“Hurwitz's Irrational Number Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hurwitz's Irrational Number Theorem

释义

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 ApproximationTheorem, 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.

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