TOPICS
Search

Hurwitz's Irrational Number Theorem

As Lagrange showed, any irrational number alpha has an infinity of rational approximations p/q which satisfy

|alpha-p/q|<1/(sqrt(5)q^2).
(1)

Furthermore, if there are no integers a,b,c,d with |ad-bc|=1 and alpha=(aalpha+b)/(dalpha+c) (corresponding to values of alpha associated with the golden ratio phi through their continued fractions), then

|alpha-p/q|<1/(sqrt(8)q^2),
(2)

and if values of alpha associated with the silver ratio 1+sqrt(2) are also excluded, then

|alpha-p/q|<5/(sqrt(221))1/(q^2).
(3)

In general, even tighter bounds of the form

|alpha-p/q|<1/(L_nq^2)
(4)

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

See also

Continued Fraction, Irrationality Measure, Lagrange Number, Liouville's Approximation Theorem, Markov Number, Roth's Theorem, Segre's Theorem

Explore with Wolfram|Alpha

References

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 145, 1997.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.

Referenced on Wolfram|Alpha

Hurwitz's Irrational Number Theorem

Cite this as:

Weisstein, Eric W. "Hurwitz's Irrational Number Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html

Subject classifications