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  • Liouville Number -- from Wolfram MathWorld
    A Liouville number is a transcendental number which has very close rational number approximations An irrational number beta is called a Liouville number if, for each n, there exist integers p>0 and q>1 such that 0<|beta-p q|<1 (q^n) Note that the first inequality is true by definition, since it follows immediately from the fact that beta is irrational and hence cannot be equal to p q for any
  • Liouville number | Irrational, Transcendental, Real | Britannica
    Liouville number, in algebra, an irrational number α such that for each positive integer n there exists a rational number p q for which p q < |α − (p q)| < 1 qn All Liouville numbers are transcendental numbers—that is, numbers that cannot be expressed as the solution (root) of a polynomial
  • Liouvilles Theorem (Number Theory) - ProofWiki
    Let x x be an irrational number that is algebraic of degree n n Then there exists a constant c> 0 c> 0 (which can depend on x x) such that: for every pair p, q ∈ Z p, q ∈ Z with q ≠ 0 q ≠ 0 Liouville numbers are transcendental Let r1,r2, …,rk r 1, r 2, …, r k be the rational roots of a polynomial P P of degree n n that has x x as a root
  • 5 The Beginning of Transcendental Numbers - University of South Carolina
    For Liouville’s proof, we define Definition 4 A real number α is a Liouville number if for every positive integer n, there are integers a and b with b > 1 such that 0 < α − a b < 1 bn In a moment, we will show that Liouville numbers exist The second proof of Theorem 11 will then follow from our next result Theorem 12
  • Transcendental Numbers - Stanford University
    5 Liouville Numbers Definition:Lis a Liouville number if for every n∈Z+ there exists p q ∈Q such that, 0 < α− p q < 1 qn Proposition Lis Liouville if and only if µ(L) = ∞ Proof Let µ(L) = ∞ then for any n∈Z+ there must be a k>nsuch that L is k-approximable because µ(L) >n Therefore, there is a solution p q ∈Q (in
  • Liouvilles Theorem on Diophantine Approximation and Liouville Number
    Definition 2 1(Liouville Number) A number α∈R is called a Liouville number if for any n∈N, there exist a,b∈Z and b>1 such that 0 < α− a b < 1 bn (2) By definition, every Liouville number is approximated nicely by rational numbers to some extent In view of Theorem1 1, this suggests that every Liouville number is transcen-dental
  • Liouville number - Encyclopedia of Mathematics
    The fact that a Liouville number is transcendental (cf Transcendental number) follows from the Liouville theorem (cf Liouville theorems) These numbers were studied by J Liouville [1] Examples of Liouville numbers are: $$\alpha_1=\sum_ {n=1}^\infty2^ {-n!},$$ $$\alpha_2=\sum_ {n=1}^\infty (-1)^n2^ {-3^n},$$
  • Introduction to Liouville Numbers - University of Białystok
    The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approxi-mated “quite closely” by a sequence of rational numbers A real number xis a Liouville number iff for every positive integer n, there exist integers pand qsuch that q>1 and 0 < x− p q < 1 qn
  • Liouville number - PlanetMath. org
    A Liouville number is an irrational number l l such that for any integer n> 0 n> 0 there is a pair of integers j j and k k such that the inequality holds All Liouville numbers are transcendental numbers, but not all transcendental numbers are Liouville numbers The first example given by Joseph Liouville was a number of the form
  • Liouville number - HandWiki
    Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental However, not every transcendental number is a Liouville number The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be





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