My Math Forum  

Go Back   My Math Forum > Math Forums > Math

Math General Math Forum - For general math related discussion and news

LinkBack Thread Tools Display Modes
April 12th, 2016, 07:54 PM   #1
Joined: Apr 2016
From: uk

Posts: 3
Thanks: 0

Archimedes’ Approximation to SQRT3

Archimedes states that 1351/780 > SQRT3 > 265/153, but doesn’t say how he derived this.

It struck me that Archimedes may have used geometry to get there, and I came up with a possible solution at Archimedes and the Square Root of Three, A Geometrical Approach.

This suggests a general procedure for refining square roots, but I’m puzzled about how I might derive the starting approximation SQRT3=26/15 without knowing the general procedure. Because if I know the general procedure, I can eventually work out, by number crunching, that 1351/780 > SQRT3 > 265/153 simply by starting from SQRT3=7/4, derived from the Babylonian method of approximating square roots.

Please, does anyone know of an ancient method of deriving SQRT3 that generates 26/15 or 45/26?
gjb is offline  
April 13th, 2016, 03:48 AM   #2
Global Moderator
Joined: Dec 2006

Posts: 20,968
Thanks: 2217

I don't know how Archimedes did it. Below is a more recent method, but possibly still fairly old.

Consider the sequences (7, 26, 97, 362, 1351, ...) and (4, 15, 56, 209, 780, ...). To generate further terms, multiply the last term by 4, then subtract the previous term.

The ratios of corresponding terms approximate √3, and they include 26/15 and 1351/780.
Note that 45/26 = 97/26 - 2.
skipjack is offline  

  My Math Forum > Math Forums > Math

approximation, archimedes’, sqrt3

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Best approximation chenfire Real Analysis 2 January 13th, 2015 02:41 PM
Approximation kyry Calculus 2 January 31st, 2014 09:47 PM
approximation for the sum Anton29 Applied Math 5 December 28th, 2011 01:11 PM
I approximation aaron-math Calculus 1 October 3rd, 2011 12:34 PM
Pi Approximation Marcel777 Number Theory 5 September 27th, 2010 11:49 PM

Copyright © 2019 My Math Forum. All rights reserved.