Needless to say, as any professional mathematician or, for that matter, any person who has at least completed a course in calculus will attest, such claims cannot possibly be correct. For other examples, see this Math Scholar blog. For example, one author asserts that pi = 17 – 8 * sqrt(3) = 3.1435935394… Another author asserts that pi = (14 – sqrt(2)) / 4 = 3.1464466094… A third author promises to reveal an “exact” value of pi, differing significantly from the accepted value. Thus it is with great sadness and consternation to see a growing number of writers (mostly lacking advanced mathematical training) who reject basic mathematical theory and the accepted value of pi, claiming instead that they have found pi to be some other value. The subject of pi is dear to the present author, as he has published several papers and a book on this very subject. The most recent record of pi, as of the present date, was to 31 trillion digits (actually 31,415,926,535,897 digits, to be exact, which happens to be the first 14 digits of pi), by a researcher at Google. For example, in 1965 researchers found that the “fast Fourier transform” (FFT), a computational technique widely used in signal analysis and many other scientific fields, can be used to greatly accelerate high-precision multiplication. Even more important was the discovery of some very clever computer algorithms applicable to computing pi. For example, this paper presents a formula that permits one to calculate digits (in a binary or hexadecimal base) starting at an arbitrary starting position, without needing to calculate any of the digits that came before. Facilitating these calculations are some remarkable new formulas for pi. In the 20th century, with the advent of the computer, pi was computed first to thousands, then to millions, then to billions, and, most recently, to trillions of digits.
This culminated in 1874 with Shanks’ hand calculation of pi to 707 digits (alas, only the first 527 were correct). Beginning in the 17th century, in the wake of the discovery of calculus by Newton and Leibniz, numerous new formulas by pi were found, some permitting even more digits to be computed. In the centuries following Archimedes, ancient mathematicians worldwide used this approach to calculate pi to increasing accuracy. Archimedes (c.287–212 BCE) was the first to present a scheme for calculating pi as a limit of perimeters of inscribed and circumscribed polygons, as illustrated briefly in the graphic to the right (see this Math Scholar blog for details). Pi = 3.1415926535…, namely the ratio between the circumference of a circle and its diameter, has fascinated not only mathematicians and scientists but the public at large for centuries. Credit: Michele Vallisneri, NASA JPL Computing pi
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