History of the formulas and algorithms for pi

TL;DR

This paper reviews the historical development of formulas and algorithms for calculating pi, highlighting mathematical innovations and computational advancements over two millennia.

Contribution

It provides a comprehensive overview of the evolution of pi calculation methods, including series, products, continued fractions, and radical expansions, emphasizing computational progress.

Findings

Various formulas have been used to compute pi over history

Advances in computer technology enabled record-breaking digit calculations

Surprising and rapidly converging series have been crucial for high-precision calculations

Abstract

Throughout more than two millennia many formulas have been obtained, some of them beautiful, to calculate the number pi. Among them, we can find series, infinite products, expansions as continued fractions and expansions using radicals. Some expressions which are (amazingly) related to pi have been evaluated. In addition, a continual battle has been waged just to break the records computing digits of this number; records have been set using rapidly converging series, ultra fast algorithms and really surprising ones, calculating isolated digits. The development of powerful computers has played a fundamental role in these achievements of calculus.