Sep 7, 20205 min read

Pi: The Eternal Journey

Just like you and me who are a part of an adventurous story, Pi being just a ratio has its own magical story which began about 3500 years ago. Ranging from Egyptians to Greeks to Indians, pi has laid its presence in varying cultures and texts.

You might be surprised that the earliest forms of pi were found in Egypt (1650 B.C.). The Rhind papyrus written by the scribe named Ahme formulated the area of a circle that made pi equal to (16/9)2 ≈ 3.1605.
In 1600 B.C. Babylon a clay tablet had a geometrical statement stating that pi is equal to 25/8 = 3.1250.
The earliest form of pi in the Indian text were the Shulba Sutras dated 600 B.C. that made pi equal (9785/5568)2 ≈ 3.088.
Moving ahead in the timeline the Hebrew Bible also known as the Old Testament (400B.C.) had a very accurate description of pi that held the record for the greatest number of correct digits for several hundred years afterwards.
It described the construction of The Temple of Solomon, saying that the ceremonial pool's dimensions include a diameter of 10 cubits and circumference of 30 cubits. By basic math we can know that the value of pi comes out to be 3. This has made it one of the most debatable topics but interestingly a far more accurate approximation for pi lies deep within the mathematical code of the Hebrew language.
In Hebrew, each letter equals a certain number and a word's value is equal to the sum of its letters. In 1 Kings 7:23, the word “line” is written Koph Vau Heh, but “Heh” does not need to be there and is not pronounced.
With the extra letter, the word has a value of 111, but without it the value is 106. (Koph=100, Vau=6, He=5). The ratio of pi to 3 is very close to the ratio of 111 to 106. In other words, pi/3 = 111/106 approximately; solving for pi we find pi = 3.1415094.
Archimedes is credited to be the first to calculate the value of pi in 250B.C. He proved that pi equaled 223/71 < π < 22/7 or (3.1408 < π < 3.1429).
Liu Hui Created a polygon based algorithm and used to calculate a 3,072 sided polygon to figure out the value of pi 3.1416 in 256 A.D.
The next accurate assumption of pi was made by Zu Chongzhi in 480 A.D. he calculated pi equaled around 355/133 by using Hui's algorithm, applied it to a 12,288 sided polygon. Resulting in a correct assumption of the first 7 digits of pi in the form of "3.141592920"
During the same time book "Aryabhatiya" written by Aryabhata bore a reference to pi. Aryabhata said that circumference of a circle with a diameter of 20000 is (4+100) x8 +62000= 62832. And we know that the value of pi is the ratio of the circumference to the diameter, so in this case 62832/20000, which is incredibly 3.1416. This is the value of pi accurate to five figures.
The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha. He attributes the series to an earlier Indian mathematician Madhava who used infinite series to estimate π to 11 digits around 1400. The proofs of this series were presented in the Indian work Yuktibhāṣā.

Following this was a time when mathematicians attempted to find decimal digits of pi. Francois Viete achieved 9 digits, Adriaan von Roomenus computed pi to 17 digits after the decimal, of which 15 were correct, Willebrord Snellius reached pi value up to 34 places, Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

Ludolph van Ceulen spent a great part of his life hunting for pi, and by the time he died in 1610, he had accurately found 35 digits. His accomplishments were considered so extraordinary that the digits were cut into his tombstone in St. Peter's Churchyard in Leyden. Still today, Germans refer to pi as the Ludolphian Number to honor the man who had such great perseverance.
John Machin, a professor of astronomy in London knowing that arctan x + arctan y = arctan (x+y) / (1-xy), discovered the wonderful formula: pi/4 = 4 arctan (1/5) - arctan (1/239). Machin calculated pi with his new formula, and computed 100 places by hand.
As the number of accurate decimal digits increased the need for a particular symbol to represent the ratio was clearly felt.
The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in his 1706 work Synopsis Palmariorum Matheseos or A New Introduction to the Mathematics unfortunately; his notation was not immediately adopted by others.
The symbol ’π’ was popularized by Leonhard Euler. He used π = 3.14... in his 1736 work Introduction to analysis infinitorium , since Euler corresponded heavily with other mathematicians in Europe, the use of the Greek letter spread rapidly and the practice was universally adopted thereafter in the Western world
Thereafter numerous mathematicians revealed the various properties of pi. In 1761 Johann Heinrich Lambert Proved that pi was irrational. His argument was, in its simplest for m, that if x is a rational number, then tan x cannot be rational; since tan pi/4 = 1, pi/4 cannot be rational, and therefore pi is irrational.
Adrien-Marie Legendre proved that pi squared is also irrational in 1794, Ferdinand von Lindemann proved pi is transcendental, or not a non-constant polynomial in 1882.
Modern π calculators do not use iterative algorithms exclusively. New infinite series were discovered in the 1980s and 1990s that are as fast as iterative algorithms, yet are simpler and less memory intensive.[122] The fast iterative algorithms were anticipated in 1914, when the Indian mathematician Srinivasa Ramanujan published dozens of innovative new formulae for π, remarkable for their rapid convergence
The development of computers in the mid-20th century again revolutionized the hunt for digits of π. Mathematicians John Wrench and Levi Smith reached 1,120 digits in 1949 using a desk calculator. Using an inverse tangent (arctan) infinite series, a team led by George Reitwiesner and John von Neumann that same year achieved 2,037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. This continued until 1 million digits were discovered in 1973. Till date pi has been found up to 31 trillion digits.
And such is the nature of pi ever evolving and dynamic that takes us on an amazing journey of the prestigious past and provides an insight to a mysterious future.