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| In [[mathematics]], a '''transcendental number''' is a (possibly [[complex number|complex]]) number that is not [[algebraic number|algebraic]]—that is, it is not a [[root of a function|root]] of a non-zero [[polynomial]] equation with [[rational number|rational]] [[coefficient]]s. The most prominent examples of transcendental numbers are [[Pi|π]] and ''[[E (mathematical constant)|e]]''. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, [[almost all]] [[real number|real]] and complex numbers are transcendental, since the algebraic numbers are [[countable]] while the sets of real and complex numbers are both [[uncountable]]. All real transcendental numbers are [[irrational number|irrational]], since all rational numbers are algebraic. The [[Conversion (logic)|converse]] is not true: not all irrational numbers are transcendental; e.g., the [[square root of 2]] is irrational but not a transcendental number, since it is a solution of the polynomial equation ''x''<sup>2</sup> − 2 = 0.
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| ==History==
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| The name "transcendental" comes from [[Gottfried Leibniz|Leibniz]] in his 1682 paper where he proved [[Sine|Sin]] ''x'' is not an [[algebraic function]] of ''x''.<ref>{{cite book|title=Leibnizens mathematische Schriften|author=Gottfried Wilhelm Leibniz, Karl Immanuel Gerhardt, Georg Heinrich Pertz|publisher=A. Asher & Co.|year=1858|volume=5|pages=97–98}}[http://books.google.com/books?id=ugA3AAAAMAAJ&pg=PA97]</ref><ref>{{cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994|page=74}}</ref> [[Leonhard Euler|Euler]] was probably the first person to define transcendental ''numbers'' in the modern sense.<ref>{{cite journal|doi=10.2307/2690369|title=Some Remarks and Problems in Number Theory Related to the Work of Euler|author=[[Paul Erdős]], Underwood Dudley|journal=Mathematics Magazine|volume=76|issue=5|date=December 1943|pages=292–299|jstor=2690369}}</ref>
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| [[Joseph Liouville]] first proved the existence of transcendental numbers in 1844,<ref>{{cite journal|title=On Transcendental Numbers|author=Aubrey J. Kempner|journal=Transactions of the American Mathematical Society|volume=17|issue=4|date=October 1916|pages=476–482|doi=10.2307/1988833|publisher=American Mathematical Society|jstor=1988833}}</ref> and in 1851 gave the first decimal examples such as the Liouville constant
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| :<math>\sum_{k=1}^\infty 10^{-k!} = 0.1100010000000000000000010000\ldots</math>
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| in which the ''n''th digit after the decimal point is 1 if ''n'' is equal to ''k''! (''k'' [[factorial]]) for some ''k'' and 0 otherwise.<ref>[http://mathworld.wolfram.com/LiouvillesConstant.html Weisstein, Eric W. "Liouville's Constant", MathWorld]</ref> Liouville showed that this number is what we now call a [[Liouville number]]; this essentially means that it can be more closely approximated by [[rational number]]s than can any irrational algebraic number. Liouville showed that all Liouville numbers are transcendental.<ref>{{cite journal|title=Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques|author=J. Liouville|journal=J. Math. Pures et Appl.|volume=16|year=1851|pages=133–142|url=http://www-mathdoc.ujf-grenoble.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf}}</ref>
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| [[Johann Heinrich Lambert]] conjectured that ''[[E (mathematical constant)|e]]'' and [[Pi|π]] were both transcendental numbers in his 1761 paper proving the number π is [[irrational number|irrational]]. The first number to be proven transcendental without having been specifically constructed for the purpose was ''e'', by [[Charles Hermite]] in 1873.
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| In 1874, [[Georg Cantor]] proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a [[Cantor's first uncountability proof|new method]] for constructing transcendental numbers.<ref>{{cite journal|title=Über eine Eigenschaft des Ingebriffes aller reelen algebraischen Zahlen|author=Georg Cantor|journal=J. Reine Angew. Math.|volume=77|year=1874|pages=258–262|url=http://www.digizeitschriften.de/main/dms/img/?PPN=GDZPPN002155583}}</ref> In 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers.<ref>{{cite journal|title=Ein Beitrag zur Mannigfaltigkeitslehre|author=Georg Cantor|journal=J. Reine Angew. Math.|volume=84|year=1878|pages=242–258|url=http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15}} (Cantor's construction builds a [[one-to-one correspondence]] between the set of transcendental numbers and the set of real numbers. In this article, Cantor only applies his construction to the set of irrational numbers. See p. 254.)</ref> Cantor's work established the ubiquity of transcendental numbers.
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| In 1882, [[Ferdinand von Lindemann]] published a proof that the number π is transcendental. He first showed that ''e'' to any nonzero algebraic power is transcendental, and since ''e''<sup>''i''π</sup> = −1 is algebraic (see [[Euler's identity]]), ''i''π and therefore π must be transcendental. This approach was generalized by [[Karl Weierstrass]] to the [[Lindemann–Weierstrass theorem]]. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving [[compass and straightedge]], including the most famous one, [[squaring the circle]].
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| In 1900, [[David Hilbert]] posed an influential question about transcendental numbers, [[Hilbert's seventh problem]]: If ''a'' is an algebraic number, that is not zero or one, and ''b'' is an irrational [[algebraic number]], is ''a''<sup>''b''</sup> necessarily transcendental? The affirmative answer was provided in 1934 by the [[Gelfond–Schneider theorem]]. This work was extended by [[Alan Baker (mathematician)|Alan Baker]] in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).<ref>J J O'Connor and E F Robertson: [http://www-history.mcs.st-andrews.ac.uk/Biographies/Baker_Alan.html Alan Baker]. The MacTutor History of Mathematics archive 1998.</ref>
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| ==Properties==
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| The set of transcendental numbers is [[uncountable|uncountably infinite]]. Since the polynomials with integer coefficients are [[countable]], and since each such polynomial has a finite number of [[root of a function|zeroes]], the [[algebraic number]]s must also be countable. But [[Cantor's diagonal argument]] proves that the real numbers (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable.
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| No [[rational number]] is transcendental and all real transcendental numbers are irrational. A rational number can be written as ''p''/''q'', where ''p'' and ''q'' are integers. Thus, ''p''/''q'' is the root of ''qx'' − ''p'' = 0. However, some [[irrational number]]s are not transcendental. For example, the [[square root of 2]] is irrational and not transcendental (because it is a solution of the polynomial equation ''x''<sup>2</sup> − 2 = 0). The same is true for the square root of other non-perfect squares.
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| Any non-constant [[algebraic function]] of a single variable yields a transcendental value when applied to a transcendental argument. For example, from knowing that π is transcendental, we can immediately deduce that numbers such as 5π, (π − 3)/√{{overline|2}}, (√{{overline|π}} − √{{overline|3}})<sup>8</sup> and (π<sup>5</sup> + 7)<sup>1/7</sup> are transcendental as well.
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| However, an algebraic function of several variables may yield an algebraic number when applied to transcendental numbers if these numbers are not [[algebraically independent]]. For example, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether π + ''e'', for example, is transcendental, though at least one of π + ''e'' and π''e'' must be transcendental. More generally, for any two transcendental numbers ''a'' and ''b'', at least one of ''a'' + ''b'' and ''ab'' must be transcendental. To see this, consider the polynomial (''x'' − ''a'')(''x'' − ''b'') = ''x''<sup>2</sup> − (''a'' + ''b'')''x'' + ''ab''. If (''a'' + ''b'') and ''ab'' were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an [[algebraically closed field]], this would imply that the roots of the polynomial, ''a'' and ''b'', must be algebraic. But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental.
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| The [[computable number|non-computable numbers]] are a [[strict subset]] of the transcendental numbers.
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| All [[Liouville number]]s are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its [[continued fraction]] expansion. Using a [[counting argument]] one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.
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| Using the explicit continued fraction expansion of ''e'', one can show that ''e'' is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). [[Kurt Mahler]] showed in 1953 that π is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms that are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals).<ref>{{cite journal|title=On the complexity of algebraic numbers, II. Continued fractions|author=Boris Adamczewski and Yann Bugeaud|journal=Acta Mathematica|volume=195|issue=1|date=March 2005|pages=1–20|doi=10.1007/BF02588048}}</ref>
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| A related class of numbers are [[closed-form number]]s, which may be defined in various ways, including rational numbers (and in some definitions all algebraic numbers), but also allow exponentiation and logarithm.
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| ==Numbers proven to be transcendental==
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| Numbers proven to be transcendental:
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| * ''[[e (mathematical constant)|e]]<sup>a</sup>'' if ''a'' is [[Algebraic number|Algebraic]] and nonzero (by the [[Lindemann–Weierstrass theorem]]).
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| *[[Pi|π]] (by the [[Lindemann–Weierstrass theorem]]).
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| * ''e''<sup>π</sup>, [[Gelfond's constant]], as well as ''e''<sup>−π/2</sup>=''i'' <sup>i</sup> (by the [[Gelfond–Schneider theorem]]).
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| * ''a<sup>b</sup>'' where ''a'' is algebraic but not 0 or 1, and ''b'' is irrational algebraic (by the Gelfond–Schneider theorem), in particular:
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| ::<math>2^\sqrt{2},</math>
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| :the [[Gelfond–Schneider constant]] (or Hilbert number).
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| *The Continued Fraction Constant, [[Carl Ludwig Siegel]] (1929)
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| :<math>
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| {1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}}
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| </math>
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| *[[trigonometric function|sin]](''a''), cos(''a'') and tan(''a''), and their multiplicative inverses csc(''a''), sec(''a'') and cot(''a''), for any nonzero algebraic number ''a'' (by the Lindemann–Weierstrass theorem).
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| *[[natural logarithm|ln]](''a'') if ''a'' is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
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| *''[[Lambert W Function|W]]''(''a'') if ''a'' is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem).
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| *[[gamma function|Γ]](1/3),<ref>Le Lionnais, F. Les nombres remarquables (ISBN 2-7056-1407-9). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, [http://mathworld.wolfram.com/TranscendentalNumber.html Transcendental Number]</ref> Γ(1/4),<ref name = "Chudnovsky">{{cite book | last=Chudnovsky | first=G. V. | title=Contributions to the Theory of Transcendental Numbers | isbn=0-8218-1500-8 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1984 }} via Wolfram Mathworld, [http://mathworld.wolfram.com/TranscendentalNumber.html Transcendental Number]</ref> and Γ(1/6).<ref name = "Chudnovsky"/>
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| *0.12345678910111213141516..., the [[Champernowne constant]].<ref>{{cite journal|author=K. Mahler|title=Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen|journal=Proc. Konin. Neder. Akad. Wet. Ser. A.|issue=40|year=1937|pages=421–428}}</ref><ref>Mahler (1976) p.12</ref>
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| *Ω, [[Chaitin's constant]] (since it is a non-computable number).<ref>{{cite book | title=Information and Randomness: An Algorithmic Perspective | series=Texts in Theoretical Computer Science | first=Cristian S. last=Calude | edition=2nd rev. and ext. | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-43466-6 | zbl=1055.68058 | page=239 }}</ref>
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| * The '''Fredholm number'''<ref>Allouche & Shallit (2003) pp.385,403</ref><ref name=Sha1999>{{cite book | editor1-first=Dennis A. | editor1-last=Hejhal | editor1-link=Dennis Hejhal | editor2-last=Friedman | editor2-first=Joel | editor3-last=Gutzwiller | editor3-first=Martin C. | editor3-link=Martin Gutzwiller | editor4-last=Odlyzko | editor4-first=Andrew M. | editor4-link=Andrew Odlyzko | title=Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15--26, 1996 | series=The IMA volumes in mathematics and its applications | volume=109 | publisher=[[Springer-Verlag]] | year=1999 | isbn=0-387-98824-6 | last=Shallit | first=Jeffrey | author1-link=Jeffrey Shallit | chapter=Number theory and formal languages | pages=547–570 }}</ref>
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| ::<math>\sum_{n=0}^\infty 2^{-2^n}</math>
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| :more generally, any number of the form
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| ::<math>\sum_{n=0}^\infty \beta^{2^n}</math>
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| :with 0 < |β| < 1 and β algebraic.<ref name=Lox1988>{{cite book | first=J. H. | last=Loxton | chapter=13. Automata and transcendence | title=New Advances in Transcendence Theory | editor1-link=Alan Baker (mathematician) | editor1-first=A. | editor1-last=Baker | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-33545-0 | zbl=0656.10032 | pages=215–228 }}</ref>
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| * The aforementioned Liouville constant
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| ::<math>\sum_{n=1}^\infty 10^{-n!};</math>
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| :more generally any number of the form
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| ::<math>\sum_{n=1}^\infty \beta^{n!}</math>
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| :with 0 < |β| < 1 and β algebraic
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| * The [[Prouhet–Thue–Morse constant]].<ref>{{cite journal | first=Kurt | last=Mahler | authorlink=Kurt Mahler | title=Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen | journal=[[Math. Annalen]] | volume=101 | year=1929 | pages=342–366 | jfm=55.0115.01 }}</ref><ref>Allouche & Shallit (2003) p.387</ref>
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| * Any number for which the digits with respect to some fixed base form a [[Sturmian word]].<ref>{{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}</ref>
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| * For β > 1
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| ::<math>\sum_{k=0}^\infty 10^{-\left\lfloor \beta^{k} \right\rfloor};</math>
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| :where <math>\beta\mapsto\lfloor \beta \rfloor</math> is the [[floor function]].
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| ==Numbers which may or may not be transcendental==
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| Numbers for which it is currently unknown whether they are transcendental: they have neither been proven to be algebraic, nor proven to be transcendental:
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| * Most sums, products, powers, etc. of the number π and the [[E (mathematical constant)|number ''e'']], e.g. π + ''e'', π − ''e'', π''e'', π/''e'', π<sup>π</sup>, ''e''<sup>''e''</sup>, π<sup>''e''</sup>, π<sup>√{{overline|2}}</sup>, ''e''<sup>π<sup>2</sup></sup> are not known to be rational, algebraic irrational or transcendental. Notable exceptions are π + ''e''<sup>π</sup>, π''e''<sup>π</sup> and ''e''<sup>π√''{{overline|n}}''</sup> (for any positive integer ''n'') which have been proven to be transcendental.<ref>{{MathWorld|IrrationalNumber|Irrational Number}}</ref><ref>[http://mr.crossref.org/iPage/?doi=10.1070%2FSM1996v187n09ABEH000158 Modular functions and transcendence questions, Yu. V. Nesterenko, Sbornik: Mathematics(1996), 187(9):1319]</ref>
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| * The [[Euler–Mascheroni constant]] ''γ'' (which has not been proven to be irrational).
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| * [[Catalan's constant]], also not known to be irrational.
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| * [[Apéry's constant]], ζ(3) (which [[Apéry]] proved is irrational)
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| * The [[Riemann zeta function]] at other odd integers, ζ(5), ζ(7), ... (not known to be irrational.)
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| * The [[Feigenbaum constants]], δ and α.
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| Conjectures:
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| * [[Schanuel's conjecture]],
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| * [[Four exponentials conjecture]].
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| ==Sketch of a proof that ''e'' is transcendental==
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| The first proof that [[E (mathematical constant)|the base of the natural logarithms, ''e'']], is transcendental dates from 1873. We will now follow the strategy of [[David Hilbert]] (1862–1943) who gave a simplification of the original proof of [[Charles Hermite]]. The idea is the following:
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| Assume, for purpose of finding a contradiction, that ''e'' is algebraic. Then there exists a finite set of integer coefficients ''c''<sub>0</sub>, ''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' satisfying the equation:
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| :<math>c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0, \qquad c_0, c_n \neq 0.</math>
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| Now for a positive integer ''k'', we define the following polynomial:
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| :<math> f_k(x) = x^{k} \left [(x-1)\cdots(x-n) \right ]^{k+1},</math>
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| and multiply both sides of the above equation by
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| :<math>\int^{\infty}_{0} f_k e^{-x}\,dx,</math>
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| to arrive at the equation: | |
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| :<math>c_{0} \left (\int^{\infty}_{0} f_k e^{-x}\,dx\right )+ c_1e\left ( \int^{\infty}_{0}f_k e^{-x}\,dx\right )+\cdots+ c_{n}e^{n} \left (\int^{\infty}_{0}f_k e^{-x}\,dx\right ) = 0.</math>
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| This equation can be written in the form
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| :<math>P+Q=0</math>
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| where
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| :<math>P =c_{0}\left ( \int^{\infty}_{0}f_k e^{-x}\,dx\right )+ c_{1}e\left (\int^{\infty}_{1}f_k e^{-x}\,dx\right )+ c_{2}e^{2}\left (\int^{\infty}_{2}f_k e^{-x}\,dx\right ) +\cdots+ c_{n}e^{n}\left (\int^{\infty}_{n}f_k e^{-x}\,dx\right ) </math>
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| :<math>Q=c_{1}e\left (\int^{1}_{0} f_k e^{-x}\,dx\right )+c_{2}e^{2} \left (\int^{2}_{0} f_k e^{-x}\,dx\right )+\cdots+c_{n}e^{n}\left (\int^{n}_{0} f_k e^{-x}\,dx \right ) </math>
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| '''Lemma 1.''' For an appropriate choice of ''k'', <math>\tfrac{P}{k!}</math> is a non-zero integer.
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| <blockquote>'''Proof.''' Each term in ''P'' is an integer times a sum of factorials, which results from the relation
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| :<math>\int^{\infty}_{0}x^{j}e^{-x}\,dx=j!</math>
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| which is valid for any positive integer ''j'' (consider the [[Gamma function]]).
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| It is non-zero because for every ''a'' satisfying 0< ''a'' ≤ ''n'', the integrand in
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| :<math>c_{a}e^{a}\int^{\infty}_{a} f_k e^{-x}\,dx</math>
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| is ''e<sup>−x</sup>'' times a sum of terms whose lowest power of ''x'' is ''k''+1 after substituting ''x'' for ''x'' - ''a'' in the integral. Then this becomes a sum of integrals of the form
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| :<math>\int^{\infty}_{0}x^{j}e^{-x}\,dx</math>
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| with ''k''+1 ≤ ''j'', and it is therefore an integer divisible by (''k''+1)!. After dividing by ''k!'', we get zero [[Modular arithmetic|modulo]] (''k''+1). However, we can write: | |
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| :<math>\int^{\infty}_{0} f_k e^{-x}\,dx = \int^{\infty}_{0} \left ([(-1)^{n}(n!)]^{k+1}e^{-x}x^k + \cdots \right ) dx</math>
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| and thus
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| :<math>{\frac{1}{k!}}c_{0}\int^{\infty}_{0} f_k e^{-x}\,dx = c_{0}[(-1)^{n}(n!)]^{k+1} \qquad \mod (k+1).</math>
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| By choosing ''k'' so that ''k''+1 is prime and larger than ''n'' and |''c''<sub>0</sub>|, we get that <math>\tfrac{P}{k!}</math> is non-zero modulo (''k''+1) and is thus non-zero.</blockquote>
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| '''Lemma 2.''' <math>\left|\tfrac{Q}{k!}\right|<1</math> for sufficiently large ''k''.
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| <blockquote> '''Proof.''' Note that
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| :<math>f_k e^{-x} = x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x} = \left ([x(x-1)\cdots(x-n)]^k \right ) \left ((x-1)\cdots(x-n)e^{-x}\right )</math>
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| Using upper bounds G and H for <math>|x(x-1)\cdots(x-n)|</math> and <math>|(x-1)\cdots(x-n)e^{-x}|</math> on the [[interval (mathematics)|interval]] [0,''n''] we can infer that
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| :<math>|Q|<G^{k}H(|c_{1}|e+2|c_{2}|e^{2}+\cdots+n|c_{n}|e^{n})</math>
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| and since
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| :<math>\lim_{k\to\infty}\frac{G^k}{k!}=0</math>
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| it follows that | |
| :<math>\lim_{k\to\infty}\frac{Q}{k!}=0</math>
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| which is sufficient to finish the proof of this lemma.
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| </blockquote>
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| Noting that one can choose ''k'' so that both Lemmas hold we get the contradiction we needed to prove the transcendence of ''e''.
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| ===The transcendence of π===
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| A similar strategy, different from Lindemann's original approach, can be used to show that the [[Pi|number π]] is transcendental. Besides the [[gamma-function]] and some estimates as in the proof for ''e'', facts about [[symmetric polynomial]]s play a vital role in the proof.
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| For detailed information concerning the proofs of the transcendence of π and ''e'' see the references and external links.
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| ==Mahler's classification==
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| [[Kurt Mahler]] in 1932 partitioned the transcendental numbers into 3 classes, called '''S''', '''T''', and '''U'''.<ref name=Bug250>Bugeaud (2012) p.250</ref> Definition of these classes draws on an extension of the idea of a [[Liouville number]] (cited above).
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| ===Measure of irrationality of a real number===
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| One way to define a Liouville number is to consider how small a given real number '''x''' makes linear polynomials |''qx'' − ''p''| without making them exactly 0. Here ''p'', ''q'' are integers with |''p''|, |''q''| bounded by a positive integer ''H''.
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| Let ''m''(''x'', 1, ''H'') be the minimum non-zero absolute value these polynomials take and take:
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| :<math>\omega(x, 1, H) = - \frac{\log m(x, 1, H)}{\log H}</math>
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| :<math>\omega(x, 1)= \limsup_{H\to\infty} \omega(x,1,H).</math>
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| ω(''x'', 1) is often called the '''measure of irrationality''' of a real number ''x''. For rational numbers, ω(''x'', 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. [[Roth's theorem]] says that irrational real algebraic numbers have measure of irrationality 1.
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| ===Measure of transcendence of a complex number===
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| Next consider the values of polynomials at a complex number ''x'', when these polynomials have integer coefficients, degree at most ''n'', and [[Height of a polynomial|height]] at most ''H'', with ''n'', ''H'' being positive integers.
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| Let ''m''(''x'',''n'',''H'') be the minimum non-zero absolute value such polynomials take at ''x'' and take:
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| :<math>\omega(x, n, H) = - \frac{\log m(x, n, H)}{n\log H}</math>
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| :<math>\omega(x, n)= \limsup_{H\to\infty} \omega(x,n,H).</math>
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| Suppose this is infinite for some minimum positive integer ''n''. A complex number ''x'' in this case is called a '''U number''' of degree ''n''.
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| Now we can define
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| :<math>\omega (x) =\limsup_{n\to\infty}\omega(x,n).</math>
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| ω(''x'') is often called the '''measure of transcendence''' of ''x''. If the ω(''x'',''n'') are bounded, then ω(''x'') is finite, and ''x'' is called an '''S number'''. If the ω(''x'',''n'') are finite but unbounded, ''x'' is called a '''T number'''. ''x'' is algebraic if and only if ω(''x'') = 0.
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| Clearly the Liouville numbers are a subset of the U numbers. [[William LeVeque]] in 1953 constructed U numbers of any desired degree.<ref name="Baker, p. 86">Baker (1975) p. 86.</ref><ref name=LV172>LeVeque (2002) p.II:172</ref> The [[Liouville numbers]] and hence the U numbers are uncountable sets. They are sets of measure 0.<ref>Burger and Tubbs, p. 170.</ref>
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| T numbers also comprise a set of measure 0.<ref>Burger and Tubbs, p. 172.</ref> It took about 35 years to show their existence. [[Wolfgang M. Schmidt]] in 1968 showed that examples exist. It follows that [[almost all]] complex numbers are S numbers.<ref name=Bug251/> Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:<ref>LeVeque (2002) pp.II:174–186</ref><ref>Burger and Tubbs, p. 182.</ref> this shows that ''e'' is an S number and gives a proof of the transcendence of π. The most that is known about π is that it is not a U number. Many other transcendental numbers remain unclassified.
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| Two numbers ''x'', ''y'' are called '''algebraically dependent''' if there is a non-zero polynomial ''P'' in 2 indeterminates with integer coefficients such that ''P''(''x'', ''y'') = 0. There is a powerful theorem that 2 complex numbers that are algebraically dependent belong to the same Mahler class.<ref name=LV172/><ref>Burger and Tubbs, p. 163.</ref> This allows construction of new transcendental numbers, such as the sum of a Liouville number with ''e'' or π.
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| It is often speculated that S stood for the name of Mahler's teacher [[Carl Ludwig Siegel]] and that T and U are just the next two letters.
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| ===Koksma's equivalent classification===
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| [[Jurjen Koksma]] in 1939 proposed another classification based on approximation by algebraic numbers.<ref name=Bug250/><ref name=Baker87>Baker (1975) p.87</ref>
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| Consider the approximation of a complex number ''x'' by algebraic numbers of degree ≤ ''n'' and height ≤ ''H''. Let α be an algebraic number of this finite set such that |''x'' − α| has the minimum positive value. Define ω*(''x'',''H'',''n'') and ω*(''x'',''n'') by:
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| :<math>|x-\alpha| = H^{-n\omega^*(x,H,n)-1}.</math>
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| :<math>\omega^*(x,n) = \limsup_{H\to\infty} \omega^*(x,n,H).</math>
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| If for a smallest positive integer ''n'', ω*(''x'',''n'') is infinite, ''x'' is called a '''U*-number''' of degree ''n''.
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| If the ω*(''x'',''n'') are bounded and do not converge to 0, ''x'' is called an '''S*-number''',
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| A number ''x'' is called an '''A*-number''' if the ω*(''x'',''n'') converge to 0.
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| If the ω*(''x'',''n'') are all finite but unbounded, ''x'' is called a '''T*-number''',
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| Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.<ref name=Baker87/> The ''A*''-numbers are the algebraic numbers.<ref name=Bug251>Bugeaud (2012) p.251</ref>
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| ===LeVeque's construction===
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| Let
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| :<math>\lambda= \tfrac{1}{3} + \sum_{k=1}^\infty 10^{-k!}</math>
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| It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n.<ref>Baker(1979), p. 90.</ref>
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| This construction can be improved to create an uncountable family of U-numbers of degree ''n''. Let ''Z'' be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of ''Z'' is uncountable. Deleting any of the subsets of ''Z'' from the series for λ creates uncountably many distinct Liouville numbers, whose nth roots are U-numbers of degree ''n''.
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| ===Type===
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| The [[supremum]] of the sequence {ω(''x'', ''n'')} is called the '''type'''. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.<ref name="Baker, p. 86"/>
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| ==See also==
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| * [[Transcendence theory]], the study of questions related to transcendental numbers
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| ==Notes==
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| {{reflist|30em}}
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| ==References==
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| {{inline citations|date=June 2013}}
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| * [[David Hilbert]], "Über die Transcendenz der Zahlen ''e'' und <math>\pi</math>", ''Mathematische Annalen'' '''43''':216–219 (1893).
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| * A. O. Gelfond, ''Transcendental and Algebraic Numbers'', Dover reprint (1960).
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| * {{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 }}
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| * {{cite book | last=Mahler | first=Kurt | authorlink=Kurt Mahler | title=Lectures on Transcendental Numbers | series=Lecture Notes in Mathematics | volume=546 | publisher=[[Springer-Verlag]] | year=1976 | isbn=3-540-07986-6 | zbl=0332.10019 }}
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| * {{cite book | zbl=0482.10047 | last=Sprindzhuk | first=Vladimir G. | title=Metric theory of Diophantine approximations | others=Transl. from the Russian and ed. by Richard A. Silverman. With a foreword by Donald J. Newman | series=Scripta Series in Mathematics | publisher=John Wiley & Sons | year=1979 | isbn= }}
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| * {{cite book | last = LeVeque | first = William J. | authorlink = William J. LeVeque | title = Topics in Number Theory, Volumes I and II | publisher = Dover Publications | location = New York | year = 2002 |origyear = 1956 | isbn = 978-0-486-42539-9 }}
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| * {{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
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| * {{cite book | last1=Burger | first1=Edward B. | last2=Tubbs | first2=Robert | title=Making transcendence transparent. An intuitive approach to classical transcendental number theory | location=New York, NY | publisher=[[Springer-Verlag]] | year=2004 | isbn=0-387-21444-5 | zbl=1092.11031 }}
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| * [[Peter M Higgins]], "Number Story" Copernicus Books, 2008, ISBN 978-1-84800-001-8.
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| * {{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-11169-0 | zbl=pre06066616 }}
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| ==External links==
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| * {{en icon}} [http://planetmath.org/encyclopedia/EIsTranscendental.html Proof that ''e'' is transcendental]
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| * {{en icon}} [http://deanlm.com/transcendental/ Proof that the Liouville Constant is transcendental]
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| * {{de icon}} [http://www.mathematik.uni-muenchen.de/~fritsch/euler.pdf Proof that ''e'' is transcendental (PDF)]
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| * {{de icon}} [http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf Proof that <math>\pi</math> is transcendental (PDF)]
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| {{Number Systems}}
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| {{DEFAULTSORT:Transcendental Number}}
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| [[Category:Transcendental numbers|*]]
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| [[Category:Articles containing proofs]]
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