Continued Fractions & Rational Approximations

Continued fractions can be thought of as an alternative to digit sequences for representing numbers, based on division rather than multiplication by a base. Studied occasionally for at least half a millennium, continued fractions have become increasingly important through their applications to dynamical systems theory and number theoretic algorithms.

*Mathematica* has highly efficient original algorithms for finding large numbers of terms in continued fractions, as well as for handling exact continued fractions for quadratic irrationals.

ContinuedFraction — continued fraction expansion

FromContinuedFraction — construct exact or inexact numbers from continued fractions

Convergents — a list of successive convergents of a continued fraction

Rationalize — find rational approximations

QuadraticIrrationalQ — test for a quadratic irrational (repeating continued fraction)

ContinuedFractionK — construct a continued fraction from a symbolic formula

Khinchin — Khinchin's constant characterizing random continued fractions

Objects with Notable Continued Fractions