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RMP 38, RMP 82 and the Akhmim Wooden Tablet
Posted:
Mar 11, 2009 4:10 PM


Dear forum members.
Concerning RMP 38, having studied Peet, Chace and Ahmes' versions of the problem, a central point needs to made. Chace and Ahmes report that 35/11 was multiplied by ONE HEKAT, written as 320, both skipping over the final 1/10 multiplication to obtain 101 9/11. Ahmes actually multiplied 320 x 7/22, a critical fact that Peet, Chace, nor Ahmes explicitly mentioned.
Peet only discussed Ahmes jumping to the proof side of the discussion, by twice showing that 101 9/11 was multiplied by 22/7 to obtain 320. Peet did not confirm the initial calculation of 101 9/11 as 320 times 7/22 (written as 35/110 by Ahmes), a critical oversight.
Yes, in RMP 38 Ahmes shows that his division operation was the inverse of his multiplication operation. Both arithmetic operations should be written as we think of multiplication of fractions as n/p x m/q = mn/pq, and the inverse operation mn/pq x q/m = n/p, today.
More over, Ahmes accepted the Ahmimim Wooden Tablet's 1900 BCE's pair of division and multiplication operations by writing: [(hekat)/n times (hekatn) = hekat] 29 times in RMP 82. Ahmes multiplied ONE HEKAT, written as (64/64), by 1/n, with n limited to the range 1/64 < n < 64. In these 29 examples, Ahmes did not offer inverse operations to return (64/64). That work had been completed 250 years earlier.
The Akhmim Wooden Tablet scribe returned (64/64) five times with respect to divisors 3, 7. 10, 11 and 13, a fact that was likely known to Ahmes. Hana Vymazalova proved in 2001 that the all five AWT divisors returned (64/64). My view is that hekat multiplications ( as 320 ro) and hekat divisions ( as 64/64) were returned to a hekat by an inverse arithmetic operation.
It is important to note that Vymazalova corrected Darssy's 1906 paper in which divisors 11 and 13 were not reported as returning to a hekat unity. Peet disagreed with Daressy's 1906 incomplete analysis of the AWT in 1923.
I'd be happy to forward to Vymazalova's, Peet's and/or Daressy's RMP 82 background papers. Send an email to milogardner@juno.com, and one, two or all three papers will be forwarded.
In RMP 82, Ahmes offered reliable information on multiplication and division as inverse operations. Ahmes converted (64/64)/n = Q/64 + (5R/n)ro twopart numbers to onepart hin (1/10 hekat) numbers by a simple formla 10/n hin. RMP 82 data is found in the writing of Peet, Chace, Gillings, and Ahmes.
Scholars should no longer muddle RMP 82 and RMP 38 data, or any RMP problem that contains arithmetic operations. Ahmes' used only one arithmetic definition for division, an inverse to multiplication.
Best Regards,
Milo Gardner



