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Re: Archimedes (and Ahmes) square root of 5, 6 and 7
Posted:
Dec 28, 2012 10:58 AM


Peter,
my work contains typos .. the problem you cited was not in context. a quotient 5 with remainder 1/4 was not possible on a STEP 1.
had a quotient 5 appeared ... I.E
ESTIMATING the square root of 29.
step 1 would have found R = (2925)/2Q = 4/10 = 2/5
(5 + 2/5)^2 = 29 + 4/25
STEP 2
reduce error 4/25 by dividing by 2( 5 + 2/5)
4/25 x 5/54 = 2/27
hence a final unit fraction series would replace
(5 + 2/5  2/27)^2 by considering
(5 + 1/5 + (2710)/135)^2 = (5 + 1/5 + 1/9 + 2/135)^2
was accurate to (2/27)^2
NOTE: the conversion of 2/135 to a unit fraction series follows Ahmes 2/n table rules
(2/5)(1/27) = (1/3 + 1/15)(1/27) = 1/51 + 1/405
MEANT THE SQUARE ROOT OF 29 WAS ESTIMATED IN TWO STEPS BY
(5 + 1/5 + 1/9 + 1/51 + 1/405)^2



