Approximation of Irrational Numbers

Abstract

Our story begins with one of the oldest questions in number theory: How well can a real number be approximated by rational numbers? Phrased in this way, the answer is “arbitrarily well,” since every real number α is the limit of a sequence $(\frac{p_n}{q_n})$ of rationals. But in such a convergent sequence, e.g., the decimal expansion of an irrational number α, the denominators usually grow very fast