Babylonian mathematicians estimated a circle's area (πr^{2}) to be 3 times the the square of its radius, giving pi its first known approximation of 3. A stone tablet from 1900-1680 BCE revealed the additional use of 3 ⅛ (3.125).

# Cooperative Computation

Help compute as many digits of pi as possible!

Depending on your hardware, each round can take 3-30 seconds. Please be patient. If it takes longer than 2 minutes, reload the page.

We have calculated

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digits of π.

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You have calculated

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digits of π.

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All of the computers involved are calculating terms, assigned by a central server, of the series in the Chudnovsky algorithm.

After a series of operations, the sum of the terms provides an approximation of pi. The more terms calculated, the more digits determined. This series is rapidly convergent; each new term provides approximately 14.18 more digits of pi.

The Chudnovsky algorithm is considered the fastest method for calculating pi, and Google used it to achieve their 2022 record of 100 trillion digits.

# History of Pi

Click on the dots to reveal information.

2000 BCE

~2000 BCE

~1550 BCE

The *Rhind Mathematical Papyrus*, an ancient Egyptian scroll, helped scribes learn to solve particular math problems. Ahmes, the scribe who authored it, wrote "Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle," which implies π = 4 * (8/9)^2 ≅ 3.16045: a fair approximation.

1500 BCE

1000 BCE

500 BCE

287-212 BCE

Archimedes, a Greek mathematician, used the Method of Exhaustion and the perimeters of inscribed and circumscribed regular polygons to calculate pi. Using a 96-sided polygon, he found pi to be between 3.1408 (3 1/7) and 3.1429 (3 10/71). Learn more

0 CE

429-501 CE

Zu Chongzhi used a 24,576-sided polygon to achieve the approximation of 355/113 (3.141592), correct to 6 decimal places, a record that would last for the next 800 years.

500 CE

1000 CE

~1400 CE

Madhava, a mathematician-astronomer from medieval India, is thought to have discovered the Madhava-Leibniz power series for pi (picture) and a rapidly convergent series that obtained 11 decimal places of pi from 21 terms. He also determined a 4-term approximation formula for sine remarkably close to the Taylor series approximation two centuries before the discovery of the Taylor expansion.

1500 CE

1615 CE

Ludolph van Ceulen, inspired by Archimedes' methods, spent 25 years calculating pi to 35 decimal places using a polygon with 2^62 sides (~4.6 quintillion). Ceulen published 20 decimal places in his lifetime, and the other 15 were published posthumously in 1615.

1631 CE

William Oughtred published *Clavis Mathematicae* in 1631, the first book to use the greek letter π in relation to circles. However, pi denoted circumference, unlike the modern-day use of the symbol, which is "the ratio between a circle's circumference and its diameter" (before the symbol caught on, referring to the ratio was verbose). William Jones' 1706 *A New Introduction to the Mathematics* introduced the modern role of pi.

1761 CE

In 1761, Johann Lambert proved that pi was irrational, which means it cannot be expressed as a ratio of two integers. Using a continued fraction expansion, he showed that if x is rational, tan(x) cannot be rational. Since tan(π/4) = 1 and 1 is rational, π/4 and thus π cannot be rational.

1882 CE

Ferdinand von Lindemann, in 1882, proved that pi is transcendental: not a solution to any algebraic equation. This deemed the quadrature of the circle, an ancient problem that caused thousands of years of mental toil, impossible. This problem is also called "squaring the circle," which has become an idiom for trying to solve something impossible.

20th Century CE

D. F. Ferguson found 808 digits of pi after a year of calculations on a mechanical calculator. In 1949, the ENIAC (Electronic Numerical Integrator and Computer) (picture) was complete and could calculate 2037 digits of pi in 70 hours. The IBM 7090 computer surpassed 100,000 digits in 1961, and the CDC 7600 reached pi's millionth decimal place in 1973.

2000 CE

21st Century CE

In 2021, scientists at the University of Applied Sciences of the Grisons calculated pi to 62.8 trillion decimal places. Last year, Google used the Chudnovsky algorithm and their Cloud Compute Engine to achieve the current record of 100 trillion digits.

# Find a Number, Picture, or Text in Pi

Find the position of the first occurrence in the first billion digits of pi.

Choose one to search for:

Result:

# Archimedes' Method of Exhaustion

## Visualization

The red polygon represents the lower bound of the circle's perimeter/area while the blue polygon represents the upper bound.

## Calculator

Read on to see how Archimedes performed this calculation without trigonometric functions!

Start with circle and a regular hexagon inscribed in the circle. A hexagon can be split into six equilateral triangles, so the circle's radius (r=1 for simplicity) is equal to the hexagon's side lengths.

For each "iteration" in this method, the number of sides the polygon has doubles. In each iteration, the goal is to calculate the side length of the next polygon. Inscribe a regular 12-gon whose vertices meet the hexagon's.

Connect the center of the polygons to a vertex of the 12-gon; this segment bisects the hexagon's side.

Apply Pythagoras' Theorem on the right triangle outlined in purple. The hypotenuse is the radius, one leg is half of the side length (S_{1}), and the other leg is the radius minus a small distance NC from the 12-gon's vertex.

Now NC and NB, two legs of the right triangle outlined in black, are defined in terms of S_{1}, and the hypotenuse is the side length of the 12-gon (S_{2}).

Apply Pythagoras again and solve for S_{2}, then find the ratio between circumference and diameter (or circumference and twice the radius) to determine the approximation. At 12 sides, the approximation is 3.106, but this is better than the approximation with the hexagon, 3.

Continuing to double the sides and recursively apply the formula results in better approximations. Archimedes went to a 96-gon (6, 12, 24, 48, 96) and determined three digits.

To find the "upper bound" of each step, Archimedes used a circumscribed polygon drawn around the circle with the same number of sides as the inscribed one but a slightly larger side length. This length is found easily using similar triangles.

The side length or perimeter of the inscribed polygon is multiplied by a scale factor to get those of the circumscribed polygon.

# Fun Facts

- π = C/d. Pi is the ratio between any circle's circumference (perimeter) and its diameter.
- People suspect π to contain every conceivable combination of the digits 0-9. This seems likely but hasn't been proven.
- A number with this characteristic is called a normal number. This is an irrational number for which all (finite) sequences of numbers of a certain length occur with the same frequency. For instance, each pair of digits 00-99 will occur 1% (1/100) of the time.
- In other words, the digits of a normal number are uniformly distributed across the infinite decimal expansion, and no particular number or sequence is more frequent. All normal numbers are irrational, but not all irrational numbers are normal.
- Looking at the first 30 million digits of pi makes it appear to have this property, but there is no known way to prove this for pi.
- Go to Find a Number, Picture, or Text in Pi

- The decimal part of all positive multiples of pi is uniformly (evenly) distributed from 0 to 1.
- y = n*pi mod 1 -> 0 < y < 1, y is uniform between 0 and 1
- For more info, see Weyl's equidistribution theorem

- Pi has applications in physics, astronomy, mathematics, and even architecture and construction.
- Pi emerges in nature through the meandering of rivers.
- A river's bendiness is determined by its “meandering ratio," or the ratio of the river's actual length to the direct distance from its source to its mouth.
- Numberphile video on meandering and pi

- Akira Haraguchi, in 16 hours and 30 minutes, recited 100,000 digits of π from memory at a public event in 2006.
- Pi Day is March 14th, 3/14, because π ≅ 3.14. Pi Approximation Day is celebrated on July 22nd because 22/7 (≅3.143) is a popular approximation for pi.
- Albert Einstein was born on March 14th, 1879, and Stephen Hawking died on March 14th, 2018.
- In 2009, Congress passed Resolution 224, which designated March 14th as Pi Day, in hopes of fostering enthusiasm for math and science among U.S. students.
- Pi provides accurate calculations even at modest precisions. Using π with just 9 decimal digits can give a figure for the Earth's circumference that only errs 0.25 inches in 25,000 miles.
- Still, the competition to calculate more digits of π continues.
- In 2010, we knew ~5 trillion digits, and since 2022, we know 100 trillion digits.
- These computation feats primarily demonstrate advancements in computation, storage, and networking. Google used its 100-trillion-digit record to endorse its cloud infrastructure services.
- The calculation of pi is a common stress test for a computer processor.

# Credits & Info

Pi Day 2023 is an educational initiative by the Applied Programming Club with guidance from the Math Club.