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Re: Who discovered irrational numbers?
Posted:
Mar 27, 2009 9:41 AM
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Occam's Razor does not report one algorithm within a modern sense of the term in the Middle Kingdom -- even with many scholars sensing fragments thereof. Old KingdOm Horus-Eye numeration, of course, was cursive based -- and algorithmic in scope, hence a sense of algorithm was realistically sensed in the older context.
In other words, I know of one Old Kingdom arithmetic operations that was continued in the RMP and Middle Kingdom texts. It was the multiplication (proof) rule. Other algorithmic arithmetic operations likely existed in the Old Kingom -- but none are found in the RMP or any text when read - line by line.
Old Kingdom Round-off errors that generally dropped a 1/64 unit, in Horus-Eye numeration, were exactly corrected by Middle Kingdom and later scribes. Egyptian fraction numeration generally converted rational numbers to quotients and exact remainders without using an algorithm(OF COURSE MULTIPLICAITION PROOFS DID EMPLOY AN AN ALGORITHM).
Example, convert 22/7 to an Egyptian fraction series, as listed in RMP 38 shows:
quotient: 3
exact remainder: 1/7
An interesting aspect of RMP 38 is that 7/22 was created by first converting 35/11
written by:
quotient 3
exact remainder 4/11
(which was further redued to unit fractions)
and then multiplying by 1/10 creating a vulgar fraction
statement: 35/110 = 7/22
which was multiplied by 320 ro, obtaining 101 9/11 ro
320 x 7/22 = 101 9/11 was then returned to 320 ro
by multiplying 101 9/11 by 22/7 = 320 ro
As a hyptothesis, Ahmes may have been discussing a well known buIlt in loss contained in 1 hekat of grain. Ahmes consdered pi = 256/8. A correction may have been instituted by considering 22/7, the details of which were not recorded by Ahmes.
Best Regards,
Milo Gardner
On the irrational number level, rounding off was required by most often -- but n ot always using 256/81.
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