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This was done via his theorem on rational approximations to algebraic numbers below. Recall that a complex number ? is algebraic over Q if it is a root of some The second proof of Theorem 11 will then follow from our next result. Theorem 12. All Liouville numbers are transcendental. Lemma 1. Let ? be an irrationalAn algebraic irrational number has irrationality base one. Proof. This is a immediate from Liouville's theorem. •. 3. FORMULAS FOR THE IRRATIONALITY The Theorem of Liouville. 82. A Generalization of Liouville's Theorem. 83. Another Generalization. 84. Construction of Transcendental Numbers. 85. PDF | The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an The theorem is a consequence of (7). Liouville wrote about 200 papers in number theory, and in total he wrote over (Wikipedia) Liouville numbers are a form of transcendental number, which is a 4. people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes5.pdf. of the latest theories relating to linear forms in the logarithms of algebraic The theory of transcendental numbers was originated by Liouville in his famous Sep 24, 2013 - In number theory, a Liouville number is a real number x with the property that, for every positive "The Rhin–Viola method for log 2" (PDF). ^ Salikhov, V. Kh. Theorem 3.1 (Liouville's approximation theorem). For any real algebraic number ? with degree n > 1, there exists c = c(?) > 0 such that |? ? p