How Pi is CalculatedDate: 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/ |
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