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Badly Approximable


Every irrational number x has an approximation constant c(x) defined by

 c(x)=liminf_(q->infty)q|qx-p|,

where p=nint(qx) is the nearest integer to qx and liminf is the infimum limit. The quantity q|qx-p| measures how well x is approximated by the rational number p/q.

x is said to be badly approximable if c(x)>0. An irrational number is badly approximable iff the terms a_n of its continued fraction [a_0;a_1,...] are bounded. Since quadratic surds have periodic continued fractions, they are badly approximable. The maximum possible value of c(x) is 1/sqrt(5), attained for example at x=(sqrt(5)-1)/2=1/phi, where phi is the golden ratio.


See also

Continued Fraction, Irrational Number, Rational Approximation, Rational Number

This entry contributed by Keith Briggs

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References

Burger, E. B. Exploring the Number Jungle: A Journey Into Diophantine Analysis. Providence, RI: Amer. Math. Soc., 2000.

Referenced on Wolfram|Alpha

Badly Approximable

Cite this as:

Briggs, Keith. "Badly Approximable." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BadlyApproximable.html

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