Date: Dec 3, 2012 9:33 PM
Author: Milo Gardner
Subject: Re: Egyptian and Greek sqare root

2. Based on Parker's Coptic reporting of the approximate square root of 200 as (14 + 1/7) it s clear that a 'quick and dirty' method was available to construction workers and student scribes. The easy method inspected the quotient 14, doubled it, and multiplied the inverse 1/28 by 4 (200 - 196) = 4/28 = 1/7. The easy method considered (200 - Q^2), with Q = 14, hence, the 'quick and dirty' method estimated the square root of 200 as:

(14 + 1/7)^2 = 196 + 14/7 + 14/7 + 1/49 = 200 + 1/49

accurate to 1/49.

Modern vulgar fraction proof:

(14 + 1/7)^2 = (98/7 + 1/7)^2 = (99/7)^2 = 9801/49 = (9800 + 1)/49 = 200 + 1/49

C. GREEK ERA SQUARE ROOT: The Hibeh Papayrus data offers examples of Greek numeration based on Egyptian numeration as Gillings, following the work of R.C. Archibald reported unified rational number subjects connected to a 200 BCE text discussed in MATH IN THE TIME OF THE PHARAOHS on PAGES 214-217

Square root of 164 denoted as [164]^1/2 estimated [12 + 2/3 + 1/15 + 1/26 + 1/32]

without citing alternative precise historical methods, was this estimate the best Egyptian and Greek square root method? That is, was an accuracy of (1/160)^2 the best that was available?

1. An advanced square root method follows Occam's razor says:

a. "Quick and dirty" Method

(12 + R)^2 considered 1/24 and(164 - 144) = 20 meant an E(QD) = 20/24 = 5/6

an error that the scribe reduced by considering

b. 164^1/2 was nearly (12 + 4/5)^2 stated in terms of (Q + R)^2, quotient (Q) and remainder (R) form, or,

(1) 164^ 2 = (144 + 48/5 + 48/5 + 16/25) = (163 + 21/25), an error as high as 4/25

(2) to reduce the lesser 4/25 error,

(a) multiplied 1/28 (4/25) 1/7(1/25) = 1/175

(b) the scribe converted 4/5 to a 24/24 scaled unit fraction series:

(4/5)(24/24) = 96/120 = (80 + 8 + 5 + 3)/120 = 2/3 + 1/15 + 1/24 + 1/40

the last term 1/40 was increased by at least 1/175.

Actually 1/160 was selected by considering

(1/24 - 1/40) = 16/(24)(40) = 1/80, not enough

(1/30 - 1/40) = 1/120, not enough

(1/32 - 1/40) = 1/160, about right , hence replace 1/40 with 1/32

Pretty neat, to my eyes.

Any comments?