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### Square roots by hand: Divide and Average

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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?

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/
```
Associated Topics:
High School Square & Cube Roots
Middle School Square Roots

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