From the Math history list:

The 1650 BCE Rhind Mathematical Papyrus report three of its 87 problems, RMP 47, 82 and 83 problems written within 36 division of hekat problems. The 36 examples are added to 6 Akhmim Wooden Tablet hekat examples. The RMP and AWT hekat problems were solved in 2005 and published in 2006. RMP 47 divided 100 hekat, written as (6400/64), by 70. The quotient 91/64 and remainder 30/64 were written in binary (Horus-Eye)(64 + 16 + 8 + 2 + 1)/64 units and scaled by 5/5 to 1/320, (150/70)ro unit, quotient 2 and remainder 1/7, respectively.

Robins-Shute reported aspects of RMP 47 in 1987 by finding the correct final answer; however, garbling the beginning and intermediate steps. Robins-Shute missed Ahmes' divisors n that were allowed to increase beyond 64 as hekat numerators increased beyond 64.

For example, RMP 82 listed 29 examples that divided one hekat, replaced by (64/64). The 29 examples ranged from n = 1/64 < n < 64. The 29 problems reported binary quotients and scaled remainders to a 1/320 hekat unit that followed another example, RMP 47, that allowed numerators to increase (by scaling factors) as divisors increased.

The table of 29 answers also were converted to unscaled equivalent hin, 1/10 of a hekat unit showing that Ahmes generally reported hin units by writing 10/n hin statements.

Tanja Pemmerening pointed out corrected aspects of the unscaled 64/n dja and 320/n ro statements in 2002 and 2005.

RMP 83, the bird-Feeding rate problem, was not correctly read by Chace, nor by other 20th century scholars. Today the 1900 BCE Akhmim Wooden Tablet, and its six division problems, allows a clear view of the division of one hekat, written as 64/64, divided by n. In RMP 83 n= 6, 20 and 40. Division took place by multiplying 1/6, 1/20 and 1/40. Ahmes' answer calculated 5/8 of a hekat, the amount of grain eaten by six birds in one day.

Ahmes usually began hekat division discussions with a complex example, and proceeded to simpler examples. An exception was RMP 35-38. RMP 35 was the simplest example. It is important to note Egyptian weights and measures are decoded by stripping away Rhind Mathematical Papyrus 2/n table to reveal modern rational numbers and modern arithmetic operations.

In RMP 35-38 and RMP 66, hekat division replaced a hekat by 320 ro. Ahmes' unit fraction shorthand notations is translated to modern arithmetic as follows:

1. 320 ro was multiplied by 3/10 = 96 ro (RMP 35)

Proof: 3/10 + 6/10 + 1/10 = 1, and 96 ro + 192 ro + 32 ro = 320 ro = 1 hekat

2. 320 ro was multiplied by 1/90 = 3 + 1/2 + 1/18 = 64/18 (RMP 37)

Proof: 64/72 + 64/567 = 1

3. 320 ro was divided by 7/22 = 101 9/11 (RMP 38)

was proven by 101 9/11 times 22/7 = 320 ro

It may be interesting to note that the initial divisor 7/22 was proven from binary steps yielding 35/11 times 1/10.

4. 10 hekats of fat, 3200 ro, was divided by 365 = 8 + 280/365 (RMP 66)

was proven by the quotient 8 created by the binary steps 1 - 365, 2 - 730, 4 - 1460 and 8 - 2920 and the remainder 280 (3200 less 2920) by the addition of

246 1/3 + 36 1/2 + 1/6 = 320 ro = 1 hekat

since the unit fractions contained in the answer (2/3 + 1/10 + 1/2190) were each multiplied by 365.

Summary: Egyptian volume weights and measures replaced a hekat by two equivalents, a unity (64/64), and 320 ro. Division of the hekat, be it (64/64), or 320 ro, Ahmes' arithmetic operations anticipated modern multiplication and division as inverse operations.

A scribal method of finding areas of triangles and other shapes is reported in RMP 53-55 confirms several connections to the 3,650 year old Egyptian weights and measures topic. More importantly, RMP 38 shows that the tradition hekat, defined within a cylinder, using pi = 256/81, corrected for grain losses by considering pi = 22/7.

Bibliography

1

A.B. Chace, Bull, L., Manning, H.P. and Archibald, R.C. The Rhind Mathematical Papyrus, Mathematical Association of America, Vol 1, 1927, vol 2, 1929, and reprint 1979 (NCTM).

2

Georges Daressy, "Calculs Egyptiens du Moyan EmpireÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃâ, Recueil de Travaux Relatifs De La Philologie et al Archaelogie Egyptiennes Et Assyriennes XXVIII, 1906, 62ÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃâ72, Paris, 1906.

3

Milo Gardner, An Ancient Egyptian Problem and its Innovative Solution, Ganita Bharati, MD Publications Pvt Ltd, 2006.

4

Milo Gardner, The Egyptian Mathematical Leather Roll, Attested Short Term and Long Term, History of the Mathematical Sciences, Editors: Ivor Gratton-Guinness, and B.S. Yadav, Hindustan Book Agency, 119-134, 2002.

5

Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books, 1992.

6

T.E. Peet, Arithmetic in the Middle Kingdom, Journal Egyptian Archeology, 1923.

7

Tanja Pommerening, "Altagyptische Holmasse Metrologish neu Interpretiert" and relevant phramaceutical and medical knowledge, an abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die Altagyptschen Hohlmass, Buske-Verlag, 2005.

8

Gay Robins, Charles Shute "Rhind Mathematical Papyrus", London, British Museum, British Museum Press, 1987.

9

Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.