
	Where results make sense

Topic: Liouville number

Related Topics

Joseph Liouville

Pi

Madhava

Aryabhatta

In the News (Mon 13 Feb 12)

EXECUTIVE POWER

Iran

ANALYSIS

Pakistan

Family compact

	Liouville number - Wikipedia, the free encyclopedia
		A Liouville number can thus be approximated "quite closely" by a sequence of rational numbers.
		In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, and he provided an example of a Liouville number, thus establishing the existence of transcendental numbers for the first time.
		Establishing that a given number is a Liouville number provides a useful tool for proving a given number is transcendental.
	en.wikipedia.org /wiki/Liouville_number (840 words)

	Transcendental number - Wikipedia, the free encyclopedia
		Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be particularly well approximated by rational numbers.
		All Liouville numbers are transcendental, however not all transcendental numbers are Liouville numbers.
		Any Liouville number must have unbounded terms in its continued fraction expression, and so using a counting argument one can show that there exist transcendental numbers which are not Liouville.
	en.wikipedia.org /wiki/Transcendental_number (1130 words)

	PlanetMath: example of transcendental number
		The following is a classical application of Liouville's approximation theorem.
		Next we use the theorem to construct a transcendental number.
		This is version 4 of example of transcendental number, born on 2005-02-16, modified 2005-02-16.
	planetmath.org /encyclopedia/ExampleOfTranscendentalNumber.html (175 words)

	Liouville biography (Site not responding. Last check: 2007-10-08)
		Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France.
		In 1837 Liouville was appointed to lecture at the Collège de France as a substitute for Biot.
		Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33.
	www-groups.dcs.st-and.ac.uk /history/Biographies/Liouville.html (1850 words)

	Joseph Liouville
		In 1831, Liouville was appointed to a number of private schools and to the Ecole Centrale.
		In 1838, Liouville was appointed Professor of Analysis and Mechanics at the Ecole Polytechnique.
		Liouville contributed to differential geometry studying conformal transformations.
	www.stetson.edu /~efriedma/periodictable/html/Lu.html (742 words)

	Irrational number (Site not responding. Last check: 2007-10-08)
		The discovery of irrational numbers is usually to Pythagoras more specifically to the Pythagorean Hippasus of Metapontum who produced a (most likely geometrical) of the irrationality of the square root of 2.
		For the nineteenth century it remained complete the theory of complex numbers to separate irrationals into algebraic and to prove the existence of transcendental numbers and to make a scientific study a subject which had remained almost dormant Euclid the theory of irrationals.
		Continued fractions closely related to irrational numbers (and to Cataldi 1613) received attention at the of Euler and at the opening of the century were brought into prominence through the of Joseph Louis Lagrange.
	www.freeglossary.com /Irrational_number (1734 words)

	Math Forum - Ask Dr. Math
		I know that a transcendental number is a number such that it can't be put in normal algeabraic forms, such as pi.
		In spite of the fact that most numbers are transcendental (a fact proved by Cantor), proving that any particular number is transcendental is usually quite difficult.
		The number e is also transcendental, and so is a^b, if a and b are algebraic and b is irrational (like 2 to the square root of 2 power, for example).
	mathforum.org /library/drmath/view/53910.html (427 words)

	Re: Irrationality / transcendence of these numbers. How to find out?
		Note that Liouville numbers are required to be irrational.
		A number can be approximated by rationals to order k if there exists a constant C>0 and infinitely many pairs of integers (m,n) such that 0
		Liouville numbers are those numbers which can be appoximated to all orders.
	www.usenet.com /newsgroups/sci.math/msg23456.html (358 words)

	Ivars Peterson's MathTrek - Absolutely Abnormal
		A number is said to be "absolutely normal" if its digits are normal not only to base 10 but also to every integer base greater than or equal to 2.
		At the same time, although it is known that almost all real numbers are absolutely normal, no one has yet proved even a single, "naturally occurring" real number to be absolutely normal.
		Liouville had introduced such numbers as examples of transcendental numbers--real numbers that are not roots of polynomial equations with integer coefficients.
	www.maa.org /mathland/mathtrek_11_05_01.html (583 words)

	Liouville, Joseph (Site not responding. Last check: 2007-10-08)
		He was the first to prove (1844) the existence of transcendental numbers, and he constructed an infinite class of such numbers.
		The Liouville theorem concerning the measure-preserving property of the Hamiltonian dynamics is basic to statistical mechanics and measure theory.
		In analysis Liouville was the first to deduce the theory of doubly periodic functions from general theorems (including his own) in the theory of analytic functions of a complex variable.
	www.phy.bg.ac.yu /web_projects/giants/liouvi~1.htm (309 words)

	Irrational number In mathematics (Site not responding. Last check: 2007-10-08)
		In mathematics, an irrational number is any real number that is not a rational number, i.e., one that cannot be written as a fraction,a/b, with a and b integers, and b not zero.
		Some irrational numbers are algebraic numbers, such as √2, the square root of two, and 3√5, the cube root of 5; others are transcendental numbers such as π and e.
		Continued fractions, closely related to irrational numbers (and due to Cataldi, 1613), received attention at the hands of Euler, and at the opening of the nineteenth century were brought into prominence through the writings of Joseph Louis Lagrange.
	www.metu.edu.tr /~e128327/Irras.htm (1677 words)

	E (mathematical constant) - Gurupedia
		The exponential function is important because it is, up to multiplication by a scalar, the unique function which is its own derivative and is hence commonly used to model growth or decay processes.
		The number e is known to be irrational and even transcendental.
		Liouville number); the proof was given by Charles Hermite in 1873.
	www.gurupedia.com /e/e_/e_(mathematical_constant).htm (608 words)

	Search Results for Transcendental (Site not responding. Last check: 2007-10-08)
		These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.
		Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
		Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers".
	www-gap.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Transcendental&CONTEXT=1 (2156 words)

	Classification of numbers : overview
		In 1844, Joseph Liouville (1809-1882) [12] was the first to prove the existence of such numbers and he gave an important and elementary characterization of a category of transcendental numbers, the Liouville numbers:
		This number is not rational because it's clearly not periodic and satisfies the characterization theorem because there so many 0 between the 1 that it can be approximated by a rational number very easily.
		is finite (the number of roots of a polynomial is finite).
	numbers.computation.free.fr /Constants/Miscellaneous/classification.html (1196 words)

	Discrete Math, First Problem Set (June 23) REU 2003
		A transcendental number is a number that is not algebraic.
		In contrast to Liouville, Cantor did not produce any explicit transcendental numbers; yet he proved that the overwhelming majority of real numbers are transcendental, by introducing the hierarchy of infinite of cardinalities.
		Definition 2 An algebraic integer is an algebraic number which is a zero of a monic polynomial with integer coefficients.
	people.cs.uchicago.edu /~laci/reu03/notes1 (460 words)

	E (mathematical constant) (Site not responding. Last check: 2007-10-08)
		The mathematical constant e (occasionally called Euler's number after the Swiss mathematician Leonhard Euler or Napier's constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function.
		Alongside the number π and the imaginary unit i e is one of the most important mathematical constants.
		The exponential function is important because it up to multiplication by a scalar the function which is its own derivative and is hence commonly used to growth or decay processes.
	www.freeglossary.com /Eulers_number (1030 words)

	PlanetMath: decimal expansion
		In every case, the period length is a factor of the number
		Also any irrational number has a unique decimal expansion, but it is non-periodic; for example the Liouville's number
		Cross-references: transcendental, irrational number, rational, digit, base, Euler's totient function, factor, prime factors, power, denominator, fraction, coprime, satisfy, series, integers, positive, rational number
	planetmath.org /encyclopedia/DecimalExpansion.html (219 words)

	Search Results for Liouville (Site not responding. Last check: 2007-10-08)
		Liouville was in many ways someone who Libri should not have competed with, for he was an outstanding mathematician who could usually come up with a more elegant proof of Libri's results than he could himself.
		Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
		However Liouville addressed the meeting after Lame and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true.
	www-history.mcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Liouville&CONTEXT=1 (3102 words)

	About "The 15 Most Famous Transcendental Numbers" (Site not responding. Last check: 2007-10-08)
		Numbers like e = 2.718..., pi = 3.1415..., and 2^(sqrt 3) are transcendental.
		These numbers are not the solutions of polynomial expressions having rational coefficients.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/view/6382.html (112 words)

	[No title]
		In particular the existence of an actually infinite number of finite numbers, the concept of countability, and the different cardinality of natural numbers and real numbers are contradicted.
		Sequence of all natural numbers and its diagonal number 12212310341004510005......N has not an infinite number of places and, therefore, is a natural number not contained in the original list.
		Concluding, the natural numbers do not include their diagonal number N. So every diagonal number R constructed from the real numbers can be paired with a diagonal number N constructed from the natural numbers.
	www.fh-augsburg.de /~mueckenh/Infinity/ZE040327.doc (1148 words)

	id:A092874 - OEIS Search Results
		The famous Liouville number is defined so that its n-th fractional decimal digit is 1 iff there exists k, such that k!=n.
		The binary Liouville number is defined similarly, but as a binary number:its n-th fractional binary digit is 1 iff there exists k, such that k!=n.
		According to the definitions introduced in A092855 and A051006, this number is "the binary mapping" of the sequence of factorials (Axxx).
	www.research.att.com /~njas/sequences/A092874 (136 words)

	Math G Mission College Santa Clara
		His goal in his research was to "renew arithmetic as Plato had understood it, as the doctrine of whole numbers and their properties." To this end Fermat worked to develop a stricter method of mathematical solutions and disregarded the methods of Diophantus.
		Liouville became best known for his work in fractional calculus and discovered transcendental numbers that removed the dependence on continued fractions.
		Liouville failed to prove the uniqueness of factorization, however this failure was instrumental to discoveries made by Kummer.
	www.missioncollege.org /depts/math/jackson.htm (2225 words)

	Liouville's Theorem (Site not responding. Last check: 2007-10-08)
		PlanetMath: proof of the fundamental theorem of algebra (Liouville's theorem)...
		Liouville's theorem: a bounded analytic function is constant...
		Liouville's Theorem -- from Eric Weisstein's World of Physics...
	www.scienceoxygen.com /math/461.html (193 words)

	Canisius College - Abstracts (Site not responding. Last check: 2007-10-08)
		Abstract: The French mathematician Joseph Liouville, in a series of eighteen papers published between 1858 and 1865, announced without proof a number of amazing elementary arithmetic formulae, from which many results in elementary number theory can be deduced.
		Williams has published many research papers, mostly in number theory, and is the coauthor or coeditor of eight books including "The Collected Papers of Sarvadaman Chowla" in three volumes (with James G. Huard of Canisius College), and most recently "Introductory Algebraic Number Theory" (with Saban Alaca of Carleton University).
		Gonek's main research interests are in the field of analytic number theory, particularly multiplicative number theory, the theory of the Riemann zeta-function, L-functions, and the distribution of prime numbers.
	www.canisius.edu /maa/abstracts.asp (4220 words)

	Mathematical Constants (Site not responding. Last check: 2007-10-08)
		It is a shame that complex numbers have dropped out of some A level syllabuses because they open the door to a whole new world Basic Definitions, Introduction to complex numbers.
		It is also intimately connected to the Fibonnaci numbers Fibonacci Numbers and the Golden Section 1, 1, 2, 3, 5, 8, 13, 21, 34 where each term is the sum of the previous two.
		A googol is larger than any known estimate of the number of atoms in the universe and thus it would be physically impossible to write out the even larger googolplex since there are aren't the atoms available to make up the zeroes.
	dspace.dial.pipex.com /town/way/po28/maths/constant.htm (2115 words)

	Math Trek: Absolutely Abnormal, Science News Online, Nov. 3, 2001 (Site not responding. Last check: 2007-10-08)
		Mathematician Greg Martin of the University of Toronto recently turned his attention to the opposite extreme—real numbers that are normal to no base whatsoever.
		Martin's formulation of this number and proof of its absolute abnormality involved so-called Liouville numbers, named for Joseph Liouville (1809—1882).
		Liouville had introduced such numbers as examples of transcendental numbers—real numbers that are not roots of polynomial equations with integer coefficients.
	www.sciencenews.org /articles/20011103/mathtrek.asp (664 words)

Try your search on: Qwika (all wikis)