Square roots by hand: Divide and AverageDate: 8/31/96 at 8:10:55 From: Janet Rodgers Subject: Calculating square roots Dear Dr. Math, My 7th grade son has been trying to find out how to calculate square roots by hand. I remember learning to do this in the old days before calculators, but I can't find the technique in any of my books. Can you find out how to do this? Thanks! Date: 9/1/96 at 0:29:0 From: Doctor Jerry Subject: Re: Calculating square roots There are several methods, but the most ancient (dating back to the Babylonians, ca. 1700 B.C.) and best is the divide-and-average method. Here's the method. Let A be the number whose square root is wanted. Determine in some way a first guess g of the square root of A. The remaining part of the method is to repeat the following step: Divide A by the current guess g and then average the quotient with g. The average is the new guess. Repeat until you have calculated the square root to desired accuracy. Roughly, the number of decimal places doubles with each repetition. Suppose we want to calculate the square root of 2. Take the first guess to be 1.5 (or 1 or ...). I'm going to copy the results directly from my calculator screen. In a "production line" calculation the intermediate results need not be written down - just keep them in the calculator. 1. 2/1.5 = 1.33333333333 (1.5+1.33333333333)/2 = 1.41666666666 2. 2/1.41666666666 = 1.41176470589 (1.41666666666+1.41176470589)/2 = 1.41421568628 Since the square root of 2 is (to calculator accuracy) 1.41421356237, the algorithm is doing rather well. One more step gives the result to 11 significant figures! -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 9/1/96 at 0:32:23 From: Doctor Ken Subject: Re: Calculating square roots Hello! There are a couple of other methods to calculate square roots that you can find in our archives, or look under "Square roots..." in our Dr. Math FAQ at http://mathforum.org/dr.math/faq/ -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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