Today is π day, so I thought I should give a brief history on both.
Origins of π:
π is quite the most elusive and yet famous number of all. It's a simple number that even the ancients thought about it. And yet it's concept is quite easy to explain. It is the ratio between the circumference and the diameter. The reason why the diameter was chosen rather than the radius was because it was the only thing they could directly measure [though I argue that the circle constant should Tau. It's understandable once you get to angle measurements]. Some well known approximations are as follow: π pops up in the Bible, with a value of 3. The Bablylonians wrote that π was approximately 25/8 which is 3.125.
But things got interesting with the ancient Greeks. Archimedes attempted to use what is best described as an early for on integration called a method of exhaustion, and it's actually rather simple.
Have you ever wondered as a kid about regular polygons? Notice that they seem to get closer and closer to a circle the more sides you add onto them. By using this property, Archimedes created an upper and lower bound for π. 223/71<π<22/7.
Eventually, the Chinese mathematicians would take this and attempt to get a much more accurate approximation, setting the record at 7 digits, which lasted for 800 years.
Rebirth of Greco-Roman culture and the Scientific Revolution:
The study of mathematics was less relevant during the 1000 years of darkness in Europe. The attitude towards mathematics began to change during the Renaissance age, which led to the scientific revolution. Many major development happened after the Renaissance such as Cartesian coordinates, which showed the connection between Algebra and Geometry. Newton and Leibniz created calculus for the first time, and with it came a new age of discovering π. Tools such as limits and infinite series allowed for faster rates of convergence onto the value of π. We start to see a different way of thinking from the Greek, where problems with infinity can be much easier to solve. Many mathematicians in Europe raced to see who can create the fastest converging series for π. In 1789, a Slovene mathematician, Jurij Vega, calculated π correctly to 126 digits.
Things are getting interesting:
However, not all studies into π was about trying to get better approximations. Leonhard Euler is probably a name many know. He wrote many things about π, such as Euler's identity or solving the Basel problem. With solving the Basel problem, he created basis for Riemann Zeta function, which is still being studied today in the field of analytic continuation. The Riemann Zeta function also became important for number theory when talking about probability of coprimality.
Carl Fredrick Gauss worked on π as well. He used π in Gaussian integration, which led to what we now know as the error function. We also see Lambert's proof of π irrationality using a continued fraction of the tangent function. And in 1882, Ferdinand von Lindemann proved π was not a solution to any algebraic equation, thus establishing π transcendental property. This also showed that it is impossible to square a circle, a problem from antiquity. [And for anyone who wants a funny story, look up Indiana Pi Bill]
In 1910, Srinivasa Ramanujan found a formula that rapidly converges to π. His formula would be the basis for the fastest current algorithm to calculate π. By the mid-20th centuries, computers started to be used to calculate π, with the current world record being set at 31.4 trillion digits of π by Emma Haruka Iwao in March 2019.
So, is this the end of π's story? Since only 30 digits of π is required to hypothetically calculate the circumference of the universe down to an error of a hydrogen atom, would we ever need to investigate π anymore? The study of π, in my view was never really about getting more and more precise formula. I do think we have more than enough digits of π. Ironically, I think I am more alike to the Greeks, but in a different way. The Pythagoreans admire the relation between integers and ratios of integers. They viewed it as numbers sharing a connection. In a sense, π started off as a simple concept and a simply conquest to find the ratio of 2 lengths. And the journey that mathematicians took down the stream of mathematics reveals things about the nature of geometry, algebra, calculus, probability, number theory, complex numbers, and the universe itself. And I don't believe in one myTake, I could do justice for the nature of π.