Approximations for the mathematical constant 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 is held by William Shanks, who calculated 527 digits correctly in 1853. Since the middle of the 20th century, the approximation of has been the task of electronic digital computers (for a comprehensive account, see Chronology of computation of ). On 8 June 2022, the current record was established by Emma Haruka Iwao with Alexander Yee's y-cruncher with 100 trillion () digits.
Early history
The best known approximations to 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 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 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 as = 3.125, about 0.528% 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 as ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon.
Astronomical calculations in the Shatapatha Brahmana (c. 6th century BCE) use a fractional approximation of .
The Mahabharata (500 BCE – 300 CE) offers an approximation of 3, in the ratios offered in Bhishma Parva verses: 6.12.40–45.
In the 3rd century BCE, Archimedes proved the sharp inequalities < < , by means of regular 96-gons (accuracies of 2·10−4 and 4·10−4, respectively).
In the 2nd century CE, Ptolemy used the value , the first known approximation accurate to three decimal places (accuracy 2·10−5). It is equal to which is accurate to two sexagesimal digits.
The Chinese mathematician Liu Hui in 263 CE computed to between and by inscribing a 96-gon and 192-gon; the average of these two values is (accuracy 9·10−5). He also suggested that 3.14 was a good enough approximation for practical purposes. He has also frequently been credited with a later and more accurate result, π ≈ = 3.1416 (accuracy 2·10−6), although some scholars instead believe that this is due to the later (5th-century) Chinese mathematician Zu Chongzhi. Zu Chongzhi is known to have computed to be between 3.1415926 and 3.1415927, which was correct to seven decimal places. He also gave two other approximations of : π ≈ and π ≈ , which are not as accurate as his decimal result. The latter fraction is the best possible rational approximation of using fewer than five decimal digits in the numerator and denominator. Zu Chongzhi's results surpass the accuracy reached in Hellenistic mathematics, and would remain without improvement for close to a millennium.
In Gupta-era India (6th century), mathematician Aryabhata, in his astronomical treatise Āryabhaṭīya stated:
Approximating to four decimal places: π ≈ = 3.1416, Aryabhata stated that his result "approximately" ( "approaching") gave the circumference of a circle. His 15th-century commentator Nilakantha Somayaji (Kerala school of astronomy and mathematics) has argued that the word means not only that this is an approximation, but that the value is incommensurable (irrational).