Finding square roots

Date: 6/17/96 at 14:42:9
From: Lori Brennan
Subject: Finding square roots

Jessica Baker wants to know how to find a square root of a number.  
She's a fifth grader.

Date: 6/19/96 at 11:8:11
From: Doctor Dave
Subject: Re: Finding square roots

Dear Jessica,

The best way to find the square root is to use a calculator, but I 
assume this is not allowed. So, you begin with a guess.

Say you want to find the square root of 24. 

This means what number can be multiplied by itself to give 24? 

Try 5: 5x5 = 25. This is too large, so try 4.  4x4 = 16, which is
too small, so you know that the answer is somewhere between 4 and 5.

Since 25 is closer to 24 than 16, your number is probably closer
to 5. You can try 4.5 and 4.7. When you calculate 4.9x4.9 you get 24.01,
which is very close. If you want to get closer, try 4.91 or 4.89.

You keep going like this until you get as close as you want. 
I'm sorry we don't have a simpler way but don't worry, you may 
get to use a calculator in the sixth grade.  Good luck!

-Doctor Dave,  The Math Forum
 Check out our web site!

Date: 6/19/96 at 11:6:59
From: Doctor Jerry
Subject: Re: Finding square roots

Humans have known how to find square roots by the divide and average 
method for nearly 4,000 years!  Here's how to do it.  To find the 
square root of 10, for example, first make a guess.  We'll guess 3.  
Then divide 10 by the guess. This gives 10/3. Average 10/3 and the 
guess.  This means add 3 and 10/3 and then divide by 2.  This gives 
19/6, which is about 3.167. This is already quite close to the square 
root of 10, which, according to my calculator, is 3.162277660...

If you want to improve the accuracy, just repeat this process.  The 
number 19/6 is the new guess.  Divide 10 by 19/6 and then average the 
quotient with guess.  

Each divide-and-average will double the number of correct decimal 
places. 

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/