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Topic: Archimedes (and Ahmes) square root of 5, 6 and 7
Replies: 8   Last Post: Jan 16, 2013 6:58 PM

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Milo Gardner

Posts: 1,105
Registered: 12/3/04
Re: Archimedes (and Ahmes) square root of 5, 6 and 7
Posted: Dec 26, 2012 10:11 AM
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Premise: Egyptian numeration scaled rational numbers to concise unit fraction series in a wide range of applications. The inverse proportion aspect of Egyptian square root formed a finite math foundation for the pesu.

The pesu scaled Middle Kingdom Egyptian commodities that measured exact portions of grain used to produce one loaf of bread, one glass of beer, and one of other products.

Two and three step Middle Kingdom, Greek and medieval square roots were continually in use from 2050 BCE to 1202 AD. The common method was decoded in reverse chronological order, solved one of the oldest unsolved problems in the history of mathematics (Kevin Brown)

http://www.mathpages.com/home/kmath038/kmath038.htm

per:

step 1. guess (1 + 2/3)^2 = 1 + 4/3 + 4/9 = 2 + 3/9 + 4/9) = 2 7/9, error 2/9

step 2 reduce error 2/9 (3/10) = 1/15 (by dividing 2/9 by 1/2 (1 + 2/3)

(1 + 2/3 + 1/15)^2, error (1/15)^2 = 1/225, knowing (1 + 11/15) = 26/15

step 3 reduced error 1/225 (15/52) = 1/15(52) = 1/780

reached

(26/15 + 1/780)^2 = (1353/780)^2 in modern fractions

Archimedes recorded a final unit fraction series that likely began with step 2 data and added 1/780

( 1 + 2/3 + 1/15 + 1/780)^2

Archimedes may have added

1/15 + 1/780 = 53/780,

such that

(1 + 2/3 + 1/30 + 1/52 + 1/130 + 1/160)^2

was recorded?

Anyone know the actual series Archimedes recorded?



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