105 trillion digits of pi
The number π (3.1415926…) has now been computed to 105 trillion decimal places! The computational background behind this feat is discussed by Jordan Ranous in yesterday’s article 105 Trillion Pi Digits: The Journey to a New Pi Calculation Record. The calculation, which took 75 days, was achieved using the Chudnovsky algorithm, which is based on formulas for π discovered by Srinivasa Ramanujan (1887–1920).
In his 1914 paper Modular equations and approximations to π, Ramanujan listed a large number of formulae for the number 1/π, including the one shown below. The infinite sum converges very quickly, and the first term alone (i.e., k=0) shows that π is approximately equal to 9801/(1103×2√2), or 3.14159273. This is already accurate to six decimal places.
In 2002, Takeshi Sato discovered a way to generalize Ramanujan’s formula for π. These more general formulae for π are now known as Ramanujan–Sato series. One particularly useful Ramanujan–Sato series is the one below, which was discovered by the Chudnovsky brothers (David and Gregory Chudnovsky) in 1988. Just the first term of the Chudnovsky formula computes π to 13 decimal places. Almost all of the record breaking computations of the digits of π since 2009 have been based on the Chudnovsky algorithm.
An issue that needs to be dealt with when computing large numbers of digits of π is proving that the answer is correct. One way to do this might be to perform the computation twice using different algorithms, but this is very computationally expensive. A much better method of checking digits is Bellard’s formula (shown below) which was developed by Fabrice Bellard in 1997. Bellard’s formula is a faster version of the earlier BBP formula discovered by David H. Bailey, Peter Borwein, and Simon Plouffe in 1995.
What is remarkable about Bellard’s formula and the BBP formula is that they give a way to compute the nth digit of π without first computing all the earlier digits. This was a surprising discovery, since the problem of computing the nth digit of π does not sound as if it would be significantly easier than computing the first n digits. The original versions of Bellard’s formula and the BBP formula compute the nth digit of π in hexadecimal (base 16), but Simon Plouffe developed a version of the formula in 2022 that works in base 10. Recent state of the art calculations of the digits of π have taken about 100 days to compute, but only about 20 hours to check using BBP-type algorithms.
In case you were wondering, the 105,000,000,000,000th digit of π after the decimal place is a 6.
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Picture credits and relevant links
The first and last graphics come from the Wikipedia page on the chronology of the computation of π.
The photograph of Ramanujan is his 1913 passport photo. It appears on Wikipedia.
Ramanujan’s formula appears on the Wikipedia page on Ramanujan–Sato series.
The Chudnovsky brothers’ formula for π appears on the Wikipedia page on the Chudnovsky algorithm.
Bellard’s formula also appears on Wikipedia.
My post on the number 163 gives a little more insight into the miracle that makes the Chudnovsky formula work so well.
Call me irrational, but I don’t think “pi day” is a real thing. In most countries, March 14 is “14.3”, not “3.14”.
Substack management by Buzz & Hum.
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