Tag Archives: pi

Pi and the AGM

In celebration of Pi Day 2024, I would like to explain how the “Arithmetic-Geometric Mean” of Gauss and Legendre can be used to give a rapid method for computing the digits of \pi. By “rapid” here, I mean that the algorithm exhibits quadratic convergence: the number of correct digits roughly doubles with each iteration. I will mainly follow the exposition in Donald J. Newman’s 1985 paper “A Simplified Version of the Fast Algorithms of Brent and Salamin”.

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A motivated and simple proof that pi is irrational

Liana Tang pieToday is 3/14/15 — Super Pi Day — so was I telling my 7-year-old son all about the number \pi this afternoon.  When I told him that \pi keeps on going forever and ever he asked “How do you know that?”  Although I don’t know a proof that I could explain to a 7-year-old, I wanted to record the following proof which uses only basic calculus.  It is essentially Niven’s famous proof, as generalized by Alan Parks, but I have tried to write it in a way which is more motivated than the usual treatments.  As a bonus, the proof also shows that e is irrational.

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