Everyone (at least most) has a favorite number that they love. However, the number that we can easily say that it is the world champion in terms of interesting numbers is perhaps the most famous number of numbers, “pi”.
This mathematical constant is literally used as a benchmark for computing power, or forms the basis of a worldwide never-ending struggle over who can list the most random digits in the correct order. In the meantime, let’s say that the current record is 111,700.
The reason pi can affect our imagination in this way is because it is an irrational number. In other words, it is a number whose decimal expansion is never ending and is completely random. It is said that any sequence of numbers you can think of can be found somewhere in the expansion of pi, but still knowing a particular sequence anywhere in the expansion gives you no information about the next digit’s future.
But almost unbelievably, about a year ago it turned out that there was a way to find any pi digit you were curious about.
Of course, there is an important detail here: This method is based on the estimates made to calculate the Euler and Bernoulli numbers. Both of these numbers are sequences that take a lot of time and effort to calculate and grow so fast that it will be very difficult to even fit them into your calculator, let alone successfully using the 14th digit of pi.
But that’s not exactly the point of their result, as mathematician Simon Plouffe, who quietly uploaded his formula to the ArXiv preprint server in January 2022, noted: “The formula is not only correct, it’s elegant and simple. It’s a nice formula, especially for the 2nd base. So we can say that the formula is pretty cool.”
We can say that the second base Pi is actually Plouffe’s specialty. Plouffe is the P in the BBP algorithm, a method of calculating the nth digit of the binary expansion of pi, which he discovered in 1995. Now, he says this result can be extended to any base: “By adjusting for base 10 or base 2, it applies to all n’s. It can be done on any basis if we want, for which I can adjust the formula quite simply.”
In a conversation with IFLScience, Plouffe says the new formula is based on results that have been “known for centuries” and yet it has rarely been reexamined by working mathematicians. Therefore, the most interesting thing in the new article, apart from the result itself, is how short it is. The entire article consists of only six pages, excluding a short reference section. The paper lacks lengthy calculations or abstract proofs, and Plouffe’s conclusion is based on his ability to look at something old in a new way.
Plouffe said, “They are so interconnected that if we separate pi or pi from the nth power, we get a formula with the nth Bernoulli number; [and] so precise that if we cut at the nth position, we get enough precision to confirm that it’s the nth decimal number.”
It is unlikely that there will be many practical applications for this discovery, as are many results that solve the most deceptive mathematical constants. Even NASA’s absolute highest accuracy calculations for missions like interplanetary navigation require expansion to only about 16 significant figures. So it’s pretty hard to imagine a scenario where you might need to know the 143rd digit of Pi but know nothing else about the number.
In short, it can be said that the most important point of this solution is that the only thing that needs to be done for the result to emerge is to require a new perspective on an old problem, including old solutions.
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