This is a nice article, and gently (with great explanation) introduces: • The idea of approximating a real number by rational numbers (e.g. √2 ≈ 99/70), • Dir

This is a nice article, and gently (with great explanation) introduces: • The idea of approximating a real number by rational numbers (e.g. √2 ≈ 99/70), • Dirichlet's approximation theorem, that every real number x has a rational approximation p/q such that |qx - p| < 1/(N+1) using a denominator q ≤ N, for any integer N, • The Duffin–Schaeffer conjecture, recently announced to be proved. BTW, I got the above "√2 ≈ 99/70" (≈ 1.4142857…, better than 1414/1000 which has a much larger denominator) from a webpage I made a while ago, to generate the best rational approximations (with increasing denominators) for any number: https://shreevatsa.net/fraction/best https://shreevatsa.net/fraction/best (Of course the continued fraction for √2 has a very simple form so convergents could be written down by hand.)