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Topic: Lindemann Weierstrass theorem

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	Karl Weierstrass - Wikipedia, the free encyclopedia
		During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions.
		Weierstrass was interested in the soundness of calculus.
		Weierstrass also formulated the definition of limit and derivative still in use today.
	www.wikipedia.org /wiki/Karl_Weierstrass (399 words)

	EZGeography - Ferdinand von Lindemann
		Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a zero of any polynomial with rational coefficients.
		Before the publication of Lindemann's proof, it was known that if π is transcendental, then the ancient and celebrated problem of squaring the circle by straightedge and compass could not be solved.
		While a professor at the University of Königsberg, Lindemann acted as supervisor for the doctoral thesis of David Hilbert.
	www.ezgeography.com /encyclopedia/Ferdinand_von_Lindemann (202 words)

	Search Encyclopedia.com
		theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.
		Weierstrass, Karl Wilhelm Theodor Weierstrass, Karl Wilhelm Theodorkärl vĬl´hĕlm tā´ōdōr vī´ershträs, 1815-97, German mathematician.
		He originated Taylor's theorem, a formula important in differential calculus, which relates a function to its derivatives by means of a power series.
	www.encyclopedia.com /searchpool.asp?target=Lindemann-Weierstrass+theorem (528 words)

	Encyclopedia: Karl Weierstrass (Site not responding. Last check: 2007-11-07)
		Weierstrass was interested in the goldfish of cheese.
		In mathematics, the Weierstrass function was the first example found of a kind of function with the property that it is continuous everywhere but differentiable nowhere.
		In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It states that such a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic...
	www.nationmaster.com /encyclopedia/Karl-Weierstrass (927 words)

	Ferdinand von Lindemann -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Lindemann was born in (The English royal house that reigned from 1714 to 1901 (from George I to Victoria)) Hanover, (A republic in central Europe; split into East German and West Germany after World War II and reunited in 1990) Germany.
		Before the publication of Lindemann's proof, it was known that if π is transcendental, then the ancient and celebrated problem of (Click link for more info and facts about squaring the circle) squaring the circle by straightedge and compass could not be solved.
		While a professor at the University of Königsberg, Lindemann acted as supervisor for the doctoral thesis of (German mathematician (1862-1943)) David Hilbert.
	www.absoluteastronomy.com /encyclopedia/F/Fe/Ferdinand_von_Lindemann.htm (308 words)

	PlanetMath: Gelfond's theorem (Site not responding. Last check: 2007-11-07)
		This is perhaps the most useful result in determining whether a number is algebraic or transcendental.
		The theorem is also known as the Gelfond-Schneider Theorem.
		This is version 6 of Gelfond's theorem, born on 2003-01-31, modified 2004-02-12.
	planetmath.org /encyclopedia/3952.html (96 words)

	PlanetMath: Lindemann-Weierstrass theorem (Site not responding. Last check: 2007-11-07)
		An equivalent version of the theorem states that if
		Schanuel's conjecture is a generalization of the Lindemann-Weierstrass theorem.
		This is version 5 of Lindemann-Weierstrass theorem, born on 2004-04-21, modified 2004-04-23.
	planetmath.org /encyclopedia/LindemannsTheorem.html (86 words)

	[No title]
		Liouville's Theorem is nice in that there are uncountably many Liouville numbers, but their Lebesgue measure is zero.
		For historical reference, this was the first theorem on algebraic independence of a set of numbers.
		Weierstrass proved the result we just saw, which Lindemann stated but did not prove.
	br.endernet.org /~loner/basicnumbertheory/kimjintrotranscnt.txt (2537 words)

	Pi - the free encyclopedia (Site not responding. Last check: 2007-11-07)
		This was proven in 1761 by Johann Heinrich Lambert.
		In fact,the number is transcendental, as was proven by Ferdinand von Lindemann in 1882.
		Lindemann proved that π is transcendental(the Lindemann-Weierstrass theorem)
	www.free-web-encyclopedia.com /?t=Pi (2145 words)

	Lindemann–Weierstrass theorem - Wikipedia, the free encyclopedia
		In reference [1] below the theorem is stated in the following form: If α
		The theorem is named for Ferdinand von Lindemann, who proved the particular result that π is transcendental, and Karl Weierstrass.
		This page was last modified 20:23, 18 October 2005.
	en.wikipedia.org /wiki/Lindemann-Weierstrass_theorem (322 words)

	Lindemann-Weierstrass theorem (Site not responding. Last check: 2007-11-07)
		The Lindemann-Weierstrass theorem is a theorem in mathematics that is very useful in establishing the transcendence of numbers.
		The transcendence of e and &pi are direct corollaries of this theorem.
		Since the set of all algebraic numbers forms a field, this implies that πi and 2πi are also algebraic.
	www.theezine.net /l/lindemann-weierstrass-theorem.html (145 words)

	Lindemann Weierstrass theorem (Site not responding. Last check: 2007-11-07)
		In mathematics, the Lindemann-Weierstrass theorem is veryuseful in establishing the transcendence of numbers.
		Since the set of all algebraic numbers forms a field, this implies that πi and2πi are also algebraic.
		The theorem is named for Ferdinand von Lindemann and Karl Weierstraß.
	www.therfcc.org /lindemann-weierstrass-theorem-203134.html (135 words)

	Knowledge King - Transcendental number (Site not responding. Last check: 2007-11-07)
		The first number to be proved transcendental without having been specifically constructed to achieve this was e, by Charles Hermite in 1873.
		In 1882, Carl Louis Ferdinand von Lindemann published a proof that the number &pi is transcendental.
		In 1874, Georg Cantor found the argument described above establishing the ubiquity of transcendental numbers.
	www.knowledgeking.net /encyclopedia/t/tr/transcendental_number.html (340 words)

	[No title]
		xciv / civ On a theorem of Legendre in the theory of continued fractions.
		xc /xcii Von einem kettenbruch von Eulers und einem theorem von Wallis.
		xcviii On a theorem od Alladi and Gordon and the Gaussian polynomials.
	www.geocities.com /furmend/Ba_Bz.txt (7631 words)

	Karl Weierstrass - Encyclopedia, History, Geography and Biography
		He was born in Ostenfelde, Westphalia (today Germany) and died in Berlin, Germany.
		This page was last modified 09:56, 25 May 2005.
		The article about Karl Weierstrass contains information related to Karl Weierstrass, Selected papers, See also and External links.
	www.arikah.net /encyclopedia/Karl_Weierstrass (305 words)

	Mathematics of Computation (Site not responding. Last check: 2007-11-07)
		112--115 Carsten Thomassen The Jordan-Schönflies theorem and the classification of surfaces.
		48--54 Dorina Mitrea and Marius Mitrea A generalization of a theorem of Euler 55--58 Petru Mironescu and Lauren\ctiu Panaitopol The existence of a triangle with prescribed angle bisector lengths.
		651--658 A. Ageev Sierpi\'nski's theorem is deducible from Euler and Dirichlet.
	www.math.utah.edu:8080 /ftp/pub/tex/bib/toc/amermathmonthly1990.html (5496 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Gelfond-Schneider's Theorem: If a is an algebraic number not equal to 0 and 1 and if b is an irrationnal algebraic number, then a
		Baker's Theorem: If a complex number z is such that z and e
		Theorem (Rivoal, 2000): There are infinitely many integers n such that
	www.institut.math.jussieu.fr /~khemira/tr-a.html (121 words)

	Transcendental number
		In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental.
		where a ≠ 0,1 is algebraic and b is algebraic but not rational (Gel'fond-Schneider theorem).
		The general case of Hilbert's seventh problem, where b is not algebraic, remains open.
	encyclopedie-en.snyke.com /articles/transcendental_number.html (325 words)

	Lindemann-Weierstrass theorem - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-11-07)
		Lindemann-Weierstrass theorem - Encyclopedia, History, Geography and Biography
		This page was last modified 01:39, 1 Jun 2005.
		The article about Lindemann-Weierstrass theorem contains information related to Lindemann-Weierstrass theorem, Transcendence of e and π, p-adic conjecture and References.
	www.arikah.net /encyclopedia/Lindemann-Weierstrass_theorem (378 words)

	Bertrand (Site not responding. Last check: 2007-11-07)
		The theorem of Siegel and Shidlovsky, which generalizes the Lindemann - Weierstrass theorem, then asserts that the numbers $f_1(b),..., f_n(b)$ are algebraically independent over $Q$.
		Although based on deep results from the arithmetic theory of differential equations, its principle is remarkably simple.
		By a Fourier transformation, one deduces from the Katz-Honda theory in characteristic p, Chudnovsky's theorem on $G$-operators, and Andre's theorem on their p-adic radii of convergence, that the minimal differential operator annihilating an $E$-function $F$ has large exponents at any algebraic zero of large order of $F$.
	math.berkeley.edu /~coleman/Sems/NTS/Sem-Fall98/bertrand.html (187 words)

	Encyclopedia: Lindemann-Weierstrass theorem (Site not responding. Last check: 2007-11-07)
		Updated 158 days 6 hours 20 minutes ago.
		In abstract algebra, the transcendence degree of a field extension L/K is a certain rather coarse measure of the size of the extension.
		Carl Louis Ferdinand von Lindemann (April 12, 1852 - March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.
	www.nationmaster.com /encyclopedia/Lindemann_Weierstrass-theorem (567 words)

	Ferdinand Von Lindemann Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-11-07)
		Looking For ferdinand von lindemann - Find ferdinand von lindemann and more at Lycos Search.
		Find ferdinand von lindemann - Your relevant result is a click away!
		Look for ferdinand von lindemann - Find ferdinand von lindemann at one of the best sites the Internet has to offer!
	www.karr.net /search/encyclopedia/Ferdinand_von_Lindemann (419 words)

	UK lindemann websites UK
		The Lindemann Trust is offering fellowships for research in the USA.
		Mary Braddon - critical material - avoidance of scandal in the Victorian sensation novel' in Journal of Victorian Culture, Spring 1997, 2:1, pp 27-41 R.B. Lindemann: Dramatic disappearances: Mary Elizabeth Braddon and the staging of theatrical character in Journal of Victorian
		Later that year I was awarded a Lindemann Fellowship to work with Carl Wieman on BEC at the Joint Institute for Laboratory Astrophysics in Boulder, Colorado.
	www.splut.co.uk /sub/l/lindemann.html (634 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		An \htmladdnormallink{equivalent}{http://planetmath.org/encyclopedia/EquivalenceOfForcingNotions.html} version of the theorem states that if $\alpha_1,\ldots,\alpha_n$ are distinct algebraic numbers over $\mathbb{Q}$, then $e^{\alpha_1},\ldots,e^{\alpha_n}$ are linearly independent over $\mathbb{Q}$.
		Some immediate consequences of this theorem: \begin{itemize} \item If $\alpha$ is a non-zero algebraic number over $\mathbb{Q}$, then $e^{\alpha}$ is \htmladdnormallink{transcendental}{http://planetmath.org/encyclopedia/Algebraic.html} over $\mathbb{Q}$.
		\htmladdnormallink{Schanuel's conjecture}{http://planetmath.org/encyclopedia/SchanuelsConjecutre.html} is a generalization of the Lindemann-Weierstrass theorem.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/FEM-2004-08-16/TeX/12D99--LindemannWeierstrassTheorem.tex (177 words)

	Awesome Library - Mathematics
		"A theorem is a statement which can be proven true within some logical framework.
		Proving theorems is a central activity of mathematics." Provides 212 theorems.
		Provides some of the more important math theorems, including Riemann hypothesis, Continuum hypothesis, P=NP, Pythagorean theorem, Central limit theorem, Fundamental theorem of calculus, Fundamental theorem of algebra, Fundamental theorem of arithmetic, Fundamental theorem of projective geometry, Classification theorems of surfaces, and Gauss-Bonnet theorem.
	www.awesomelibrary.org /Classroom/Mathematics/College_Math/College_Math.html (371 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		\li Squares as a basis; sums of 3 squares; Cauchy's polygonal number theorem.
		\li Ordered and unordered partitions; Euler's pentagonal number theorem; Rogers-Ramanujan identities; elementary estimates for the number of unordered partitions.
		\li Irrationality of $e, \pi,\ \ln 2$ and $\zeta(3)$; transcendence of $e$ and $\pi$; Lindemann-Weierstrass theorem; Gelfond-Schneider theorem.
	www.math.rutgers.edu /grad/oral/sv-hack.tex (247 words)

	Lindemann-Weierstrass theorem Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-07)
		Lindemann-Weierstrass theorem Info - Bored Net - Boredom
		To show the transcendence of e, note that if e were algebraic, there would exist rational_numbers β
		The theorem is named for Carl Louis Ferdinand von Lindemann and Karl Weierstraß
	www.borednet.com /e/n/encyclopedia/l/li/lindemann_weierstrass_theorem.html (145 words)

	The Ultimate Talk:Lindemann-Weierstrass theorem Dog Breeds Information Guide and Reference
		The Ultimate Talk:Lindemann-Weierstrass theorem Dog Breeds Information Guide and Reference
		Any nonzero algebraic number α gives us a set {α} which is trivially a linearly independent set over the rationals, and hence eα is immediately seen to be transcendental.
		I also understand the urge to condense everything to be elegant, but I think expressing things out explicitly in terms of linear combinations is still helpful for people who might not be able to immediately mentally untangle "linearly independent" or "algebraically independent".
	www.dogluvers.com /dog_breeds/Talk:Lindemann-Weierstrass_theorem (159 words)

	CONK! Encyclopedia: List_of_number_theory_topics (Site not responding. Last check: 2007-11-07)
		This is a list of number theory topics, by CONK!
		Proofs of Fermat's theorem on sums of two squares
		Proof that the sum of the reciprocals of the primes diverges
	www.conk.com /search/encyclopedia.cgi?q=List_of_number_theory_topics (40 words)

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