Square Roots: Guess, Divide, Average

Date: 6/23/96 at 12:10:58
From: Anonymous
Subject: Square Roots

What is the formula for square roots?

Date: 6/24/96 at 10:45:59
From: Doctor Ethan
Subject: Re: Square Roots

I am not sure what you mean by the "formula for square roots."
I am going to assume that you want a formula for finding the square 
root of a number and I am afraid that no such formula exists. There 
are methods you can use to approximate the square root. I will teach 
you my favorite.

Remember, the square root of a number is just the number which when 
multiplied by itself gives the first number.

So 2 is the square root of 4 because 2 * 2 = 4.

Now here is the method:

Start with the number you want to find the square root of.
Let's use 12. There are three steps:

1. Guess
2. Divide
3. Average.

Then you just keep repeating steps 2 and 3.

So let's do it.

First, start by guessing a square root value. It helps if your guess 
is a good one but it will work even if it is a terrible guess. Just to 
show you that I will guess that 2 is the square root of 12.

Now in step two, I divide 12 by my guess of 2 and I get 6.

Now in step three, I average 6 and 2: (6+2)/2 = 4

Now I repeat step two with the new guess of 4. So 12/4 = 3

Now average 4 and 3: (4+3)/2 = 3.5  

Now repeat step two: 12/3.5 = 3.43

Now average:   (3.5 + 3.43)/2 = 3.465

We could keep going forever, getting a better and better approximation
but let's stop here to see how we are doing.

3.465 * 3.465 = 12.006225

That is pretty close to 12, so we are doing pretty well.

Hope that this helps you.

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