21 to an Eye of Horus binary quotient by parsing 21/64 such that:

:$frac\{16\; +\; 4\; +\; 1\}\{64\}\; =\; frac\{1\}\{4\}\; +\; frac\{1\}\{16\}\; +\; frac\{1\}\{64\}.$

And the intermediate remainder re-stated 1/192 to 5/960 such that 1/320 was factored and renamed $ho$.

Note that the final remainder converted the vulgar fraction 5/3 to an Egyptian fraction (1 + 2/3), scaled to 1/320 of a hekat. The complete final remainder was written as $(1\; +\; 2/3)\; ho$.

In summary,the AWT's first problem reported a final two-part answer by dividing (64/64) by 3 detailing:

:$frac\{1\}\{4\}\; +\; frac\{1\}\{16\}\; +\; frac\{1\}\{64\}\; +\; left(1\; +\; frac\{2\}\{3\}\; ight)\; ho.$

Hana Vymazalova's 2002 Master's paper shows that the AWT scribe was required to prove that all five division answers were multiplied by the initial divisor, returning all answers to its starting point 64/64. That is, the division by 3 data was processed by the scribe such that

:$left(frac\{1\}\{4\}\; +\; frac\{1\}\{16\}\; +\; frac\{1\}\{64\}\; ight)\; imes\; 3\; =\; frac\{21\}\{64\}\; imes\; 3\; =\; frac\{63\}\{64\},$

for the first part; and

:$left\; [left(1\; +\; frac\{2\}\{3\}\; ight)frac\{1\}\{320\}\; ight]\; imes\; 3\; =\; frac\{5\}\{960\}\; imes\; 3\; =\; frac\{15\}\{960\}\; =\; frac\{1\}\{64\}$

for the second part. That is, adding the first and second parts: 63/64 + 1/64 = 64/64. The other four two-part answers also calculated (64/64), the beginning hekat unity value. These five division statements and proofs were reported by Vymazalova in 2002. Vymazalova thereby corrected Daressy's 1906 paper that found only three answers to be exact.

The Rhind Mathematical Papyrus contains over 30 examples of this type of hekat division. There are 29 examples of two-part numbers listed in RMP 80. Gillings's analysis of RMP 80 data was shallow. Gillings noted the small fraction divisors, from 1/2 to 1/64, and the large divisors, from 1 to 64, and other raw data. Gillings provides no arithmetic explanation of the data's respective meanings. Hekat two-part answers, quotient and remainders, were re-written into one-part remainders, 1/10 of a hekat unit, data that Gillings includes in RMP's appendix, again with no arithmetic analysis. The tenth of a hekat, a hin, was cited the hinu system, the extent of Gillings' analysis. The 30th example, RMP #34, followed AWT rules that divided 100 hekat (6400/64) by 70. The intermediate division steps followed the method set down in the AWT, dividing (6400/64) by 70, finding an intermediate quotient 91/64, and an intermediate remainder (30/70)*(1/64), (150/70)*1/320, and so forth, reaching a final two-part answer 1 + 1/4 + 1/8 + 1/32 + 1/64 + (2 + 1/7)ro. Ahmes did not double check his answer by finding the initial 100 hekat unity, skipping over the AWT's proof step.

The Ebers Papyrus is a famous late Middle Kingdom medical text. Its raw data was written in hekat one-parts suggested by the AWT, handling divisors greater than 64. Tanja Pommerening and others accurately read information from this text, for the first time, in 2002, by understanding the one-part divisions of a hekat.

References

*Daressy, G. " _fr. Cairo Museum des Antiquities Egyptiennes." Catalogue General Ostraca, Volume No. 25001-25385, 1901.

*Daressy, Georges, “ _fr. Calculs Egyptiens du Moyan Empire”, Recueil de Travaux Relatifs De La Phioogie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, 62–72.

*Gardener, Milo, "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157-173.

*Gillings, R. Mathematics in the Time of the Pharaohs. Boston, MA: MIT Press, pp. 202-205, 1972. ISBN 0-262-07045-6. (Out of print)

*Peet, T. E. "Arithmetic in the Middle Kingdom." J. Egyptian Arch. 9, 91-95, 1923.*

*Pommerening, Tanja, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass" in studien zur Altagyptischen Kulture, Beiheft, 10, Hamburg, Buske-Verlag, 2005

*Vymazalova, H. "The Wooden Tablets from Cairo: The Use of the Grain Unit HK3T in Ancient Egypt." Archiv Orientalai, Charles U., Prague, pp. 27-42, 2002.

External links

*http://mathworld.wolfram.com/AkhmimWoodenTablet.html, Scaled AWT Remainders
*http://www.mathorigins.com/image%20grid/awta.htm, Breaking the AWT Code
*http://rmprectotable.blogspot.com/, Breaking the RMP 2/n Table Code
*http://planetmath.org/encyclopedia/RemainderArithmetic.html, Quotient and remainder arithmetic in weights and measures
*http://www.whonamedit.com/synd.cfm/443.html,