Diophantine approximation

Diophantine approximation studies how closely real numbers can be approximated by rational numbers, often with bounds on denominators and errors.

Rational approximation: The use of rational numbers to approximate real or other numerical values. Diophantine approximation is the number-theoretic study of rational approximation with explicit error and denominator bounds.
Rational number: A number expressible as the quotient of two integers with nonzero denominator. Rational numbers are the approximants whose closeness to a target is measured.
Pietro Cataldi: An Italian mathematician who published an early systematic treatment of continued fractions in 1613. His work helped establish continued fractions as a tool for approximating numbers.
Irrationality of the square root of two: The theorem that no ratio of integers has square equal to two. Approximation arguments illuminate why rational fractions cannot equal this classic irrational number.
Hurwitz's theorem (number theory): A theorem stating that every irrational real number has infinitely many rational approximations with error less than 1 over the square root of five times the denominator squared. Its constant is sharp, prompting questions about approximation under stricter conditions.
Continued fraction: An expansion of a number as an integer part followed by a sequence of fractional reciprocals. Its convergents give exceptionally accurate rational approximations and describe their quality.
Irrational number: A real number that cannot be expressed as the quotient of two integers. Irrational targets cannot be matched exactly by rational approximants.
Christiaan Huygens: A Dutch mathematician and scientist who used continued fractions to compute planetary distances and construct a planetarium. His practical calculations exemplified the value of accurate rational approximations.
Irrationality of pi: The theorem that pi is not a rational number. Strong rational-approximation bounds have been used to establish irrationality.
Lagrange spectrum: The set of values measuring the best asymptotic quadratic rational-approximation constants of real numbers. Its intricate structure organizes how approximation quality varies across irrationals.

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