π has always captivated me. The basic idea is easy to undestand and frequently used. Pi is the ratio of a circle’s circumference to its diameter and used to calculate or measure anything round. It tells us how much pie filling goes with how much pie crust. Yet few numbers hide as much depth. No matter how precisely you try to determine that simple ratio, you will never find an exact answer. The study of π reveals deep truths about man’s relationship to the universe and connects the mundane to the divine.
Infinite, irrational and non-algebraic, π is a very special kind of number, a transcendental number. Its digits extend infinitely and can never be all written down. Unlike infinitely repeating rational numbers like its infinite digits have no pattern so there’s no way to use our ordinary counting numbers to describe them. Transcendental numbers rise above even irrational numbers like and the golden ratio phi () — no algebraic expression can describe them. Math simply cannot define these numbers without the infinite.
Transcendental numbers relate to each other in mysterious ways. Another, Euler’s constant () is the base of the natural logarithm and describes the rate of continuous growth. That doesn’t seem like it should have anything to do with circles, but it and π have a close relationship in what has been called the most beautiful formula in mathematics:
People have been trying to decipher π for a very long time in many different ways. From the simple ; to Archimedes’s geometric construction ; to the Madhava–Leibniz infinite series ; to the Monte Carlo method of counting randomly scattered dots and seeing where they land in a circle inscribed in a square. As new math is invented, new methods of calculating π come along and vice versa. The current record for calculating digits stands at 22,459,157,718,361.
All those extra digits aren’t of any practical use in making calculations more accurate. Just 39 digits are sufficient to calculate the circumference of the observable universe to within one atom’s diameter.
So why chase this unsolvable problem? No matter how sophisticated the method or how fast the computers there will always be more digits out there.
Pi is a humbling reminder of human limitations. No matter how we strive the universe will always hold things out of man’s reach, forever beyond our ken. We are finite, mortal. Pi is infinite, transcendental. Only God knows all the digits of π.
One of the newest methods for calculating π lets us find the value of an arbitrary digit of π without calculating all the previous. Want the billion and one-th digit? It’s now possible to quickly calculate just that one without knowing any preceding digits. These much faster methods for calculating far-off digits of transcendental numbers were first discovered in the mid-90s and are called Bailey-Borwein-Plouffe-type formulas.
That’s an interesting wrinkle, a paradox.
We can’t know all the digits of π, but there are no specific digits we can’t know should we decide to calculate them. Our potential is also infinite. Pi is our connection to the infinite and divine.