Manual Method for Finding Square Roots

Date: 9/11/95 at 12:31:22
From: gilman
Subject: square roots

I'm looking for a manual method to find a square root of a number.  Many years
ago, we were taught how to do it, but my memory is failing me, and with the
calculator it's too easy.  I believe I would have learnt this subject in 
school year 5 or 6.  many thanks to the person(s) that can help me.

Date: 9/13/95 at 16:46:57
From: Doctor Ethan
Subject: Re: square roots

Well,

   I know of a way but I think you may not mind the calculator after you use
it for a while.  It is based on an algorithm called Newton's method, but you 
don't have to understand that to be able to do the method.

My idea is to first make a good guess, then see how close you are, and then
make a better guess.

We'll do it with 6, which according to my computer is

2.4494897427831780982

We will see how close we can come.

We start by guessing.  2 is a pretty good guess.  So we check

2^2 is 4  - too low  so we guess 3  
3^2 is 9  - too high  

So now we need to refine our guess.  

The number we are looking for is 6.   4 is 2 lower than six and 9 is
3 higher than six.

So ( here is the important part)  to make our next guess we take the total
difference in the results of our first two guesses (9-4 = 5) and then we 
find how much the desired answer is greater than the smaller guess (6-4 = 2).

Then we add the quotient to our guess, so our next guess will be 2 and 2/5
or 2.4  

2.4 ^2 is 5.76  - too low,  so try 2.5 ^2 = 6.25   

Now we repeat  
6.25 - 5.76 = .49
6 - 5.76 = .24

.24/.49 = .44898  the level of precision here is actually quite small,
so you should always just use the tenths place, which is .4 - so our next 
guess would be 2.44

Notice that I did not add .4 to get our next guess; I merely added the digit 
four to the end of the number that we are using as our guess.

This method should lead to greater and greater refinement.
As you can see, we already have the first three digits right.

I will do one more example to let you see how this would work for a large
number.  Let us consider 700.

My initial guess will be 20
20^2 = 400  - too low so I go to 30
30^2 = 900  - too high, so we go to the method

900 - 400 = 500
700 - 400 = 300

So 300/500 is .6  and our next guess will be 26. Do you see why?

26^2 = 676   - too low, so we try 27
27 ^ 2 = 729

so 729 - 676 = 53
and 700 - 676 = 24

and 24/53 = .4538...  but all we need is the tenths place, which is .4

So our next guess is 26.4
26.4^2 = 696.96  - too low, so try  26.5
26.5 ^ 2  = 702.5  

702.5 - 696.96 = 5.54
700 - 696.96 = 3.04

3.04/5.54 = .5487

But again we will just take the 5 and add it to the end.
This makes our next guess 26.45.

and 26.45^2 = 699.603.

You can see that we are very close now and that we could get as close as 
we wanted to with this method. My computer gives this answer.
26.457513110645905905

so we didn't do too bad, huh?

Hope this helps.

- Doctor Ethan,  The Geometry Forum