Pi (the Greek character is π) is the symbol we use to designate the ratio of a circle’s circumference to its diameter. Ancient people, among whom there were many master builders and surveyors, seemed quite interested, and they expended considerable mental energy in attempting to calculate its value. They endeavored also to “square the circle” by geometric means, i.e. with a compass and straight edge.
Undoubtedly the earliest efforts to ascertain the value of π involved drawing a circle with its diameter shown, then taking measurements. Subsequent efforts were not instrument based, but instead the idea was to calculate the value of π using inscribed polygonal calculations, infinite series computation, iterative algorithms including Fourier transform-based methods and finally distributed computing projects.
It is not possible to express π as a ratio of two integers. That is because it is an irrational number. As such, it is not a repeating decimal. Cantor’s proof states that real numbers are uncountable and rational numbers are countable. This implies that most real numbers are irrational, so it is not surprising that π is a member of that set.
Since π is an irrational number, it cannot be written as a fraction, except as an approximation. For many practical applications, 22/7 is sufficiently precise.
Notwithstanding Carl Sagan’s fascinating novel Contact (released in 1997 as a movie starring Jodie Foster), the digits that comprise π have no discernible pattern according to numerous tests that have been performed for statistical randomness. Some non-mathematicians attach great significance to the Feynman Point, where beginning at the 762nd decimal place six consecutive nines occur, but actually that and similar sequences are inevitable in the fullness of time.
Ancient calculations of the value of π ranged between 3.1250 and 3.1605. By the mid-first millennium, Chinese and Indian mathematicians had achieved greater accuracy, but it was not until the 1500’s that European researchers achieved 20- and 35-digit resolution, using polygonal methods.
Major breakthroughs emerged in the 16th and 17th centuries when infinite series techniques were developed, most notably by Gottfried Wilhelm Leibniz, anticipated in original Indian documents written in Sanskrit.
In 1706, John Machin discerned 100 digits with an algorithm based on Leibniz’s work. Such methods were widely used prior to the computer age, the greatest achievement being 620 digits.
In the computer age, the use of iterative algorithms at first and then new infinite series methods permitted accelerated machine number crunching. Spigot algorithms were an important innovation that stood in contrast to infinite series and iterative algorithms which used earlier digits to achieve the end result.
Near the end of 2014, the value of π had been computed to 13,300,000,000,000 digits. Inasmuch as the number of particles in the observable universe is considered to be 1080, greater resolution of the value of π will eventually be of value only as an abstraction without reference to the real world, but that won’t stop this journey that has no endpoint.