# How to calculate π [duplicate]

Possible Duplicate:
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
–  Argon Sep 15 '12 at 18:16
A nice article of Jonathan Borwein "The Life of Pi: From Archimedes to Eniac and Beyond" –  Raymond Manzoni Sep 15 '12 at 18:39

## marked as duplicate by Argon, Jyrki Lahtonen, Jennifer Dylan, Alex Becker♦, SashaSep 15 '12 at 18:59

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
• 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