10 hours ago
- 22/7 and 355/113 are surprisingly accurate rational approximations of π, with errors below 1/b².
- Dirichlet's approximation theorem proves that for any real number, there exists a rational a/b with error < 1/b².
- Rational numbers have at most a finite number of such '2-good' approximations, limited by their denominator in lowest terms.
- Irrational numbers have infinitely many 2-good approximations, as shown by the pigeonhole principle and repeated application of Dirichlet's theorem.
- The approximation score s = ε·b determines if an approximation is 1-good (s < 1) or 2-good (s < 1/b).
- Geometric intuition: irrational numbers as a line through an integer lattice never hit lattice points but can get arbitrarily close.
- The study of these approximations, Diophantine approximation, is deep and includes results like the irrationality measure of algebraic numbers being 2.