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Liouville's Approximation Theorem

For any algebraic number x of degree n>2, a rational approximation p/q to x must satisfy

|x-p/q|>1/(q^n)

for sufficiently large q. Writing r=n leads to the definition of the irrationality measure of a given number. Apostol (1997) states the theorem in the slightly modified but equivalent form that there exists a positive constant C(x) depending only on x such that for all integers p and q with q>0,

|x-p/q|>(C(x))/(q^n).

See also

Algebraic Number, Dirichlet's Approximation Theorem, Irrationality Measure, Lagrange Number, Liouville's Constant, Liouville Number, Markov Number, Roth's Theorem, Siegel's Theorem

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References

Apostol, T. M. "Liouville's Approximation Theorem." §7.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 146-148, 1997.Courant, R. and Robbins, H. "Liouville's Theorem and the Construction of Transcendental Numbers." §2.6.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 104-107, 1996.

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Liouville's Approximation Theorem

Cite this as:

Weisstein, Eric W. "Liouville's Approximation Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LiouvillesApproximationTheorem.html

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