For any algebraic number of degree , a rational approximation to must satisfy

for sufficiently large . Writing 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 depending only on such that for all integers and with ,

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.

CITE THIS AS:

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