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Simple numerical methods for calculating the digits of Pi

How do people/computers calculate π?

Im sure long ago, someone just took a measurement of the circumference of circles and their radii and came up with 3. Then as measurements got more accurate 3.14... etc...

But how do modern people, especially computers calculate π to trillions of digits?

If π is an irrational number, then how do we know the calculations are correct?

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You should google things. –  Karolis Juodelė Sep 15 '12 at 18:01
    
    
A nice article of Jonathan Borwein "The Life of Pi: From Archimedes to Eniac and Beyond" –  Raymond Manzoni Sep 15 '12 at 18:39
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marked as duplicate by Argon, Jyrki Lahtonen, Jennifer Dylan, Alex Becker, Sasha Sep 15 '12 at 18:59

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

2 Answers

If you really want the answer then this paper might be helpful

http://www.pnas.org/content/86/21/8178

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Archimedes considered a 96-sided regular polygon inscribed in a circle and another circumscribed about the circle, and found their perimeters, and thereby showed that $3+\dfrac{10}{71} < \pi < 3+ \dfrac 1 7$.

More recently (the past two or three centuries) algorithms derived from the power series for the arctangent function have been used.

In Wikipedia's List of topics related to $\pi$ I find these:

Wikipedia's List of topics related to pi --- actually a list of Wikipedia articles on those topics --- is quite interesting:

  • 2π theorem
  • Approximations of π
  • Arithmetic-geometric mean
  • Bailey–Borwein–Plouffe formula
  • Basel problem
  • Borwein's algorithm
  • Buffon's needle
  • Cadaeic Cadenza
  • Chronology of computation of π
  • Circle
  • Euler's identity
  • Feynman point
  • Gauss–Legendre algorithm
  • Gaussian function
  • History of π
  • A History of Pi (book)
  • Indiana Pi Bill
  • Leibniz formula for pi
  • Lindemann–Weierstrass theorem (Proof that π is transcendental)
  • List of circle topics
  • List of formulae involving π
  • Liu Hui's π algorithm
  • Mathematical constant (sorted by continued fraction representation)
  • Method of exhaustion
  • Milü
  • Pi
  • Pi (letter)
  • Pi Day
  • PiFast
  • PiHex
  • Pilish
  • Pimania (computer game)
  • Piphilology
  • Proof that π is irrational
  • Proof that 22/7 exceeds π
  • Proof of Wallis product
  • Rabbi Nehemiah
  • Radian
  • Rhind Mathematical Papyrus
  • Salamin–Brent algorithm
  • Software for calculating π
  • Squaring the circle
  • Tau (2π)
  • Turn (geometry)
  • Viète's formula
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