Today March 14 is so-called ‘pi day’ where in the month/day format in the US for the date gives 3/14. The number pi (the ratio of the circumference to the diameter of a circle) is a source of endless fascination because it is so basic and yet so ubiquitous in all areas of science and mathematics and everyday life. Here is a fascinating article by Xiaojing Ye on the search for obtaining increasingly precise values of pi. It contained many nuggets of information that were quite new to me.
Various techniques have been used to calculate pi. The earliest written approximations were obtained between 3,000 and 4,000 BC in Babylon and Egypt using trial-and-error methods and they obtained values close to 3.1. The next level of precision was reached by a technique developed by Archimedes around 250 BC based on geometrical approximations using circles with inscribed and circumscribed regular polygons to set lower and upper limits and he got a value of 3.14. Ptolemy extended this method in 150 AD to get a value of 3.1416.
A Chinese mathematician named Liu Hui in 265 AD used a polygonal algorithmic technique to also get four digits but that method was extended by Zu Chongzhi in 480 AD to get seven digits. That record held for 800 years. Austrian mathematician Christoph Grienberger in 1630 used that same technique to get 38 digits “which is the most accurate approximation manually achieved using polygonal algorithms.”
The next breakthroughs came with the development of infinite series in the 16th and 17th centuries.
In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 – the best approximation achieved without the aid of a calculator or computer.
The development of iterative algorithms for calculating pi, combined with the arrival of computers, saw an explosion in the number of digits achievable.
Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!
We are well beyond the level of accuracy necessary for purely practical purposes and the search now serves other needs.
It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.
What I also found interesting were the theorems that proved things about what pi was not.
Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number – it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (such as pi2=10 or 9pi4-240pi2+1492=0).
Update: A friend of mine informs me of the Bailey–Borwein–Plouffe formula which is an algorithm that enables you to directly calculate any given binary digit of pi without having to calculate any of the preceding digits. The catch is that the further out the digit is in the sequence, the longer it takes to calculate it.