Approximations of π

Approximations for the mathematical constant pi (pi) in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. 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 (through the efforts of 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 pi is held by William Shanks, who calculated 527 digits correctly in 1853. Since the middle of the 20th century, the approximation of pi has been the task of electronic digital computers (for a comprehensive account, see Chronology of computation of pi). On 8 June 2022, the current record was established by Emma Haruka Iwao with Alexander Yee's y-cruncher with 100 trillion (e14) digits. The best known approximations to pi 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 Egyptologists have claimed that the ancient Egyptians used an approximation of pi as = 3.142857 (about 0.04% too high) from as early as the Old Kingdom. This claim has been met with skepticism. Babylonian mathematics usually approximated pi 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 pi as = 3.125, about 0.528% below the exact value.