Pi
Pi is a mathematical constant. The lowercase Greek letter π represents the symbol pi. It is an acronym of the periphery. The word pi was derived from the first letter of the Greek word perimetros. The meaning of perimetros in English is circumference. The word pi is pronounced as Pie or Py. Normally, we use the value 3.14. Every year, on March 14, Pi Day is celebrated around the world.
History of Pi
The importance of pi come into existence around 4000 years ago. The Babylonians and Egyptians were already aware of the existence and significance of the constant π. They proved that the circle has the same ratio of circumference to diameter.
The Babylonians suggest the value 25/8. While at the same the ancient Chinese were used integer 3.
In 1665, the scientist Sir Issac Newton had calculated the value of pi up to 16 decimal places. In 1719, the French mathematician Thomas Fantet de Lagny had calculated the value of pi up to 127 decimal places.
The value of pi is calculated by many other scholars. In the following table, we have summarized the ancient value of pi suggested by different scholars.
| Calculated By | Year | Calculated Value |
|---|---|---|
| Egyptians | 1650 B.C. | 19 / 6 = 3.16666 |
| Babylonians | 2000 B.C. | 25 / 8 = 3.125 |
| Archimedes | 287-212 B.C | 3.14084 < π < 3.14285 |
| Apollonius | 250-175 B.C. | 3.1416 |
| Caludius Ptolemy | 100-178 A.D. | 377 / 120 = 3.14166…. |
| Romans | – | 25 / 8 = 3.125 |
| Hindu | 500 A.D. | √10 = 3.1416 |
| Arabic | 833 A.D. | 62,832 / 20,000 = 3.1416 |
| Chinese | 100 B.C. | 3.1415926 < π < 3.1415927 |
In 1767, The Swiss mathematician Johann Heinrich Lambert proved that pi is an irrational number. In 1882, the German mathematician Ferdinand von Lindemann proved that pi is a transcendental number. It means that pi cannot be a solution to a polynomial with relational coefficients.
The Greek mathematician Archimedes had used an algorithm to calculate the value of pi up to 16 digits. Since, pi is also known as Archimedes’s constant. For calculating the value of pi, he drew two polygons, one inside the circle and another outside the circle. He repeated this process and found that the sides of the polygon getting more closer to the shape of the circle. When he achieves 96-sided polygons, he proved that:
In ancient time the calculation of pi was based on the measurement. For example, the Egyptians calculated the value of pi (16 / 9)2, which is approx. 3.16. Similarly, the Babylon rope stretchers calculated the value of pi 25 / 8 = 3.125, while they were marking the location of the building and boundaries.
Pi is an irrational number because the value of pi neither ends nor repetitive. So, we cannot calculate the accurate value of pi. The value of pi in decimal form is approximately 3.14. Sometime pi is also represented in the fractional form that is 22/7.
Definition of Pi
The circumference (C) of a circle divided by its diameter (d). It means that for any size of the circle, the value of Pi will remain constant.
The ratio of circumference and diameter is constant (pi).
Pi symbol
Pi symbol was suggested by mathematician William Jones (an Anglo-Welsh philologist) in 1706. After 30 years, it became standard mathematical notation. But it was popularized by the mathematician Leonhard Euler. When it became popular, it was used by mathematicians for denoting the ratio of the circumference of a circle to its diameter.
Lowercase representation: π
Uppercase representation: Î
In most cases, lowercase representation is used.
Facts about Pi
In the following points, we have summarized the most interesting facts about pi.
- The term pi originates in Egyptian
- It is an irrational number (all the real numbers that are no rational).
- The exact value of pi cannot be found.
- It does not follow any pattern (such as 3.14155556666).
- We can never calculate the accurate area or circumference of a circle.
- The Pi day is celebrated on March 14 or 3/14 (where 3 represents the month, and 14 represents day) at 1:59 PM (1 represents the hour, and 59 represents minutes), every year. Combining these digits, we get the pi value up to six digits i.e, 14159.
- There are six 9 in a row at 762nd place, which is known as Feynman point, as shown in the following image.
- In the first million digits of pi, the sequence 123456 does not occur.
- In Egypt, the structure of pyramids is based on the principle of pi. The word famous pyramid of Giza is the best example of it.
- Up to 50 trillion digits, the value of pi has been calculated yet.
- On the Pi day, the famous scientist Albert Einstein was born.
- Pi is also known as Ludolph’s Number, Circular constant, and Archimedes constant.
- The technique to remember the digits of pi is called
- The pi memory champion is Hiroyuki Goto, who memorizes the pi value up to 42,000 digits.
- The first one million decimal places of pi consist of:
Pi day
The Pi day was founded by physicist Larry Shaw on 14 March 1988 at San Francisco’s Exploratorium Science Museum. He is also known as the Price of Pi.
It is an annual celebration day of constant pi. The day 14 March was chosen because it represents the first three digits (3.14) of pi. It is the reason that we celebrate pi day on 14 March.
The objective of the day is to motivate the math enthusiasts to recite the endless digits of pi.
Pi Approximation Day
Pi approximation day is celebrated every year on 22 July because the fractional value of pi is 22/7 (where 22 represents the day, and 7 represents the month of July). Pi approximation day is also known as Casual Pi Day.
Uses of Pi
Pi is used in different mathematical formulas for calculating the area of circle, circumferences, area of a cylinder, etc. It is also used in different fields such as cosmology, civil engineering, etc. We have listed some common areas where pi is used:
- The statisticians use the constant for tracking population dynamics.
- In the medical field, it is used when studying the structure of the eye.
- It is also used in navigation (GPS).
- In the aeronautical field, it is used for measuring the area of skin of the aircraft.
- It is also used in spectrum analysis and signal processing.
- In the mathematics field, it is widely used for calculating the area of a circle, cylinder, etc.
There are following formulas that use pi:
- The area of circle (A) = πr2
- The circumference of a circle (C) = 2Ï€r
- The surface area of the sphere (A) = 4Ï€r2
- The volume of Sphere (V) = 4/3Ï€r3
- The surface area of cone (A) = πrl + πr2
- The volume of cone (V) = 1/3 πr2 h
- The surface area of cylinder (A) = 2Ï€r2 +h(2Ï€r)
- The volume of cylinder (V) = πr2 h
World records set for Pi
The latest world record set up by Timothy Mullican in January 2020 for calculating the most accurate value of pi up to 50 trillion digits by using the y-cruncher program. He has broken the previous world record of 31.4 trillion (31,415,926,535,897) digits set up by Emma Haruka Iwao, in March 2019.
Another world record set up for memorizing the 70,000 digits of pi by Rajveer Meena on March 21, 2015. He has broken Chao Lu record, who has memorized 67,890 digits of pi in 2005.
y-cruncher Program
y-cruncher is a multi-threaded program that is developed by Alexander J Yee. It can compute pi and other constants more than a trillion digits. It is the only program that is multi-threaded and accessible to the multi-core system. It is also a stress-testing application for testing the hardware. The program uses the Chudnovsky algorithm for calculating the value of pi. The Chudnovsky formula is given below:
The algorithm takes O(n log (n)3) time to calculate the value of pi.
It is used by many enthusiasts to set world records for computation of pi value. In the following table, we have covered the last five world records.
| Date | Computed By | Decimal Digits | Computation Time (Days) |
|---|---|---|---|
| January 29, 2020 | Timothy Mullican | 50,000,000,000,000 | 303 |
| March 14, 2019 | Emma Haruka Iwao | 31,415,926,535,897 | 121 |
| November 15, 2016 | Peter Trueb | 22,459,157,718,361 | 105 |
| October 8, 2014 | Sandon Van Ness | 13,300,000,000,000 | 208 |
| December 28, 2013 | Shigeru Kondo | 12,100,000,000,050 | 94 |
Features of the y-cruncher program
- It can compute pi and other constants such as Golden Ratio, value of Log (2), e, square root of 2,
- It uses two algorithms to compute most of the constants. The first algorithm is for computation and the second for verification.
- It allows us to use multiple hard drives for faster disk swapping.
- Its multi-threaded feature uses modern multi-core processors.
If you are interested in computing the value of pi, you can use the y-cruncher program. You can download it from http://www.numberworld.org/y-cruncher.