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The 28 pages that link to Continued fraction, each with the reason it gives.

• Rational numberRelated: Its convergents give especially accurate rational approximations.

• Diophantine equationRelated: Approximations from continued fractions produce and organize solutions to Pell equations.

• Euclidean algorithmRelated: The quotients in successive divisions are the continued-fraction coefficients of the input ratio.

• Irrational numberRelated: Irrational numbers have infinite, nonperiodic continued fraction expansions.

• PiRelated: Pi’s continued fraction generates especially useful rational approximations.

• Golden ratioRelated: The golden ratio has the repeating continued fraction [1; 1, 1, 1, …].

• Srinivasa RamanujanRelated: Ramanujan found remarkable identities and special values involving these expansions.

• Diophantine approximationRelated: Its convergents give exceptionally accurate rational approximations and describe their quality.

• Joseph LiouvilleRelated: Continued fractions provide particularly strong rational approximations, central to understanding his criterion.

• Pell equationRelated: Convergents to √D produce solutions to the Pell equation.

• Decimal expansionCompared with: Continued fractions encode numbers through nested reciprocals rather than positional powers of ten.

• Charles HermiteRelated: Hermite studied continued fractions in connection with approximation and number-theoretic problems.

• Padé approximantRelated: Many Padé approximants admit continued-fraction representations that expose their structure.

• Shor's algorithmRelated: Classical continued fractions recover the period from a measured approximation to its ratio with a power of two.

• Dirichlet's approximation theoremRelated: Convergents from continued fractions often give sharper approximations than the theorem guarantees.

• Monotone sequenceRelated: Convergents of a continued fraction split into monotone subsequences that approach its value.

• Partial fraction decompositionCompared with: Continued fractions provide a different representation of rational expressions, not a sum over denominator factors.

• Egyptian fractionCompared with: Continued fractions use nested reciprocals rather than sums of distinct unit fractions.

• Johann Heinrich LambertRelated: Lambert used continued fractions to establish properties of trigonometric functions relevant to π.

• Lowest termsRelated: Its convergents are commonly written as fractions in lowest terms.

• Extended Euclidean algorithmRelated: The quotients in the algorithm form the continued-fraction expansion of a ratio of its inputs.

• Square root of 2Related: The square root of 2 has the repeating continued fraction [1; 2, 2, 2, …].

• Zu ChongzhiRelated: The fraction 355/113 is an exceptionally accurate rational approximation to pi.

• Markov numberRelated: Continued fractions supplied the approximation problem behind Markov’s discoveries.

• Mathematical constantRelated: Continued fractions encode constants through successive rational approximations.

• Lochs's theoremNarrower topic: Its initial terms are the quantities whose growth Lochs's theorem estimates.

• Proof that π is irrationalRelated: Lambert’s proof derives irrationality from a continued fraction for tan x.

• Series expansionCompared with: It can represent or approximate functions where a power series is inefficient.