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. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental,[1] thus establishing the existence of transcendental numbers for the first time.[2]
It is known that π and e are not Liouville numbers.[3]
^Joseph Liouville (May 1844). "Mémoires et communications". Comptes rendus de l'Académie des Sciences (in French). 18 (20, 21): 883–885, 910–911.
^
Baker, Alan (1990). Transcendental Number Theory (paperback ed.). Cambridge University Press. p. 1.
In number theory, a Liouvillenumber is a real number x {\displaystyle x} with the property that, for every positive integer n {\displaystyle n} , there...
irrational algebraic number, and this class of numbers are called Liouville numbers, named in his honour. Liouville showed that all Liouville numbers are transcendental...
approximated" in Liouville's work relates to a certain exponent. He showed that if α is an algebraic number of degree d ≥ 2 and ε is any number greater than...
1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse Liouville (née Balland). Liouville gained admission to the École Polytechnique...
formula Liouville function Liouville dynamical system Liouville field theory Liouville gravity Liouville integrability Liouville measure Liouvillenumber Liouville...
Divisor function Liouville function Partition function (number theory) Integer partition Bell numbers Landau's function Pentagonal number theorem Bell series...
291,559}8528404201690728\ldots } Then α is a Liouvillenumber and is absolutely abnormal. No rational number is normal in any base, since the digit sequences...
The existence of transcendental numbers was first established by Liouville (1844, 1851). Hermite proved in 1873 that e is transcendental and Lindemann...
The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product...
we have the notion of integrability in the Liouville sense. (See the Liouville–Arnold theorem.) Liouville integrability means that there exists a regular...
the proof and showed that π is not the square root of a rational number. Liouville (1840) showed that neither e nor e2 can be a root of an integer quadratic...
theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the...
_{183212}272243682859\ldots } This sum is transcendental because it is a Liouvillenumber. Like tetration, there is currently no accepted method of extension...
from Hardy & Wright (1972, pp. 159–160, 178–179) Garibaldi 2008. Also, Liouville's theorem can be used to "produce as many examples of transcendental numbers...
Liouville numbers. The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number...
singular continuous components of the spectral measure for some Sturm–Liouville operators with square summable potential". Journal of Differential Equations...
initialism for Wentzel–Kramers–Brillouin. It is also known as the LG or Liouville–Green method. Other often-used letter combinations include JWKB and WKBJ...
also. The existence of transcendental numbers was first established by Liouville (1844, 1851). Later, Georg Cantor (1873) proved their existence by a different...
pure states evolve in time, the von Neumann equation (also known as the Liouville–von Neumann equation) describes how a density operator evolves in time...
real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number. This knowledge enabled Liouville, in...
continuous on their domains. Elementary functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. An algebraic treatment of elementary...
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