Turns out rational approximation of irrationals is related to the Hartmanis-Stearns '65 Conjecture on real-time computability, from the same paper that introduced basic complexity classes like DTIME(f(n)). And sqrt(2) makes an appearance!
The Hartmanis-Stearns real-time computability conjecture is (equivalent to):
For a real number r, if there is an algorithm that computes the first n digits in O(n) time (for all n), then r is either rational or transcendental.
And, coming from this, sqrt(2) has an even cooler connection:
If the 1st n digits of sqrt(2) can't be computed in O(n) time, then n-bit integers can't be multiplied in O(n) time. -Lipton & Regan https://rjlipton.wpcomstaging.com/2012/06/15/why-the-hartmanis-stearns-conjecture-is-still-open/
(But...sqrt(2) probably isn't the only number for which the above is true. Still a cool connection, I think.)
#math #algorithms #NumberTheory #complexity #ComputationalComplexity