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Everything you might  want to know about pi (π)

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Picture Attribution: Giuseppe Nogari, Public domain, via Wikimedia Commons.
File: URL: https://upload.wikimedia.org/wikipedia/commons/c/c3/Archimedes_by_Giuseppe_Nogari.png The author died in 1766, so this work is in the public domain in its country of origin and other countries and areas where the copyright term is the author’s life plus 100 years or fewer.
The Mysterious Pi

The mysterious Pi (often represented by the lower-case Greek letter π, one of the best-known mathematical constants) is the ratio of the circumference of a circle to its diameter. For any circle, the distance around the edge is a little more than three times the distance across[1].

Pi (π) is one of the most important and fascinating numbers in mathematics. Roughly 3.14, it is a constant that is used to calculate the circumference of a circle from that circle’s radius or diameter. It is also an irrational number, meaning that it can be calculated to an infinite number of decimal places without ever slipping into a repeating pattern, making it difficult, but not impossible, to calculate precisely[2].

If you want to calculate pi, measure the circumference of a circle by wrapping a piece of string around the edge of it and then measuring the length of the string. Then, measure the circle’s diameter (the distance between one side and the other that runs directly through the centre). Now you have the circumference, and the diameter, plug them into the formula C / D (dividing the circumference by the diameter) and hey presto, you have pi. To learn how to calculate pi using a limit or sine function, read the article at: https://www.wikihow.com/Calculate-Pi#summary_wrapper

Pi is also called Archimedes’ constant, named after the Greek mathematician Archimedes, who created an algorithm to approximate the pi value. The value of pi is irrational, which means that the count of digits after the decimal point is infinite. The symbol used to denote pi in math is π, used in Greek letters. It is known as a mathematical constant as its value does not change. The pi symbol represents 22/7 (in fractions) or 3.14 (approximately) in decimals.[3] It is used as either 3.1415929 or 22/7. In geometry, it is used in calculating the surface area, volume, and circumference of various three-dimensional shapes.

Archimedes developed the polygonal approach to approximating π.

Timeline
According to Petr Beckmann’s A History of Pi, the Greek letter π was first used by William Jones[4] in 1706, probably as an abbreviation of periphery, and became standard mathematical notation roughly 30 years later.

Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. Pi is also referred to as Archimedes’s constant[5]. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first computational formula for π, based on infinite series, was discovered a millennium later when the Madhava–Leibniz series was discovered by the Kerala school of astronomy and mathematics, documented in the Yuktibhāṣā, written around 1530.[6]

A record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of German mathematician Carl Friedrich Gauss.[7]

The invention of calculus soon led to the calculation of many hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits.[8]

Scientific American say that the importance of pi has been recognised for at least 4,000 years – A History of Pi (see above) notes that by 2000 BC:

“the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π, recognising that every circle has the same ratio of circumference to diameter. Both the Babylonians and Egyptians had rough numerical approximations to the value of pi, and later mathematicians in ancient Greece, particularly Archimedes, improved on those approximations. By the start of the 20th century, about 500 digits of pi were known. With computation advances, thanks to computers, we now know more than the first six billion digits of pi.” [9]

Six billion digits are a lot of digits. The Scientific American article looks out of date as there are several web references to ‘trillions’ of digits. You can see what the first one million digits look like if you’re interested[10], although you may like to memorise just the first ten digits: 3.1415926535

Approximations[11]
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 first century AD. 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 potential application outside of pure mathematics.

The record of manual approximation of pi is held by William Shanks, who calculated 527 digits correctly in 1853[12]. Since the middle of the 20th century, the approximation of pi has been undertaken by electronic digital computers. On 16th August 2021, the current record was established by Thomas Keller and Heiko Rölke of the Swiss University of Applied Sciences of the Grisons with 62.8 trillion digits.[13]

Methods of Calculating pi[14]


Picture Credit: “A student solves a mathematics equation at the Mfantsipim Boys School in Cape Coast” by World Bank Photo Collection is marked with CC BY-NC-ND 2.0.

Ways that pi is used every day
Honeywell[15] provides a list of how pi is used in today’s world. Engineers use pi every day as it’s a key element in a range of calculations, especially those related to round things.

  • Pi helps work out how to point an antenna toward a satellite under any circumstance.
  • Pi has a big role to play in actuation. Actuators control the flaps that move on aircraft wings and tails or the parts that open and close valves on jet engines.
  • You’ve likely already heard of RPM or Revolutions Per Minute. There’s a device on your car’s dashboard to measure how quickly your tyres are spinning, called a tachometer.
  • Pi is used to calculate the volume of cylinders. 
  • Refineries use metal vessels to make fuel, and the chemicals in those vessels must have optimal flow to complete chemical reactions in that process. Pi helps guarantee that is the case.
  • Pi is used to calculate the size of giant cylinders used from refinery equipment to rolls of paper for printers.
  • Pi is used to determine the capacity of hot water storage tanks and heaters.

Other examples[16] of how pi is used are vast and include:

  • Pi is used to measure how fast and how powerful a computer is and to check its accuracy and if there are any problems in its software or hardware.
  • It is also used to get the value of trigonometry functions like sine, cosine, tangent etc.
  • It can measure the circular velocity of things like a vehicle wheel, motor shafts, engine parts, gears and AC voltage across a coil and a capacitor.
  • Using pi, it can measure things like ocean waves, light waves, sound waves, river bends and radioactive particle distribution.
  • Electrical engineers used pi to solve problems for electrical applications.
  • Statisticians use pi to track population dynamics.
  • Medicine benefits from pi when studying the eye’s structure.
  • Biochemists see pi when trying to understand the structure/function of DNA.
  • Physicists looking into the behaviour of fluid ripples see pi and use it in their calculations.
  • Clock designers use pi when designing pendulums for clocks.
  • Aircraft designers use it to calculate areas of the skin of the aircraft.
  • Navigation, such as global positioning (GPS).
  • Calculating the number of death in a population.
  • Solving mathematical problems in Geometry like finding the area of a circle etc.

Keen space travellers among the readers of this paper will be interested to know that pi helps in studying the infinite stars in the universe[17]

  • Early astronomers used it to study the Earth and its rotation, its orbits, and in the search for new planets and their atmospheres outside our solar system. It has been used in calculating the density of a planet, which tells us about the nature of the planet (whether it is made up of rocks or gases). NASA uses  Pi to calculate the trajectory of a spacecraft (pi transfer) to measure craters and learn about the composition of asteroids.  Lately, pi was also used in calculating the amount of hydrogen present in the ocean beneath the surface of jupiter’s moon Europa. 
  • Pi is used to study how springs bounce, pendulums swing, and strings vibrate. Another important application of these sound waves is in taking an ultrasound of a baby in the womb. It also helps in studying the behaviour of ocean waves – their frequency, wavelength and amplitude. Pi comes into play even when we talk over the phone, watch television, or listen to the radio.
Sources and Further Reading

Pi number
Picture Credit: [Cropped] “Pi number” by J.Gabás Esteban is marked with CC BY 2.0.

  1. Source: https://www.piday.org/learn-about-pi/ ↑

  2. Source: https://www.wikihow.com/Calculate-Pi ↑

  3. Source: https://www.cuemath.com/geometry/what-is-pi/ ↑

  4. William Jones, FRS was a Welsh mathematician, most noted for his use of the symbol π to represent Pi, the ratio of the circumference of a circle to its diameter. He was a close friend of Sir Isaac Newton and Sir Edmund Halley. In November 1711, he became a Fellow of the Royal Society, and was later its vice-president. Source: https://en.wikipedia.org/wiki/William_Jones_(mathematician) ↑

  5. Sources: Weisstein, Eric W. “Pi”. mathworld.wolfram.com. and Bogart, Steven: “What Is Pi, and How Did It Originate?”. Scientific American. ↑

  6. Sources: Andrews, Askey & Roy 1999, p. 59, and Gupta 1992, pp. 68–71. ↑

  7. Source: Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed, pages 194-196. Springer-Verlag. ISBN 978-3-540-66572-4. English translation by Catriona and David Lischka. ↑

  8. Sources: πe trillion digits of π”. pi2e.ch. Archived from the original on 6 December 2016, and Haruka Iwao, Emma (14 March 2019). “Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes’ constant on Google Cloud”. Google Cloud Platform. ↑

  9. Source: https://www.scientificamerican.com/article/what-is-pi-and-how-did-it-originate/ ↑

  10. See: https://www.piday.org/million/ ↑

  11. Source: https://en.wikipedia.org/wiki/Approximations_of_%CF%80 ↑

  12. Source: Hayes, Brian (September 2014). “Pencil, Paper, and Pi”. American Scientist. Vol. 102, no. 5. p. 342. doi:10.1511/2014.110.342 ↑

  13. Source: Un nouveau record de calcul du nombre Pi établit en Suisse” (in French). RTS Info. 16 August 2021. ↑

  14. Source: https://www.wikihow.com/Calculate-Pi ↑

  15. See: https://www.honeywell.com/us/en/news/2020/03/5-ways-our-engineers-use-pi ↑

  16. Source: https://amazingarchimedes.weebly.com/real-life-application-of-pi.html ↑

  17. Source: http://www.greenworldinvestor.com/2016/06/05/life-is-weird-ten-real-life-applications-of-the-most-famous-number-in-the-world-pi/ ↑

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Martin Pollins

Martin Pollins is a Chartered Accountant and MBA with wide experience in corporate finance and business management. He has served on the boards of several companies, including those listed on the London Stock Exchange, AIM and OFEX. He is the Chairman and Founder of OneSmartPlace and was a Council member of the Institute of Chartered Accountants in England and Wales from 1988 to 1996. He was managing partner of PRB Martin Pollins, based in Sussex, the first Accountancy firm to advertise on British television. He went on to create and launch the CharterGroup Partnership (the UK’s first Accountancy network) and then LawGroup UK (at the time, one of the largest networks of lawyers in the UK). In recent years, he helped to raise several £millions to fund British films such as The Da Vinci Code, Bridge of San Luis Rey, Head in the Clouds and Merchant of Venice with actors such as Charlize Theron, Robert De Niro, Al Pacino, F. Murray Abraham. Kathy Bates, Gabriel Byrne, Geraldine Chaplin, Tom Hanks, Ian McKellen, Audrey Tautou, Penélope Cruz, Steven Berkoff, Lynn Collins, Jeremy Irons, Joseph Fiennes and many more. He has written several business publications (see https://onesmartplace.com/resources/)

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