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Liouville number. In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that. Liouville numbers are "almost rational ", and can thus be approximated "quite closely" by sequences of rational numbers.
May 24, 2024 · 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).
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.
- William L. Hosch
Liouville's constant, sometimes also called Liouville's number, is the real number defined by L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001... (OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial n!, and zeros everywhere else.
Mar 24, 2023 · 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: $$\alpha_1=\sum_ {n=1}^\infty2^ {-n!},$$. $$\alpha_2=\sum_ {n=1}^\infty (-1)^n2^ {-3^n},$$.
Liouville Numbers. Back in 1844, Joseph Liouville came up with this number: = 0.11000100000000000000000100…… (the digit is 1 if it is k! places after the decimal, and 0 otherwise.) It is a very interesting number because: it is irrational, and. it is not the root of any polynomial equation and so is not algebraic.
Aug 5, 2022 · They [Liouville numbers] are precisely those transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number. But what about those numbers with 2 < μ(xT) < ∞ 2 < μ ( x T) < ∞?
