Approximations of π

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 June 8, 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 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 $3+8/60+30/60^2,$ 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,How Aryabhata got the earth's circumference right
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).

Middle Ages

Further progress was not made for nearly a millennium, until the 14th century, when Indian mathematician and astronomer Madhava of Sangamagrama, founder of the Kerala school of astronomy and mathematics, found the Maclaurin series for arctangent, and then two infinite series for . One of them is now known as the Madhava–Leibniz series, based on $\pi=4\arctan(1):$

:$\pi=4\left(1-\frac 13+\frac 15-\frac 17 ...\right)$

The other was based on $\pi=6\arctan(1/\sqrt 3):$

: $\pi = \sqrt\sum^\infty_ \frac = \sqrt\sum^\infty_ \frac = \sqrt\left(1-+-+\cdots\right)$

He used the first 21 terms to compute an approximation of correct to 11 decimal places as .

He also improved the formula based on arctan(1) by including a correction:

:$\pi/4\approx 1-\frac 13+\frac 15-\frac 17+ ... -\frac\pm\frac$

It is not known how he came up with this correction. Using this he found an approximation of to 13 decimal places of accuracy when  = 75.

Jamshīd al-Kāshī (Kāshānī), a Persian astronomer and mathematician, correctly computed the fractional part of 2 to 9 sexagesimal digits in 1424, and translated this into 16 decimal digits after the decimal point:

:$2\pi \approx 6.2831853071795864,$

which gives 16 correct digits for π after the decimal point:

: $\pi \approx 3.1415926535897932$

He achieved this level of accuracy by calculating the perimeter of a regular polygon with 3 × 2²⁸ sides.

16th to 19th centuries

In the second half of the 16th century, the French mathematician François Viète discovered an infinite product that converged on known as Viète's formula.

The German-Dutch mathematician Ludolph van Ceulen (''circa'' 1600) computed the first 35 decimal places of with a 2⁶²-gon. He was so proud of this accomplishment that he had them inscribed on his tombstone.

In ''Cyclometricus'' (1621), Willebrord Snellius demonstrated that the perimeter of the inscribed polygon converges on the circumference twice as fast as does the perimeter of the corresponding circumscribed polygon. This was proved by Christiaan Huygens in 1654. Snellius was able to obtain seven digits of from a 96-sided polygon.

In 1789, the Slovene mathematician Jurij Vega calculated the first 140 decimal places for , of which the first 126 were correct, and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places, of which the first 152 were correct. Vega improved John Machin's formula from 1706 and his method is still mentioned today.

The magnitude of such precision (152 decimal places) can be put into context by the fact that the circumference of the largest known object, the observable universe, can be calculated from its diameter (93billion light-years) to a precision of less than one Planck length (at , the shortest unit of length expected to be directly measurable) using expressed to just 62 decimal places.

The English amateur mathematician William Shanks, a man of independent means, calculated to 530 decimal places in January 1853, of which the first 527 were correct (the last few likely being incorrect due to round-off errors). He subsequently expanded his calculation to 607 decimal places in April 1853, but an error introduced right at the 530th decimal place rendered the rest of his calculation erroneous; due to the nature of Machin's formula, the error propagated back to the 528th decimal place, leaving only the first 527 digits correct once again. Twenty years later, Shanks expanded his calculation to 707 decimal places in April 1873. Due to this being an expansion of his previous calculation, all of the new digits were incorrect as well. Shanks was said to have calculated new digits all morning and would then spend all afternoon checking his morning's work. This was the longest expansion of until the advent of the electronic digital computer three-quarters of a century later.

20th and 21st centuries

In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of , including

:$\frac = \frac \sum^\infty_ \frac$

which computes a further eight decimal places of with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . Even using just the first term gives

:$\pi\approx\frac\approx 3.14159273$

See Ramanujan–Sato series.

From the mid-20th century onwards, all calculations of have been done with the help of calculators or computers.

In 1944, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.

In the early years of the computer, an expansion of to decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of . For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.

In 1989, the Chudnovsky brothers computed to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of :

:$\frac = 12 \sum^\infty_ \frac.$

Records since then have all been accomplished using the Chudnovsky algorithm.
In 1999, Yasumasa Kanada and his team at the University of Tokyo computed to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of .
In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate to roughly 1.24 trillion digits in around 600 hours (25days).

Recent Records

# In August 2009, a Japanese supercomputer called the T2K Open Supercomputer more than doubled the previous record by calculating to roughly 2.6 trillion digits in approximately 73 hours and 36 minutes.
# In December 2009, Fabrice Bellard used a home computer to compute 2.7 trillion decimal digits of . Calculations were performed in base 2 (binary), then the result was converted to base 10 (decimal). The calculation, conversion, and verification steps took a total of 131 days.
# In August 2010, Shigeru Kondo used Alexander Yee's y-cruncher to calculate 5 trillion digits of . This was the world record for any type of calculation, but significantly it was performed on a home computer built by Kondo. The calculation was done between 4 May and 3 August, with the primary and secondary verifications taking 64 and 66 hours respectively.
# In October 2011, Shigeru Kondo broke his own record by computing ten trillion (10¹³) and fifty digits using the same method but with better hardware.
# In December 2013, Kondo broke his own record for a second time when he computed 12.1 trillion digits of .
# In October 2014, Sandon Van Ness, going by the pseudonym "houkouonchi" used y-cruncher to calculate 13.3 trillion digits of .
# In November 2016, Peter Trueb and his sponsors computed on y-cruncher and fully verified 22.4 trillion digits of (22,459,157,718,361 ( × 10¹²)). The computation took (with three interruptions) 105 days to complete, the limitation of further expansion being primarily storage space.
# In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 (approximately ) trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete.
# In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days.
#On August 14, 2021, a team (DAViS) at the University of Applied Sciences of the Grisons announced completion of the computation of to 62.8 (approximately ) trillion digits.
# On June 8th 2022, Emma Haruka Iwao announced on the Google Cloud Blog the computation of 100 trillion (10¹⁴) digits of over 158 days using Alexander Yee's y-cruncher.

Practical approximations

Depending on the purpose of a calculation, can be approximated by using fractions for ease of calculation. The most notable such approximations are (relative error of about 4·10⁻⁴) and (relative error of about 8·10⁻⁸).

Non-mathematical "definitions" of

Of some notability are legal or historical texts purportedly "defining " to have some rational value, such as the "Indiana Pi Bill" of 1897, which stated "the ratio of the diameter and circumference is as five-fourths to four" (which would imply "") and a passage in the Hebrew Bible".

The bill was nearly passed by the Indiana General Assembly in the U.S., and has been claimed to imply a number of different values for , although the closest it comes to explicitly asserting one is the wording "the ratio of the diameter and circumference is as five-fourths to four", which would make , a discrepancy of nearly 2 percent. A mathematics professor who happened to be present the day the bill was brought up for consideration in the Senate, after it had passed in the House, helped to stop the passage of the bill on its second reading, after which the assembly thoroughly ridiculed it before postponing it indefinitely.

Imputed biblical value

It is sometimes claimed that the Hebrew Bible

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