Continued fraction

A continued fraction is an expression built by repeatedly nesting fractions, usually with integer terms. Its finite truncations give rational numbers that can approximate the represented value.

Convergents: Rational numbers obtained by truncating a continued fraction at successive terms. Each truncation produces the next approximation to the continued fraction’s value.
Rational number: A number expressible as a ratio of two integers with a nonzero denominator. Finite simple continued fractions represent exactly the rational numbers.
Diophantine approximation: The study of approximating real numbers by rational numbers, often with bounds on numerators and denominators. Continued fractions provide systematic solutions to rational-approximation problems.
Decimal expansion: A representation of a number as an integer part and a sequence of base-ten digits. Decimals use positional digits, while continued fractions encode successive divisions.
Recurrence relation: An equation that defines a sequence of terms using one or more earlier terms. Numerators and denominators of convergents follow simple recurrences.
Irrational number: A real number that cannot be expressed as a ratio of two integers. Infinite simple continued fractions represent irrational numbers.
Pell's equation: The integer equation x² − Dy² = 1, where D is a positive nonsquare integer. Convergents to √D yield solutions to this equation.
Padé approximant: A rational function chosen to match a power series to high order near a point. Padé approximants also use ratios, but approximate functions locally rather than encode numbers through integer terms.
Euclidean algorithm: A procedure for finding the greatest common divisor by repeated division with remainder. Repeated division converts a rational number into a finite continued fraction.
Integer part: The greatest integer less than or equal to a real number. The first term of a simple continued fraction is the value’s integer part.

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