Pi is a Constant

Date: 22 Apr 1995 06:48:45 -0400
From: Karsten Koops
Subject: Pi depends on circle-diameter?

Hello!
A couple of weeks ago I got the first 21500 digits of PI by
Infobot@Infomania.com.
As an example for internet-capablilities I showed it to some friends.
One of them told me, the exact result of PI depends on the extent of the
circle diameter. This can't possibly be right, can it?

I'm looking forward to become enlightened. 
Bye, Karsten -
Innsbruck/Tyrol-Austria.
koops@magnet.at

Date: 22 Apr 1995 10:52:27 -0400
From: Dr. Ken
Subject: Re: Pi depends on circle-diameter?

Hello there!

Yes, you're right, Pi does not depend on the diameter of the circle.  What
they probably meant was that if you only have three decimal places in the
diameter of your circle, you can't hope to calculate Pi to any more than 3
decimal places.  Something like that, at least.  But rest assured, Pi is a
_constant_, and it always will be.

-K

Date: 22 Apr 1995 16:53:15 -0400
From: Karsten Koops
Subject: Pi-algorithm

Hello!
Thank you very much for your prompt reply.  Could you additionally tell me 
where I can get the algorithm - how PI is calculated?  Are there different 
ways of calculating PI?

Bye Koops@magnet.at

From: Dr. Ken
Subject: Re: Pi-algorithm
Date: 22 Apr 1995 23:35:43 -0400

Hello!

Yes, as a matter of fact there are a whole bunch of ways of calculating Pi.
If you want efficient ones, I'd recommend writing to wherever you found all
those digits; they may be able to tell you how they got them.  Here are a
couple you may find interesting, though:

 2    
---- = Cos(90/2) Cos(90/4) Cos(90/8)...
 Pi   

     =Sqrt{.5} * Sqrt{.5 + .5 Sqrt{.5}} * Sqrt{.5 + .5 Sqrt{.5 + Sqrt{.5}}}...


 4     3x3x4x4x5x5x6x6x7x7x8x8x9x...
---- = -----------------------------
 Pi    4x4x5x5x6x6x7x7x8x8x9x9x10x...


             1
 4      1 + ----------------
---- =            9
 Pi          2 + -----------------
                       25
                  2 + -------------------
                            49
                       2 + -------------------
                                 81
                            2 + --------------------
                               .
                                .
                                 .

With any of these, you can just chop off the expansion at any point when
you're tired of calculating.  These aren't all that efficient compared with
some other ways of getting the digits of Pi, but they're neat.  Oh, I almost
forgot one:

Pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - .....

Enjoy!

-K