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Rational Approximation

A rational approximation of a real number alpha is a rational number m/n chosen close to alpha. For each positive integer n, there is an integer m for which

|alpha-m/n|<=1/n.
(1)

If alpha is irrational and k is any positive integer, there is a fraction m/n with n<=k and for which

|alpha-m/n|<=1/(nk).
(2)

Furthermore, there are an infinite number of fractions m/n for which

|alpha-m/n|<=1/(n^2)
(3)

(Hilbert and Cohn-Vossen 1999, pp. 40-44).

Hurwitz has shown that for an irrational number zeta

|zeta-h/k|<1/(ck^2),
(4)

there are infinitely many rational numbers h/k if 0<c<=sqrt(5), but if c>sqrt(5), there are some zeta for which this approximation holds for only finitely many h/k.

See also

Badly Approximable, Dirichlet's Approximation Theorem, Duffin-Schaeffer Conjecture, Hurwitz's Irrational Number Theorem, Irrationality Measure, Kronecker's Approximation Theorem, Lagrange Number, Liouville's Approximation Theorem, Markov Number, Roth's Theorem, Segre's Theorem, Weak Inhomogeneous Duffin-Schaeffer Conjecture

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References

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 41, 1999.

Referenced on Wolfram|Alpha

Rational Approximation

Cite this as:

Weisstein, Eric W. "Rational Approximation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RationalApproximation.html

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