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How Pi is Calculated


Date: 08/07/97 at 07:42:58
From: Krono
Subject: How pi is calculated

If pi is calculated by the circumference of a circle divided by its 
radius, how can computers generate pi to billions of decimal places? 
Do they draw really accurate circles or what? Is there a 
straightforward function (eg. log, polynomial, trigfunction, 
exponential) to derive pi to, say, 20 places?   


Date: 08/07/97 at 08:33:10
From: Doctor Wilkinson
Subject: Re: How pi is calculated

No, nobody calculates pi by measuring circles. 

There are thousands of formulas, some simple, some very sophisticated, 
that can be used. The simplest of these, which can be derived using 
elementary calculus, comes from the series for the arctan function, 
namely

    arctan(x) = x - x^3/3 + x^5/5 - x^7/7 +-...

from which (by letting x = 1) you get

    pi/4 = 1 - 1/3 + 1/5 - 1/7 +...

Unfortunately, this series is useless for computing pi with any degree 
of accuracy. You would need billions of terms just to get your 20 
decimal places. But similar formulas involving the arctan can also be 
derived which converge much more rapidly. The best known of these is 
called Machin's formula, which says

    pi/4 = 4 * arctan(1/5) - arctan(1/239)

If you use the series above for these arctans, you can get quite good
approximations to pi fairly easily.

Much more powerful methods have been discovered, and pi has been 
calculated to several billion decimal places. For more information, 
see  http://www.cecm.sfu.ca/pi/

And for more about pi and links to other sites on the Web, see the Dr. 
Math FAQ:

     http://mathforum.org/dr.math/faq/faq.pi.html   

-Doctor Wilkinson,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Transcendental Numbers

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