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.
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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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