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Re: Why study Egyptian fraction math?
Posted:
Nov 4, 2012 7:51 AM


A longer Planetmath paper makes the same points:
http://planetmath.org/encyclopedia/EgyptianFraction2.html
Prior to 2050 BCE Old Kingdom Egyptians rounded off top sixterms binary representations stated in 1/64 units. The HorusEye recorded rational numbers in the cursive pattern:
1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + (1/64).
Note that a potential 7th term (1/64) was roundedoff and thrown way.
After 2050 BCE an exact numeration system discontinued the roundedoff Old Kingdom binary system. An exact hieratic weights and measures system reported rational numbers in 1/64 quotient and 1/320 remainder units whenever possible.
The new Middle Kingdom math system "healed" rounded off binary series by several finite methods. Two weights and measures finite systems can be reported by:
1. 1 hekat (a volume unit) used a unity (64/64)such that (32 + 16 + 8 + 4 + 2 + 1/64)hekat+ 5 ro
and (1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64)hekat + 5 ro
meant (64/64)/n = Q/64 + (5R/n)ro
Note that the hekat unity was generally divided by rational number n. To divide by 3 scribal longhand would have written out
(64/64)/3 = 21/64 hekat + 5/192 = (16 + 4 + 1)/64 hekat + 5/3 ro =
(1/4 + 1/16 + 1/64)hekat + ( 1 + 2/3)ro
2. (100hekat)/70 = (6400/64)/70 = 91/64 hekat + 30/4480 = (64 + 16 + 8+ 2 + 1)/64 hekat + 150/70 ro =
(1 + 1/4 + 1/8 + 1/32 + 1/64)hekat + (2 + 1/7)ro
meant (6400/64)/n = Q/64 + (5R/n)ro was applied for almost any hekat division problem.
The hieratic word ro meant 1/320 of a hekat in a grain weights and measures system. Note that 5 ro meant 5/320 = 1/64.
Generally, scribal shorthand recorded duplation aspects of mental calculations and fully recorded twopart hekat quotients and ro remainders.
At other times 2/64 was scaled to 10/320 such that (8 + 2)/320 = 1/40 + 2 ro
and so forth.



