## Hurwitz's Irrational Number Theorem

As Lagrange showed, any Irrational Number has an infinity of rational approximations which satisfy

 (1)

Similarly, if ,
 (2)

and if ,
 (3)

In general, even tighter bounds of the form
 (4)

can be obtained for the best rational approximation possible for an arbitrary irrational number , where the are called Lagrange Numbers and get steadily larger for each bad'' set of irrational numbers which is excluded.

See also Hurwitz's Irrational Number Theorem, Liouville's Rational Approximation Theorem, Liouville-Roth Constant, Markov Number, Roth's Theorem, Segre's Theorem, Thue-Siegel-Roth Theorem

References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 40, 1987.

Chandrasekharan, K. An Introduction to Analytic Number Theory. Berlin: Springer-Verlag, p. 23, 1968.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187-189, 1996.