Date: Nov 5, 2012 8:35 AM
Author: Milo Gardner
Subject: Re: Why study Egyptian fraction math?

On an economic level, Egyptian fraction finite arithmetic empowered a decentralized Middle Kingdom absentee landlord system:

Scribal rational numbers, LCMs, red auxiliary numbers, 2/n tables, algebra, geometry and weights and measures jump-started Middle Kingdom finite arithmetic and a partially decentralized economy. Scribes created finite quotient and exact remainders combining theoretical and practical methods to measure 1/320 portions of grain in finite arithmetic. Scribed converted rational numbers to concise unit fraction series n the AWT and RMP that scaled a hekat unity, (64/64), by quotients (Q/64) and remainders (5R/n) to obtain 1/320 (ro) units to value products.

The Egyptian fraction numeration system assisted Pharaoh and elites in controlling granary outputs and decentralized productions of bread, beer and other grain based products. Scribal algebraic geometry formulas created cubit-cubits, khar, and hekat weight and measures units. A cubit-cubit contained 3/2 khar. A khar contained 20 hekat. A hekat contained 4800 ccm. Theoretical and practical measurements of cubit-cubits, khar, 400-hekat, 100-hekat, 4-hekat, 2-hekat, 1-hekat, 4-ro, 2-ro and 1-ro reported additional hekat sub-divisions. Smaller hekat units appear in RMP 47, RMP 83, and the Kahun Papyrus.

Overall scribal recorded weights and measures units in double-entry book book keeping systems making scribes first accountants and second mathematicians. One inventory control method was discussed in one Rhind Mathematical Papyrus problem. RMP 38 exposed as an aspect of trading units by multiplying one hekat, 320 ro, by 7/22 reporting (101 + 9/11)1-ro. Ahmes proved the accuracy of the answer by inverting 7/22 and multiplying (101 + 9/11) by 22/7 returning 320 1-ro, adding a comment that an exact hekat had been found.

Inside the partially decentralized economy the Heqanakht Papers discuss two absentee landlords' family and estate production and profit concerns in four letters. Accounting for Private Estates and the Household in the 20th Century BC Middle Kingdom by Mahmoud Ezzamel appeared in the journal Abacus, Vol. 38, No. 2 (2002), pp.235-262. Ezzamel, an accountant, shows that absentee landlords relied on theoretical commodities and metals units that summed to a monetary system. Practical measurements issued payments to workers and implemented other management controls within rational number remainders written in Egyptian fractions. An abstract of Ezzamel's article follows: 2,000 BCE Accounting Article.