Hurwitz's Irrational Number Theorem gives the best rational approximation possible for an arbitrary irrational
number as

The are called Lagrange numbers and get steadily larger for each ``bad'' set of irrational numbers which is excluded.

Exclude | ||

1 | none | |

2 | ||

3 |

Lagrange numbers are of the form

where is a Markov Number. The Lagrange numbers form a Spectrum called the Lagrange Spectrum.

**References**

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

© 1996-9

1999-05-26