Hurwitz's Theorem (Number Theory)

Hurwitz's theorem, named after Adolf Hurwitz, is a result in number theory giving a bound on how well an irrational number can be approximated by fractions. It states that for every irrational number there are infinitely many pairs of relatively prime integers m and n such that the irrational number differs from the fraction m over n by less than one divided by the square root of five times n squared, and that the constant square root of five is the best possible: for the golden ratio in particular, replacing that constant with any larger one leaves only finitely many such fractions. The theorem is equivalent to the statement that the Markov constant of every irrational number is at least the square root of five. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/