SOLUTIONS

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.
ReferenceReference
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