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Topic: Archimedes square root of 3
Replies: 8   Last Post: Feb 5, 2013 9:23 AM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
Re: Archimedes square root of 3
Posted: Jan 24, 2013 10:17 AM

Archimedes estimated pi by

22/7 > pi > 223/71

1. the method scaled 22/7 by 10/10 = 220/70

2.obtained the lower limit by 22/7 10/10 = 220/70

220/71 and added 223/71

as the root of 3 was estimated by

Archimedes' square root of 3 cited limits

(1271/780)^2 > 3 > (265/153)^2

A. The higher limit 1251/780 was calculated in three steps

1. (1 + 2/3)^2 error 1 = 2/9

2. 2/9 x (3/10) = 1/15, (1 + 2/3 + 1/15)= (1 + 11/15). error 2 = 1/225

3. 1/225 x (15/52) = 1/780, (1 + 11/15 -1/780) = 1251/780

B. the lower limit 265/153 modified step 2, used

2. 1/17 rather than 1/15, (1+ 2/3 + 1/17) = (1 + 37/51)

such that (1 + 111/153) increased to (1 + 112/153) = 265/153

Q.E.D.

Date Subject Author
1/16/13 Milo Gardner
1/16/13 Milo Gardner
1/17/13 Milo Gardner
1/20/13 Milo Gardner
1/24/13 Milo Gardner
2/4/13 Milo Gardner
2/4/13 Milo Gardner
2/4/13 Milo Gardner
2/5/13 Milo Gardner