Approximations of π

Approximations for the mathematical constant pi (π) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era (Archimedes). In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 15th century (Jamshīd al-Kāshī). Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega), surpassing the accuracy required for any conceivable application outside of pure mathematics. The record of manual approximation of π is held by William Shanks, who calculated 527 digits correctly in the years preceding 1873. Since the middle of the 20th century, the approximation of π has been the task of electronic digital computers; as of November 2016, the record was 22.4 trillion digits. (For a comprehensive account, see Chronology of computation of π.) In March 2019 Emma Haruka Iwao, a Google employee from Japan calculated to a new world record length of 31 trillion digits with the help of the company's cloud computing service. The best known approximations to π dating to before the Common Era were accurate to two decimal places; this was improved upon in Chinese mathematics in particular by the mid-first millennium, to an accuracy of seven decimal places. After this, no further progress was made until the late medieval period. Some Egyptologistshave claimed that the ancient Egyptians used an approximation of π as ​22⁄7 from as early as the Old Kingdom.This claim has met with skepticism. Babylonian mathematics usually approximated π to 3, sufficient for the architectural projects of the time (notably also reflected in the description of Solomon's Temple in the Hebrew Bible).The Babylonians were aware that this was an approximation, and one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as ​25⁄8 = 3.125, about 0.5 percent below the exact value. At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of π as ​256⁄81 ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle by approximating the circle by an octagon. Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of 339/108 ≈ 3.139.