Akhmim wooden tablets
The Akhmim wooden tablets or Cairo wooden tablets (Cairo Cat. 25367 and 25368) are two ancient Egyptian wooden writing tablets. They each measure about 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The inscriptions on the first tablet includes a list of servants, which is followed by a mathematical text. The text is dated to year 38 (it was at first thought to be from year 28) of an otherwise unnamed king. The general dating to the early Egyptian Middle Kingdom combined with the high regnal year suggests that the tables may date to the reign of Senusret I, ca. 1950 BC. The second tablet also lists several servants and further contains mathematical texts.
The first half of the tablet details five multiplications of a hekat unity (64/64) by 1/3, 1/7, 1/10, 1/11 and 1/13. The answers were written in binary Eye of Horus quotients, and exact Egyptian fraction remainders, scaled to a 1/320 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 (3, 7, 10, 11 and 13) dividend that returned the ab initio hekat unity, 64/64.
In 2002, Hana Vymazalová obtained a fresh copy of the text from the Cairo Museum, and confirmed that all five two-part answers were correctly checked for accuracy by the scribe that returned a 64/64 hekat unity. Minor typographical errors in Daressy's copy of two problems, the division by 11 and 13 data, were corrected at this time. The proof that all five divisions had been exact was suspected by Daressy, but was not proven in 1906.
The first problem divides 1 hekat by writing it as + (5 ro) (which equals 1) and dividing that expression by 3.
- The scribe first divides the remainder of 5 ro by 3, and determines that it is equal to (1 + 2/3) ro.
- Next, the scribe finds 1/3 of the rest of the equation and determines it is equal to .
- The final step in the problem consists of checking that the answer is correct and the scribe multiplies by 3 and shows that the answer is (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 +) (5 ro), which he knows is equal to 1, but wrote down 64/64, as Hana Vymazalova proved in 2002.
In modern mathematical notation, one might say that the scribe showed that 3 times the hekat fraction (1/4 + 1/16 + 1/64) is equal to 63/64 and that 3 times the remainder part ((1 + 2/3) ro) is equal to 5 ro, which is equal to 1/64 of a hekat, which sums to the initial hekat unity (64/64).
The other problems on the tablets were computed by the technique. The scribe used the identity 1 hekat = 320 ro and divided 64 by 7, 10, 11 and 13. For instance in the 1/11 computation, the division of 64 by 11 gave 5 with a remainder 45/11 ro. This was equivalent to (1/16 + 1/64) hekat + (4 + 1/11) ro. Checking the work required the scribe to multiply the two-part number by 11 and showed the result 63/64) + 1/64 = 64/64, as all five proofs reported.
The computations show several minor mistakes. For instance, in the 1/7 computations was said to be 12 and the double of that 24 in all of the copies of the problem. The mistake takes place in exactly the same place in each of the versions of this problem, but the scribe manages to find the correct answer in spite of this error since the 64/64 hekat unity guided his thinking. The fourth copy of the 1/7 division contains an extra minor error in one of the lines.
The fraction 1/11 computation occurs four times and the problems appear right next to one another, leaving the impression that the scribe was practicing the computation procedure. The 1/13 computation appears once in its complete form and twice more with only partial computations. There are errors in the computations, but the scribe does find the correct answer. The 1/10 computation is the only fraction computed only once. There are no mistakes in the computations for this problem.
Hekat problems in other texts
The Rhind Mathematical Papyrus contained over 60 examples of hekat multiplication and division in RMP 35, 36, 37, 38, 47, 80, 81, 82, 83 and 84. The problems were different since the hekat unity was changed from the 64/64 binary hekat and ro remainder standard as needed to a second 320/320 standard recorded in 320 ro statements. Some examples include:
- Problems 35-38 in the Rhind Mathematical Papyrus find fractions of the hekat. Problem 38 scaled one hekat to 320 ro and multiplied by 7/22. The answer 101 9/11 ro was proven by multiplyied by 22/7 facts not mentioned by Claggett and scholars prior to Vymazalova.'
- Problem 47 scaled 100 hekat to (6400/64) and multiplied (6400/64) by 1/10, 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90 and 1/100 fractions to binary quotient and 1/1320 (ro) remainder unit fraction series.
- Problem 80 gave 5 Horus eye fractions of the hekat and equivalent fractions as expressions of another unit called the hinu. that were left unclear prior to Vymazalova. Problem 81 generally converted hekat unity binary quotient and ro remainder statements to equivalent 1/10 hinu units making it clear the meaning of RMP 80 data.
- T. Eric Peet, The Journal of Egyptian Archaeology, Vol. 9, No. 1/2 (April 1923), pp. 91–95, Egypt Exploration Society
- William K. Simpson, An Additional Fragment from the "Hatnub" Stela, Journal of Near Eastern Studies, Vol. 20, No. 1 (Jan 1961), pp. 25–30
- Daressy, Georges, Catalogue général des antiquités égyptiennes du Musée du Caire, Volume No. 25001-25385, 1901.
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- 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. http://independent.academia.edu/MiloGardner/Papers/163573/The_Arithmetic_used_to_Solve_an_Ancient_Horus-Eye_Problem
- 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)