How Rational Math Catches Slippery Irrational Numbers — Quanta Magazine

My latest column for Quanta Magazine is about a clever technique for finding rational approximations to irrational numbers. The technique, developed by the German mathematician Gustav Dirichlet, works by covering the number line with tiny intervals centered at certain rational numbers.

But Dirichlet did better. He improved this method by figuring out how to shrink the intervals around their centers while still keeping the entire number line covered. As the intervals shrink, so does the distance to any irrational number we are trying to approximate. This means we’ll get better and better rational approximations, even using relatively small denominators. But we can’t shrink the intervals too quickly: Even though there are infinitely many of them, if the intervals get too small too fast they won’t cover the entire number line. In the battle between the infinitely large and the infinitely small, Dirichlet had to find the right balance to prevent some irrationals from slipping through the cracks.

Dirichlet’s technique explains why we can always find good rational approximations to irrational numbers using small denominators, like . Developed nearly 200 years ago, this technique ultimately led to the proposal of the Duffin-Schaeffer conjecture which was finally proved this past year.

You can read the full article here.