Method to Find Square Root of Pi

Date: 03/30/97 at 18:36:36
From: Jeff Houk
Subject: Method to Find Square Root of Pi

What is the square root of Pi? I tried using a calculator, but 
that didn't work. I tried trial and error and Guess and Check, 
but neither of those helped either.  HELP!

Date: 04/01/97 at 17:23:56
From: Doctor Steven
Subject: Re: Method to Find Square Root of Pi

There is a method called the binary search method that will find the 
square root of any number N.  You'll need a calculator to do this 
method.

The first step is to pick a value you know to be too small the be the
square root of Pi.  Call this value S (for small guess).  The next
step is to pick a value you know to be too large to be the square root
of Pi.  Call this value L (for large value).

Then since Pi = 3.14159.... we'll say S = 1, and L = 2.

S = 1
L = 2

Then find S^2 and L^2 (S and L squared):

S^2 = 1
L^2 = 4

Which of these values is farther away from Pi? S^2 is farther away 
from Pi then L^2, so replace S with the average of S and L.  Now we 
have:

S = 1.5
L = 2

Continue the process:

S^2 = 2.25
L^2 = 4

Replace the small value again since its square is farther away.

S = 1.75
L = 2

Goes to:

S^2 = 3.0625
L^2 = 4

Replace the large value now:

S = 1.75
L = 1.875

Goes to:

S^2 = 3.0625
L^2 = 3.515625

Replace the large value again:

S = 1.75
L = 1.8125

Goes to:

S^2 = 3.0625
L^2 = 3.28515625

At which point we know to replace the large value again since its 
square is farther away then the small value's square.

So

S = 1.75
L = 1.78125

At this point we know the square root of Pi lies between 1.75 and 
1.78125. Continuing this process we can get finer and finer bounds for 
the value of the sqaure root of Pi.  After a while we will see that 
the square root of Pi equals 1.77245....  Since Pi is irrational we 
can never get an exact answer and we have to settle for an 
approximation.

Hope this helps.

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