Computing Square Roots Manually

Date: 03/05/98 at 23:23:36
From: Jerome Ross
Subject: Square Roots

Hello, my name is Jerome Ross and I am a student at the University of 
Phoenix. I am currently taking a college algebra class. I was given an 
assignment to learn how to compute square roots manually. If you could 
help me, I would greatly appreciate it.

Thank You,

Jerome Ross  

Date: 03/08/98 at 14:38:50
From: Doctor Sam
Subject: Re: Square Roots

Jerome,

There is an old technique that is similar to long division, but I 
can't explain it through this medium. You might check any pre-1950 
algebra textbook.  

But there is another method, the bisection method, that I think I can 
explain here. To get the idea of this method, consider the problem of 
factoring a number like 12

   12 = 1 x 12 = 2 x 6 = 3 x 4  

or a number like 64, which factors into 

   64 = 1 x 64 = 2 x 32 = 4 x 16 = 8 x 8.

In general, when a number N = AB, then when A = B, you have found the 
square root of N. When A > B, then the square root must lie between A 
and B.

Let's apply this idea to finding the square root of 500 numerically. 
Start by factoring 500 ... numbers that are more nearly equal will get 
you to the square root faster than 1 x 500, although starting there 
will get an answer eventually.

So 500 = 50 x 10. We know that 10 < sqrt(500) < 50. We also know that 
the average of 10 and 50 is between 10 and 50, as well:

   (10+50)/2 = 30.

Divide 500 by 30, resulting in 50/3, so that 500 = 30 x (50/3). Since 

   50/3 < 30

it must be true that 

   50/3 < sqrt(500) < 30.   

We started with sqrt(500) between 10 and 50 and have now narrowed the 
range to (50/3, 30).  

If we keep repeating the method, we can get sqrt(500) to any desired 
accuracy. Here are several iterations of the method:

   500 = 10 x 50         Aver.(10,50)       = 30    
                         Divide 500/30      = 50/3
 
   500 = (50/3)x 30      Aver.(50/3,30)     = 70/3  
                         Divide 500/(70/3)  = 150/7

   500 = (150/7)x(70/3)  Aver.(150/7, 70/3) = 22.381 approx.  
                   
   500 = (22.381)(22.340)

Notice that after four repetitions of the process, we have the two 
factors equal to the tenths place. Repeated applications will continue 
to halve the interval separating the factors (that's why it is called 
the bisection method).

I hope that helps.

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