July 5th, 2013, 08:37 AM  #1 
Newbie Joined: Jun 2013 Posts: 19 Thanks: 0  Algorithm for PI
I hope this is the right place to post this. I'm just wondering why there are so many algorithm for the calculation of PI. If PI is simply the ratio of the circumference of a circle to its diameter, why are there so many algorithm to compute it? How do we know that the algorithm is appropriate to ensure the accuracy of the calculation for PI? Does a perfect circle even exist to verify the accuracy of the calculations? And also, if PI is irrational, how can the ratio be so FIXED? Thanks guys. 
July 5th, 2013, 08:52 AM  #2  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Algorithm for PI Quote:
Probably there are lots of algorithms to compute lots of constants, and we've found many for pi because we're looking for them. There are surely many algorithms for computing constants we don't care so much about but haven't discovered for just that reason. Quote:
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Almost all numbers are irrational, so this quality doesn't make pi unusual in any way.  
July 6th, 2013, 06:26 AM  #3  
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Algorithm for PI Quote:
That said, you can research, for example how to estimate Pi by using Polygons of increasing number of sides and, by using some basic trigonometry, you can come up with two algorithms: n*cos(90180/n)/2 n/2*tan(90180/n) Where, as n increases, these numbers get closer to Pi. Other algortithms tend to come from the fact that Pi crops up in other parts of mathematics.  
July 7th, 2013, 05:59 AM  #4 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Algorithm for PI
... knock the 2's out of those equations! I was also thinking that if you drew a perfect circle on the surface of the Earth and measured the (curved) diameter along the Earth's surface, then you wouldn't get a ratio of C = Pi * D. In the extreme case where the circle was a great circle, then the diameter would be half a great circle, so that C = 2 * D! 
July 9th, 2013, 08:39 AM  #5 
Newbie Joined: Jun 2013 Posts: 19 Thanks: 0  Re: Algorithm for PI
Hmmm, ok, maybe I should rephrase my question. Are most algorithms derived from the fact that PI is the ratio of the circumference of a circle to its diameter or is it derived from knowing what the first few digits of PI are and then generate the rest of the digits using some algorithm that can successfully generate the first few known digits? I'm asking this because as far as I know different algorithms can have different accuracy. If so, how then do we know which algorithm is absolutely accurate? How do we judge the accuracy of the algorithm? Are the trillions of digits generated in world records absolutely correct? How can we know? And on a ruler, is the value of PI exact? Or is it moving because we don't know the exact value of PI and with increasing number of digits we get more and more precise? 
July 9th, 2013, 08:53 AM  #6  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Algorithm for PI Quote:
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I did see the verification technique for one of the worldrecord runs and it was pretty awesome. As far as I know most don't go through that much trouble. Quote:
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July 9th, 2013, 09:44 AM  #7  
Newbie Joined: Jun 2013 Posts: 19 Thanks: 0  Re: Algorithm for PI Quote:
 
July 9th, 2013, 09:49 AM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Algorithm for PI Quote:
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July 9th, 2013, 09:54 AM  #9  
Newbie Joined: Jun 2013 Posts: 19 Thanks: 0  Re: Algorithm for PI Quote:
 
July 9th, 2013, 09:58 AM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Algorithm for PI Quote:
 

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