Continued fraction

A continued fraction is a representation of a real number as a sequence of integers obtained through an iterative process of division and inversion. Every rational number has a finite continued fraction; every irrational number has an infinite one. The simplicity of this representation conceals its power: continued fractions provide the best rational approximations to real numbers, and their periodicity encodes deep algebraic structure.

The connection to Diophantine equations is immediate. The solutions to Pell's equation are generated by the periodic continued fraction of quadratic surds. The convergents of a continued fraction — the rational approximations obtained by truncating the expansion — satisfy inequalities that make them optimal in a precise sense. This is why Diophantine approximation, the study of how well irrational numbers can be approximated by rationals, is essentially the study of continued fractions.

Beyond number theory, continued fractions appear in the theory of dynamical systems, in the geometry of lattices, and in the analysis of algorithms. Their seeming obscurity is a historical accident; they are as fundamental to the structure of real numbers as prime factorization is to the integers.