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Square roots by hand: Divide and Average

Date: 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?


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 

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 

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


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


-Doctor Ken,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
Associated Topics:
High School Square & Cube Roots
Middle School Square Roots

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