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Liouville's Rational Approximation Theorem

For any Algebraic Number $x$ of degree $n>1$, a Rational approximation $x=p/q$ must satisfy

\left\vert{x-{p\over q}}\right\vert > {1\over q^{n+1}}

for sufficiently large $q$. Writing $r\equiv n+1$ leads to the definition of the Liouville-Roth Constant of a given number.

See also Lagrange Number (Rational Approximation), Liouville's Constant, Liouville Number, Liouville-Roth Constant, Markov Number, Roth's Theorem, Thue-Siegel-Roth Theorem


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.

© 1996-9 Eric W. Weisstein