We often see the value of pi(irrational) with large no of digits behind the decimal place. How is such precise value of pi calculated?
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You have rapidly converging series, which can be used. – Raphael J.F. Berger Aug 09 '16 at 17:36
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$\pi=\frac41-\frac43+\frac45-\frac47+\frac49\dots$ – barak manos Aug 09 '16 at 17:37
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1I'm pretty sure this question has been asked several times. Here is one such example where you may find some of the answers helpful. – Eff Aug 09 '16 at 17:38
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@barak manos; How is this series developed? – Lamichhane88 Aug 09 '16 at 17:46
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@Lamichhane88: Taylor series of $\arctan$? – Raphael J.F. Berger Aug 09 '16 at 17:54
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@barakmanos: That series is definitely not how large numbers of digits of $\pi$ are calculated. It converges much too slowly. – Nate Eldredge Aug 09 '16 at 18:45
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@NateEldredge: That series is definitely how large numbers of digits of $\pi$ can be calculated. The question doesn't mention anything about efficiency. – barak manos Aug 09 '16 at 18:48
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@barakmanos: I interpreted the word "is" in the title to ask about practical methods, i.e. those that actually have been used to calculate large numbers of digits. – Nate Eldredge Aug 09 '16 at 18:49
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@NateEldredge: I interpreted it as if OP could not figure out how the decimal digits of $\pi$ possibly be calculated. I would tend to guess that OP would just as well refer to any other irrational number for that matter, and just so happen had $\pi$ in mind. But that's more of an interpolation on OP ratings, plus the fact that the answer to this question could easily be found here and elsewhere... So I suppose that only OP could provide the actual meaning behind this question... – barak manos Aug 09 '16 at 18:53
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If you want an easy way to calculate $\pi$ you can divide the circumference of a circle by its diameter, but such a method involves taking precise and accurate measurements and there is always some error to be accounted for. Wolfram Mathworld has listed methods and formulas to calculate $\pi$, but if you want 5 user friendly ways, here they are.
Note: The precision to which pi can be calculated and the time taken to do so approximately depends upon the computers and algorithms used and how they have been optimised.
Faraaz Ahmad
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If its only about easy (which it isn't): A Monte-Carlo method (extremely slow conv.) is to choose points with random $x,y$ coordinates between -1 and 1. And then counting those for which $\sqrt{x^2 + y^2} \le 1$ and relating the count to the total number of points (that gives you $\pi/4$). – Raphael J.F. Berger Aug 09 '16 at 17:51
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