Approximations of pi

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. 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 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. 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).

The record of manual approximation of π is held by William Shanks, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of π has been the task of electronic digital computers (for a comprehensive account, see chronology of computation of pi). On December 11, 2025, the current record was established by StorageReview with Alexander Yee's y-cruncher with 314 trillion (3.14×1014) digits.

Early history

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.