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{{Short description|In mathematics, a non-algebraic number}}
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In [[mathematics]], a '''transcendental number''' is a [[real number|real]] or [[complex number]] that is not [[algebraic number|algebraic]]: that is, not the [[Zero of a function|root]] of a non-zero [[polynomial]] with [[integer]] (or, equivalently, [[rational number|rational]]) [[coefficient]]s. The best-known transcendental numbers are {{mvar|[[Pi|π]]}} and {{mvar|[[e (mathematical constant)|e]]}}.<ref>{{cite web |first=Cliff |last=Pickover |title=The 15&nbsp;most famous transcendental numbers |website=sprott.physics.wisc.edu |url=http://sprott.physics.wisc.edu/pickover/trans.html |access-date=2020-01-23}}</ref><ref>{{cite book |last1=Shidlovskii |first1=Andrei B. |date=June 2011 |title=Transcendental Numbers |publisher=Walter de Gruyter |isbn=9783110889055 |page=1}}</ref> The quality of a number being transcendental is called '''transcendence'''.

Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental, transcendental numbers are not rare: indeed, [[almost all]] real and complex numbers are transcendental, since the algebraic numbers form a [[countable set]], while the [[set (mathematics)|set]] of [[real numbers]] {{tmath|\R}} and the set of [[complex number]]s {{tmath|\C}} are both [[uncountable set]]s, and therefore larger than any countable set.

All '''transcendental real numbers''' (also known as '''real transcendental numbers''' or '''transcendental irrational numbers''') are [[irrational number]]s, since all [[rational numbers]] are algebraic.<ref name=numbers>{{cite book |last1=Bunday |first1=B. D. |last2=Mulholland |first2=H. |title=Pure Mathematics for Advanced Level |date=20 May 2014 |publisher=Butterworth-Heinemann |isbn=978-1-4831-0613-7 |url=https://books.google.com/books?id=02_iBQAAQBAJ |access-date=21 March 2021 |language=en}}</ref><ref>{{cite journal |last1=Baker |first1=A. |title=On Mahler's classification of transcendental numbers |journal=Acta Mathematica |date=1964 |volume=111 |pages=97–120 |doi=10.1007/bf02391010 |s2cid=122023355 |doi-access=free }}</ref><ref>{{cite arXiv |last1=Heuer |first1=Nicolaus |last2=Loeh |first2=Clara |title=Transcendental simplicial volumes |date=1 November 2019 |class=math.GT |eprint=1911.06386 }}</ref><ref>{{cite encyclopedia |title=Real number |department=mathematics |url=https://www.britannica.com/science/real-number |access-date=2020-08-11 |encyclopedia=Encyclopædia Britannica |lang=en}}</ref> The [[Converse (logic)|converse]] is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, [[Irrational number#Algebraic|algebraic irrational]], and transcendental real numbers.<ref name=numbers/> For example, the [[square root of 2]] is an irrational number, but it is not a transcendental number as it is a [[Zero_of_a_function#Polynomial_roots|root of the polynomial]] equation {{math|''x''<sup>2</sup> − 2 {{=}} 0}}. The [[golden ratio]] (denoted <math>\varphi</math> or <math>\phi</math>) is another irrational number that is not transcendental, as it is a root of the polynomial equation {{math|''x''<sup>2</sup> − ''x'' − 1 {{=}} 0}}.

==History==
The name "transcendental" comes {{ety|la|trānscendere|to climb over or beyond, surmount}},<ref>{{cite encyclopedia |title=transcendental |dictionary=[[Oxford English Dictionary]] |url=http://www.oed.com/view/Entry/204606}} ''s.v.''</ref> and was first used for the mathematical concept in [[Gottfried Leibniz|Leibniz's]] 1682 paper in which he proved that {{math|sin ''x''}} is not an [[algebraic function]] of {{mvar|x}}.<ref>{{harvnb|Leibniz|Gerhardt|Pertz|1858|pp=97–98}}; {{harvnb|Bourbaki|1994|p=74}}</ref> [[Leonhard Euler|Euler]], in the eighteenth century, was probably the first person to define transcendental ''numbers'' in the modern sense.<ref>{{harvnb|Erdős|Dudley|1983}}</ref>

[[Johann Heinrich Lambert]] conjectured that {{mvar|[[E (mathematical constant)|e]]}} and [[Pi|{{mvar|π}}]] were both transcendental numbers in his 1768 paper proving the number {{mvar|π}} is [[irrational number|irrational]], and proposed a tentative sketch proof that {{mvar|π}} is transcendental.<ref>{{harvnb|Lambert|1768}}</ref>

[[Joseph Liouville]] first proved the existence of transcendental numbers in 1844,<ref name=Kempner>{{harvnb|Kempner|1916}}</ref> and in 1851 gave the first decimal examples such as the [[Liouville number|Liouville constant]] <!-- "Decimal Liouville constant" uses 10^-n! | "Binary Liouville constant" uses 2^-n! //-->

<math display=block>
\begin{align}
  L_b &= \sum_{n=1}^\infty 10^{-n!} \\[2pt]
  &= 10^{-1} + 10^{-2} + 10^{-6} + 10^{-24} + 10^{-120} + 10^{-720} + 10^{-5040} + 10^{-40320} + \ldots \\[4pt]
  &= 0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000000000000000000000000000000000000000000000000\ \ldots 
\end{align}</math>

in which the {{mvar|n}}th digit after the decimal point is {{math|1}} if {{mvar|n}} = {{mvar|''k''!}} ({{mvar|k}} [[factorial]]) for some {{mvar|k}} and {{math|0}} otherwise.<ref>{{cite web| url = http://mathworld.wolfram.com/LiouvillesConstant.html| title = Weisstein, Eric W. "Liouville's Constant", MathWorld}}</ref> In other words, the {{mvar|n}}th digit of this number is 1 only if {{mvar|n}} is one of {{math|1=1! = 1, 2! = 2, 3! = 6, 4! = 24}}, etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by [[rational number]]s than can any irrational algebraic number, and this class of numbers is called the [[Liouville number]]s. Liouville showed that all Liouville numbers are transcendental.<ref>{{harvnb|Liouville|1851}}</ref>

The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was {{mvar|e}}, by [[Charles Hermite]] in 1873.

In 1874 [[Georg Cantor]] proved that the algebraic numbers are countable and the real numbers are uncountable. He also gave a [[Cantor's first set theory article|new method]] for constructing transcendental numbers.<ref>{{harvnb|Cantor|1874}}; {{harvnb|Gray|1994}}</ref> Although this was already implied by his proof of the countability of the algebraic numbers, Cantor also published a construction that proves there are as many transcendental numbers as there are real numbers.{{efn|
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.<ref>{{harvnb|Cantor|1878|p=254}}</ref>
}}
Cantor's work established the ubiquity of transcendental numbers.

In 1882 [[Ferdinand von Lindemann]] published the first complete proof that {{mvar|π}} is transcendental. He first proved that {{math|''e{{sup|a}}''}} is transcendental if {{mvar|a}} is a non-zero algebraic number. Then, since {{math|''e{{sup|iπ}}'' {{=}} −1}} is algebraic (see [[Euler's identity]]), {{math|''iπ''}} must be transcendental. But since {{math|''i''}} is algebraic, {{mvar|π}} must therefore be transcendental. This approach was generalized by [[Karl Weierstrass]] to what is now known as the [[Lindemann–Weierstrass theorem]]. The transcendence of {{mvar|π}} implies that geometric constructions involving [[compass and straightedge]] only cannot produce certain results, for example [[squaring the circle]].

In 1900 [[David Hilbert]] posed a question about transcendental numbers, [[Hilbert's seventh problem]]: If {{mvar|a}} is an [[algebraic number]] that is not 0 or 1, and {{mvar|b}} is an irrational algebraic number, is {{math|''a{{sup|b}}''}} 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>{{cite report |first=Alan |last=Baker |year=1998 |title=J.J. O'Connor and E.F. Robertson |type=biographies |series=The MacTutor History of Mathematics archive |publisher=[[University of St. Andrew's]] |place=[[St Andrews|St. Andrew's, Scotland]] |website=www-history.mcs.st-andrews.ac.uk |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Baker_Alan.html}}</ref>

==Properties==
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be [[irrational number|irrational]], since a [[rational number]] is the root of an integer polynomial of [[degree of a polynomial|degree]] one.<ref>{{harvnb|Hardy|1979}}</ref> The set of transcendental numbers is [[uncountable|uncountably infinite]]. Since the polynomials with rational coefficients are [[countable]], and since each such polynomial has a finite number of [[zero of a function|zeroes]], the [[algebraic number]]s must also be countable. However, [[Cantor's diagonal argument]] proves that the real numbers (and therefore also the [[complex number]]s) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both [[subset]]s to be countable. This makes the transcendental numbers uncountable.

No [[rational number]] is transcendental and all real transcendental numbers are irrational. The [[irrational number]]s contain all the real transcendental numbers and a subset of the algebraic numbers, including the [[quadratic irrational]]s and other forms of algebraic irrationals.

Applying any non-constant single-variable [[algebraic function]] to a transcendental argument yields a transcendental value. For example, from knowing that {{mvar|π}} is transcendental, it can be immediately deduced that numbers such as <math>5\pi</math>, <math>\tfrac{\pi - 3}{\sqrt{2}}</math>, <math>(\sqrt{\pi}-\sqrt{3})^8</math>, and <math>\sqrt[4]{\pi^5+7}</math> are transcendental as well.

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, {{mvar|π}} and {{math|(1 − ''π'')}} are both transcendental, but {{math|''π'' + (1 − ''π'') {{=}} 1}} is obviously not. It is unknown whether {{math|''e'' + ''π''}}, for example, is transcendental, though at least one of {{math|''e'' + ''π''}} and {{mvar|eπ}} must be transcendental. More generally, for any two transcendental numbers {{mvar|a}} and {{mvar|b}}, at least one of {{math|''a'' + ''b''}} and {{mvar|ab}} must be transcendental. To see this, consider the polynomial {{math|(''x'' − ''a'')(''x'' − ''b'') {{=}} ''x''<sup>2</sup> − (''a'' + ''b'') ''x'' + ''a b''}}&nbsp;. If {{math| (''a'' + ''b'')}} and {{mvar|a b}} 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, {{mvar|a}} and {{mvar|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.

The [[non-computable numbers]] are a strict subset of the transcendental numbers.

All [[Liouville number]]s are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its [[simple continued fraction]] expansion. Using a [[Cantor's diagonal argument|counting argument]] one can show that there exist transcendental numbers which have bounded partial quotients and hence are not Liouville numbers.

Using the explicit continued fraction expansion of {{mvar|e}}, one can show that {{mvar|e}} is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). [[Kurt Mahler]] showed in 1953 that {{mvar|π}} is also not a Liouville number. It is conjectured that all infinite continued fractions with bounded terms, that have a "simple" structure, and that are not eventually periodic are transcendental<ref>{{harvnb|Adamczewski|Bugeaud|2005}}</ref> (in other words, algebraic irrational roots of at least third degree polynomials do not have apparent pattern in their continued fraction expansions, since eventually periodic continued fractions correspond to quadratic irrationals, see [[Hermite's problem]]).

==Numbers proven to be transcendental==
Numbers proven to be transcendental:

* [[pi|{{mvar|π}}]] (by the [[Lindemann–Weierstrass theorem]]).
* {{math|<math>e^a</math>}} if {{Math|<math>a</math>}} is [[Algebraic number|algebraic]] and nonzero (by the Lindemann–Weierstrass theorem), in particular [[E (mathematical constant)|Euler's number]] {{mvar|e}}.
* {{math|<math> e^{\pi \sqrt n} </math>}} where {{math|<math>n</math>}} is a positive integer; in particular [[Gelfond's constant]] {{math|<math>e^\pi</math>}} (by the [[Gelfond–Schneider theorem]]).
* Algebraic combinations of {{math|<math>\pi </math>}} and {{math|<math> e^{\pi \sqrt n} , n\in\mathbb Z^{+}</math>}} such as {{math|<math> \pi + e^{\pi}</math>}} and {{math|<math> \pi  e^{\pi}</math>}} (following from their [[algebraic independence]]).<ref name=":2">{{Cite journal |last=Nesterenko |first=Yu V |date=1996-10-31 |title=Modular functions and transcendence questions |url=https://iopscience.iop.org/article/10.1070/SM1996v187n09ABEH000158 |journal=Sbornik: Mathematics |volume=187 |issue=9 |pages=1319–1348 |doi=10.1070/SM1996v187n09ABEH000158 |bibcode=1996SbMat.187.1319N |issn=1064-5616}}</ref>
* {{math|<math>a^b</math>}} where {{Math|<math>a</math>}} is algebraic but not 0 or 1, and {{Math|<math>b</math>}} is irrational algebraic, in particular the [[Gelfond–Schneider constant]] <math>2^{\sqrt{2}}</math> (by the Gelfond–Schneider theorem).
* The [[natural logarithm]] {{math|<math>\ln(a)</math>}} if {{math|<math>a</math>}} is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
* {{math|<math>\log_b(a)</math>}} if {{math|<math>a</math>}} and {{math|<math>b</math>}} are positive integers not both powers of the same integer, and {{math|<math>a</math>}} is not equal to 1 (by the Gelfond–Schneider theorem).
* All numbers of the form <math>\pi + \beta_1 \ln (a_1) + \cdots + \beta_n \ln (a_n)</math> are transcendental, where <math>\beta_j</math> are algebraic for all <math>1 \leq j \leq n</math> and <math>a_j</math> are non-zero algebraic for all <math>1 \leq j \leq n</math> (by [[Baker's theorem]]).

*The [[trigonometric functions]] {{math|<math>\sin(x), \cos(x), ...</math>}} and their [[Hyperbolic functions|hyperbolic counterparts]], for any nonzero algebraic number {{math|<math>x</math>}}, expressed in [[radian]]s (by the Lindemann–Weierstrass theorem).
*Non-zero results of the [[inverse trigonometric functions]] {{math|<math>\arcsin(x), \arccos(x), ...</math>}} and their [[Inverse hyperbolic functions|hyperbolic counterparts]], for any algebraic number {{math|<math>x</math>}} (by the Lindemann–Weierstrass theorem).
*<math>\pi^{-1}{\arctan(x)}</math>, for rational {{math|<math>x</math>}} such that <math>x \notin \{0,\pm{1}\}</math>.<ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |title=Transcendental Number |url=https://mathworld.wolfram.com/TranscendentalNumber.html |access-date=2023-08-09 |website=mathworld.wolfram.com |language=en}}</ref>
*The [[Fixed-point iteration#Attracting fixed points|fixed point]] of the cosine function (also referred to as the [[Dottie number]] {{math|<math>d</math>}}) – the unique real solution to the equation {{math|<math>\cos(x)=x</math>}}, where {{math|<math>x</math>}} is in radians (by the Lindemann–Weierstrass theorem).<ref name="wolfram_dottie">{{cite web|last1=Weisstein|first1=Eric W.|title=Dottie Number|url=http://mathworld.wolfram.com/DottieNumber.html|website=Wolfram MathWorld|publisher=Wolfram Research, Inc.|access-date=23 July 2016}}</ref>
*{{math|<math>W(a)</math>}} if {{math|<math>a</math>}} is algebraic and nonzero, for any branch of the [[Lambert W function|Lambert W Function]] (by the Lindemann–Weierstrass theorem), in particular the [[omega constant]] {{math|Ω}}.
* {{math|<math>W(r,a)</math>}} if both {{math|<math>a</math>}} and the order {{math|<math>r</math>}} are algebraic such that <math>a \neq 0</math>, for any branch of the generalized Lambert W function.<ref>{{Cite arXiv |eprint=1408.3999 |class=math.CA |first1=István |last1=Mező |first2=Árpád |last2=Baricz |title=On the generalization of the Lambert W function |date=June 22, 2015}}</ref>
* {{math|<math>\sqrt x _s</math>}}, the [[Tetration#Square super-root|square super-root]] of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem).
* Values of the [[gamma function]] of rational numbers that are of the form <math>\Gamma(n/2),\Gamma(n/3),\Gamma(n/4)</math> or <math>\Gamma(n/6)</math>.<ref>{{Cite book |last=Chudnovsky |first=G. |title=Contributions to the theory of transcendental numbers |date=1984 |publisher=American Mathematical Society |isbn=978-0-8218-1500-7 |series=Mathematical surveys and monographs |location=Providence, R.I |language=en, ru}}</ref>
* Algebraic combinations of {{math|<math>\pi </math>}} and {{math|<math>\Gamma(1/3)</math>}} or of {{math|<math>\pi </math>}} and {{math|<math>\Gamma(1/4)</math>}} such as the [[lemniscate constant]] <math>\varpi</math> (following from their respective algebraic independences).<ref name=":2" />
* The values of [[Beta function]] <math>\Beta(a,b)</math> if <math>a, b</math> and <math>a+b</math> are non-integer rational numbers.<ref name=":3">{{Cite web |last=Waldschmidt |first=Michel |date=September 7, 2005 |title=Transcendence of Periods: The State of the Art |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TranscendencePeriods.pdf |website=webusers.imj-prg.fr}}</ref>
* The [[Bessel function|Bessel function of the first kind]] {{math|<math>J_\nu(x)</math>}}, its first derivative, and the quotient <math>\tfrac{J'_\nu (x)}{J_\nu (x)}</math> are transcendental when ''{{math|<math>\nu</math>}}'' is rational and ''{{math|<math>x</math>}}'' is algebraic and nonzero,<ref>{{cite book |last1=Siegel |first1=Carl L. |title=On Some Applications of Diophantine Approximations |chapter=Über einige Anwendungen diophantischer Approximationen: Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1 |date=2014 |publisher=Scuola Normale Superiore |isbn=978-88-7642-520-2 |pages=81–138 |chapter-url=https://doi.org/10.1007/978-88-7642-520-2_2 |language=de |doi=10.1007/978-88-7642-520-2_2 }}</ref> and all nonzero roots of {{math|<math>J_\nu(x)</math>}} and {{math|<math>J'_\nu(x)</math>}} are transcendental when ''{{math|<math>\nu</math>}}'' is rational.<ref>{{cite journal |last1=Lorch |first1=Lee |last2=Muldoon |first2=Martin E. |title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |journal=International Journal of Mathematics and Mathematical Sciences |date=1995 |volume=18 |issue=3 |pages=551–560 |doi=10.1155/S0161171295000706 |doi-access=free }}</ref>
* The number <math>\tfrac{\pi}{2} \tfrac{Y_0 (2)}{J_0 (2)} - \gamma</math>, where {{math|<math>Y_\alpha(x)</math>}} and {{math|<math>J_\alpha(x)</math>}} are Bessel functions and {{math|<math>\gamma</math>}} is the [[Euler–Mascheroni constant]].<ref>{{Cite journal |last1=Mahler |first1=Kurt |last2=Mordell |first2=Louis Joel |date=1968-06-04 |title=Applications of a theorem by A. B. Shidlovski |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1968.0111 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=305 |issue=1481 |pages=149–173 |bibcode=1968RSPSA.305..149M |doi=10.1098/rspa.1968.0111 |s2cid=123486171}}</ref><ref>{{Cite journal |last=Lagarias |first=Jeffrey C. |date=2013-07-19 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |pages=527–628 |arxiv=1303.1856 |doi=10.1090/S0273-0979-2013-01423-X |issn=0273-0979 |doi-access=free}}</ref>
* Any [[Liouville number]], in particular: Liouville's constant.
* Numbers with large [[irrationality measure]], such as the [[Champernowne constant]] <math>C_{10}</math> (by [[Roth's theorem]]).
* Numbers artificially constructed not to be [[Period (algebraic geometry)|algebraic periods]].<ref>{{cite arXiv |eprint=0805.0349 |class=math.AG |first=Masahiko |last=Yoshinaga |title=Periods and elementary real numbers |date=2008-05-03}}</ref>
* Any [[non-computable number]], in particular: [[Chaitin's constant]].
* Constructed irrational numbers which are not simply [[Normal number|normal]] in any base.{{sfn|Bugeaud|2012|page=113}}
* Any number for which the digits with respect to some fixed base form a [[Sturmian word]].<ref>{{harvnb|Pytheas Fogg|2002}}</ref>
* The [[Prouhet–Thue–Morse constant]]<ref>{{harvnb|Mahler|1929}}; {{harvnb|Allouche|Shallit|2003|p=387}}</ref> and the related rabbit constant.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Rabbit Constant |url=https://mathworld.wolfram.com/ |access-date=2023-08-09 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[Komornik–Loreti constant]].<ref>{{citation |last1=Allouche |first1=Jean-Paul |title=The Komornik–Loreti constant is transcendental |journal=American Mathematical Monthly |volume=107 |issue=5 |pages=448–449 |year=2000 |doi=10.2307/2695302 |jstor=2695302 |mr=1763399 |last2=Cosnard |first2=Michel}}</ref>
* The [[Regular paperfolding sequence|paperfolding constant]] (also named as "Gaussian Liouville number").<ref>{{Cite web |title=A143347 - OEIS |url=https://oeis.org/A143347 |access-date=2023-08-09 |website=oeis.org}}</ref>
* The values of the infinite series with fast [[Rate of convergence|convergence rate]] as defined by Y. Gao and J. Gao, such as <math>\sum_{n=1}^\infty \frac{3^n}{2^{3^n}}</math>.<ref>{{Cite web |title=A140654 - OEIS |url=https://oeis.org/A140654 |access-date=2023-08-12 |website=oeis.org}}</ref>
* Numbers of the form <math>\sum_{k=0}^\infty 10^{-b^k}</math> and <math>\sum_{k=0}^\infty 10^{-\left\lfloor b^{k} \right\rfloor}</math> For {{math|b > 1}} where <math>b \mapsto\lfloor b \rfloor</math> is the [[floor function]].<ref name="Kempner" /><ref>{{Cite arXiv |eprint=1303.1685 |class=math.NT |first=Boris |last=Adamczewski |title=The Many Faces of the Kempner Number |date=March 2013}}</ref><ref name="Sha1999">{{harvnb|Shallit|1996}}</ref><ref>{{Cite journal |last1=Adamczewski |first1=Boris |last2=Rivoal |first2=Tanguy | authorlink2=Tanguy Rivoal |date=2009 |title=Irrationality measures for some automatic real numbers |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/irrationality-measures-for-some-automatic-real-numbers/F89F4B7BBC9A06B6E9934FB2C3AFFE4D |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |language=en |volume=147 |issue=3 |pages=659–678 |doi=10.1017/S0305004109002643 |bibcode=2009MPCPS.147..659A |issn=1469-8064}}</ref><ref name="Lox1988">{{harvnb|Loxton|1988}}</ref><ref>{{harvnb|Allouche|Shallit|2003|pp=385,403}}</ref>

* Any number of the form <math>\sum_{n=0}^\infty \frac{E_n(\beta^{r^n})}{F_n(\beta^{r^n})}</math> (where <math>E_n(z)</math>, <math>F_n(z)</math> are polynomials in variables <math>n</math> and <math>z</math>, <math>\beta</math> is algebraic and <math>\beta \neq 0</math>, <math>r</math> is any integer greater than 1).<ref>{{Cite journal |last=Kurosawa |first=Takeshi |date=2007-03-01 |title=Transcendence of certain series involving binary linear recurrences |journal=Journal of Number Theory |language=en |volume=123 |issue=1 |pages=35–58 |doi=10.1016/j.jnt.2006.05.019 |issn=0022-314X|doi-access=free }}</ref>
* The numbers <math>\alpha =  3.3003300000...</math> and <math> \alpha^{-1} = 0.3030000030...</math> with only two different decimal digits whose nonzero digit positions are given by the [[Moser–de Bruijn sequence]] and its double.<ref>{{harvnb|Blanchard|Mendès France|1982}}</ref>
* The values of the [[Rogers–Ramanujan continued fraction|Rogers-Ramanujan continued fraction]] <math>R(q)</math> where <math>{{q}} \in \mathbb C</math> is algebraic and <math>0 < |q| < 1</math>.<ref>{{Cite journal |last1=Duverney |first1=Daniel |last2=Nishioka |first2=Keiji |last3=Nishioka |first3=Kumiko |last4=Shiokawa |first4=Iekata |date=1997 |title=Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers |journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences |volume=73 |issue=7 |pages=140–142 |doi=10.3792/pjaa.73.140 |issn=0386-2194 |doi-access=free}}</ref> The lemniscatic values of [[theta function]] <math>\sum_{n=-\infty}^\infty q^{n^2}</math> (under the same conditions for <math>{{q}}</math>) are also transcendental.<ref>{{Cite journal |last=Bertrand |first=Daniel |date=1997 |title=Theta functions and transcendence |url=http://link.springer.com/10.1023/A:1009749608672 |journal=The Ramanujan Journal |volume=1 |issue=4 |pages=339–350 |doi=10.1023/A:1009749608672 |s2cid=118628723}}</ref>
* {{math|''[[j-invariant|j]]''(''q'')}} where <math>{{q}} \in \mathbb C</math> is algebraic but not imaginary quadratic (i.e, the [[Transcendental function|exceptional set]] of this function is the number field whose degree of [[Field extension|extension]] over <math>\mathbb Q</math> is 2).
* The constants <math>\epsilon_k</math> and <math>\nu_k</math> in the formula for first index of occurrence of [[Gijswijt's sequence]], where k is any integer greater than 1.<ref>{{cite arXiv |eprint=2209.04657 |first=Levi |last=van de Pol |title=The first occurrence of a number in Gijswijt's sequence|date=2022 |class=math.CO }}</ref>

==Conjectured transcendental numbers==
Numbers which have yet to be proven to be either transcendental or algebraic:
* Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational: {{mvar|eπ}}, {{math|''e'' + ''π''}}, {{mvar|π}}<sup>{{mvar|π}}</sup>, {{math|''e''<sup>''e''</sup>}}, {{math|''π''<sup>''e''</sup>}}, {{math|''π''{{sup|{{sqrt|2}}}}}}, {{math|''e''<sup>''π''<sup>2</sup></sup>}}. It has been shown that both {{math|''e'' + ''π''}} and {{math|''π''/''e''}} do not satisfy any [[polynomial equation]] of degree {{math|<math>\leq 8</math>}} and integer coefficients of average size 10<sup>9</sup>.<ref>{{Cite journal |last=Bailey |first=David H. |date=1988 |title=Numerical Results on the Transcendence of Constants Involving $\pi, e$, and Euler's Constant |url=https://www.jstor.org/stable/2007931 |journal=Mathematics of Computation |volume=50 |issue=181 |pages=275–281 |doi=10.2307/2007931 |jstor=2007931 |issn=0025-5718}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=e |url=https://mathworld.wolfram.com/e.html |access-date=2023-08-12 |website=mathworld.wolfram.com |language=en}}</ref> At least one of the numbers {{math|''e''<sup>''e''</sup>}} and {{math|''e''<sup>''e<sup>2</sup>''</sup>}} is transcendental.<ref>{{Cite journal |last=Brownawell |first=W. Dale |date=1974-02-01 |title=The algebraic independence of certain numbers related by the exponential function |journal=Journal of Number Theory |volume=6 |issue=1 |pages=22–31 |doi=10.1016/0022-314X(74)90005-5 |issn=0022-314X|doi-access=free |bibcode=1974JNT.....6...22B }}</ref> [[Schanuel's conjecture]] would imply that all of the above numbers are transcendental and [[Algebraic independence|algebraically independent]].<ref name=":12">{{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}</ref>
* The [[Euler–Mascheroni constant]] {{mvar|γ}}'':'' In 2010 it has been shown that an infinite list of [[Euler–Lehmer constants|Euler-Lehmer constants]] (which includes {{math|{{var|γ}}/4}}) contains at most one algebraic number.<ref>{{Cite journal |last1=Murty |first1=M. Ram |last2=Saradha |first2=N. |date=2010-12-01 |title=Euler–Lehmer constants and a conjecture of Erdös |journal=[[Journal of Number Theory]] |language=en |volume=130 |issue=12 |pages=2671–2682 |doi=10.1016/j.jnt.2010.07.004 |doi-access=free |issn=0022-314X}}</ref><ref>{{Cite journal |last1=Murty |first1=M. Ram |last2=Zaytseva |first2=Anastasia |date=2013-01-01 |title=Transcendence of generalized Euler constants |journal=[[The American Mathematical Monthly]] |volume=120 |issue=1 |pages=48–54 |doi=10.4169/amer.math.monthly.120.01.048 |s2cid=20495981 |issn=0002-9890}}</ref> In 2012 it was shown that at least one of {{mvar|γ}} and the [[Gompertz constant]] {{mvar|δ}} is transcendental.<ref>{{Cite journal |last=Rivoal |first=Tanguy |date=2012 |title=On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant|journal=Michigan Mathematical Journal |language=en |volume=61 |issue=2 |pages=239–254 |doi=10.1307/mmj/1339011525 |doi-access=free |issn=0026-2285 |url=https://projecteuclid.org/euclid.mmj/1339011525}}</ref>
* The values of the [[Riemann zeta function]] {{math|''ζ''(n)}} at odd positive integers <math>n\geq3</math>; in particular [[Apéry's constant]] {{math|''ζ''(3)}}, which is known to be irrational. For the other numbers {{math|''ζ''(5), ''ζ''(7), ''ζ''(9), ...}} even this is not known. 
* The values of the [[Dirichlet beta function]] {{math|''β''(n)}} at even positive integers <math>n\geq2</math>; in particular [[Catalan's constant|Catalan's Constant]] {{math|''β''(2)}}. (none of them are known to be irrational).<ref>{{Cite journal |last1=Rivoal |first1=T. |last2=Zudilin |first2=W. |date=2003-08-01 |title=Diophantine properties of numbers related to Catalan's constant |url=https://doi.org/10.1007/s00208-003-0420-2 |journal=Mathematische Annalen |language=en |volume=326 |issue=4 |pages=705–721 |doi=10.1007/s00208-003-0420-2 |issn=1432-1807 |s2cid=59328860 |hdl-access=free |hdl=1959.13/803688}}</ref>
* Values of the [[Gamma function|Gamma Function]] {{math|''Γ''(1/n)}} for positive integers <math>n=5</math> and <math>n\geq7</math> are not known to be irrational, let alone transcendental.<ref name=":0">{{cite web |title=Mathematical constants |url=https://www.cambridge.org/us/academic/subjects/mathematics/recreational-mathematics/mathematical-constants |access-date=2022-09-22 |website=Cambridge University Press |language=en |department=Mathematics (general)}}</ref><ref name=":4">{{Cite web |last=Waldschmidt |first=Michel |date=2022 |title=Transcendental Number Theory: recent results and open problems. |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs |website=Michel Waldschmidt}}</ref> For <math>n\geq2</math> at least one the numbers {{math|''Γ''(1/n)}} and {{math|''Γ''(2/n)}} is transcendental.<ref name=":3" />
* Any number given by some kind of [[Limit (mathematics)|limit]] that is not obviously algebraic.<ref name=":4" />

==Proofs for specific numbers==
===A proof that {{mvar|e}} is transcendental===
The first proof that [[E (mathematical constant)|the base of the natural logarithms, {{mvar|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:

Assume, for purpose of [[Proof by contradiction|finding a contradiction]], that {{mvar|e}} is algebraic. Then there exists a finite set of integer coefficients {{math|''c''<sub>0</sub>, ''c''<sub>1</sub>, ..., ''c<sub>n</sub>''}} satisfying the equation:
<math display=block>
  c_{0} + c_{1}e + c_{2} e^{2} + \cdots + c_{n} e^{n} = 0, \qquad c_0, c_n \neq 0 ~.
</math>
It is difficult to make use of the integer status of these coefficients when multiplied by a power of the irrational {{mvar|e}}, but we can absorb those powers into an integral which “mostly” will assume integer values. For a positive integer {{mvar|k}}, define the polynomial
<math display=block>
  f_k(x) = x^{k} \left [(x-1)\cdots(x-n) \right ]^{k+1},
</math>
and multiply both sides of the above equation by
<math display=block>
  \int^{\infty}_{0} f_k(x) \, e^{-x}\, \mathrm{d}x\ ,
</math>
to arrive at the equation:
<math display=block>
  c_0 \left (\int^{\infty}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right) + 
  c_1 e \left( \int^{\infty}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right ) 
  + \cdots + 
  c_{n}e^{n} \left( \int^{\infty}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right) = 0 ~.
</math>

By splitting respective domains of integration, this equation can be written in the form
<math display=block>
  P + Q = 0
</math>
where
<math display=block>
  \begin{align}
    P &= c_{0} \left( \int^{\infty}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right)
    + c_{1} e \left( \int^{\infty}_{1} f_k(x) e^{-x} \,\mathrm{d}x \right)
    + c_{2} e^{2} \left( \int^{\infty}_{2} f_k(x) e^{-x} \,\mathrm{d}x \right) 
    + \cdots 
    + c_{n} e^{n} \left( \int^{\infty}_{n} f_k(x) e^{-x} \,\mathrm{d}x \right) 
  \\
    Q &= c_{1} e \left(\int^{1}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right)
    + c_{2}e^{2} \left( \int^{2}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right)
    + \cdots+c_{n} e^{n} \left( \int^{n}_{0} f_k(x) e^{-x} \,\mathrm{d}x \right) 
  \end{align}
</math>
Here {{mvar|P}} will turn out to be an integer, but more importantly it grows quickly with {{mvar|k}}.

====Lemma 1====
''There are arbitrarily large {{mvar|k}} such that <math>\ \tfrac{P}{k!}\ </math> is a non-zero integer.''

'''Proof.''' Recall the standard integral (case of the [[Gamma function]])
<math display=block>
  \int^{\infty}_{0} t^{j} e^{-t} \,\mathrm{d}t = j!
</math>
valid for any [[natural number]] <math>j</math>. More generally,

: if <math> g(t) = \sum_{j=0}^m b_j t^j </math> then <math> \int^{\infty}_{0} g(t) e^{-t} \,\mathrm{d}t = \sum_{j=0}^m b_j j! </math>.

This would allow us to compute <math>P</math> exactly, because any term of <math>P</math> can be rewritten as
<math display=block>
  c_{a} e^{a} \int^{\infty}_{a} f_k(x) e^{-x} \,\mathrm{d}x =
  c_{a} \int^{\infty}_{a} f_k(x) e^{-(x-a)} \,\mathrm{d}x =
  \left\{ \begin{aligned} t &= x-a \\ x &= t+a \\ \mathrm{d}x &= \mathrm{d}t \end{aligned} \right\} =
  c_a \int_0^\infty f_k(t+a) e^{-t} \,\mathrm{d}t
</math>
through a [[Integration by substitution|change of variables]]. Hence
<math display="block">
  P = \sum_{a=0}^n c_a \int_0^\infty f_k(t+a) e^{-t} \,\mathrm{d}t
  = \int_0^\infty \biggl( \sum_{a=0}^n c_a f_k(t+a) \biggr) e^{-t} \,\mathrm{d}t
</math>
That latter sum is a polynomial in <math>t</math> with integer coefficients, i.e., it is a linear combination of powers <math>t^j</math> with integer coefficients. Hence the number <math>P</math> is a linear combination (with those same integer coefficients) of factorials <math>j!</math>; in particular <math>P</math> is an integer.

Smaller factorials divide larger factorials, so the smallest <math>j!</math> occurring in that linear combination will also divide the whole of <math>P</math>. We get that <math>j!</math> from the lowest power <math>t^j</math> term appearing with a nonzero coefficient in <math>\textstyle \sum_{a=0}^n c_a f_k(t+a) </math>, but this smallest exponent <math>j</math> is also the [[Multiplicity (mathematics)#Multiplicity of a root of a polynomial|multiplicity]] of <math>t=0</math> as a root of this polynomial. <math>f_k(x)</math> is chosen to have multiplicity <math>k</math> of the root <math>x=0</math> and multiplicity <math>k+1</math> of the roots <math>x=a</math> for <math>a=1,\dots,n</math>, so that smallest exponent is <math>t^k</math> for <math>f_k(t)</math> and <math>t^{k+1}</math> for <math>f_k(t+a)</math> with <math> a>0 </math>. Therefore <math>k!</math> divides <math>P</math>.

To establish the last claim in the lemma, that <math>P</math> is nonzero, it is sufficient to prove that <math>k+1</math> does not divide <math>P</math>. To that end, let <math>k+1</math> be any [[prime number|prime]] larger than <math>n</math> and <math>|c_0|</math>. We know from the above that <math>(k+1)!</math> divides each of <math> \textstyle c_a \int_0^\infty f_k(t+a) e^{-t} \,\mathrm{d}t </math> for <math> 1 \leqslant a \leqslant n </math>, so in particular all of those ''are'' divisible by <math>k+1</math>. It comes down to the first term <math> \textstyle c_0 \int_0^\infty f_k(t) e^{-t} \,\mathrm{d}t </math>. We have (see [[falling and rising factorials]])
<math display=block>
  f_k(t) = t^k \bigl[ (t-1) \cdots (t-n) \bigr]^{k+1} =
  \bigl[ (-1)^{n}(n!) \bigr]^{k+1} t^k + \text{higher degree terms}
</math>
and those higher degree terms all give rise to factorials <math>(k+1)!</math> or larger. Hence
<math display=block>
  P \equiv 
  c_0 \int_0^\infty f_k(t) e^{-t} \,\mathrm{d}t \equiv
  c_0 \bigl[ (-1)^{n}(n!) \bigr]^{k+1} k! \pmod{(k+1)}
</math>
That right hand side is a product of nonzero integer factors less than the prime <math>k+1</math>, therefore that product is not divisible by <math>k+1</math>, and the same holds for <math>P</math>; in particular <math>P</math> cannot be zero.

====Lemma 2====
''For sufficiently large {{mvar|k}}, <math>\left| \tfrac{Q}{k!} \right| <1</math>.''

'''Proof.''' Note that

<math display=block>\begin{align} 
  f_k e^{-x} &= x^{k} \left[ (x-1)(x-2) \cdots (x-n) \right]^{k+1} e^{-x}\\
  &= \left (x(x-1)\cdots(x-n) \right)^k \cdot \left( (x-1) \cdots (x-n) e^{-x} \right) \\
  &= u(x)^k \cdot v(x)
\end{align}</math>

where {{math|''u''(''x''), ''v''(''x'')}} are [[Continuous function|continuous functions]] of {{mvar|x}} for all {{mvar|x}}, so are bounded on the interval {{math|[0, ''n'']}}. That is, there are constants {{math|''G'', ''H'' > 0}} such that

<math display=block>\ \left| f_k e^{-x} \right| \leq |u(x)|^k \cdot |v(x)| < G^k H \quad \text{ for } 0 \leq x \leq n ~.</math>

So each of those integrals composing {{mvar|Q}} is bounded, the worst case being

<math display=block>\left| \int_{0}^{n} f_{k} e^{-x}\ \mathrm{d}\ x \right| \leq \int_{0}^{n} \left| f_{k} e^{-x} \right| \ \mathrm{d}\ x \leq \int_{0}^{n}G^k H\ \mathrm{d}\ x = n G^k H ~.</math>

It is now possible to bound the sum {{mvar|Q}} as well:

<math display=block> |Q| < G^{k} \cdot n H \left( |c_1|e+|c_2|e^2 + \cdots+|c_n|e^{n} \right) = G^k \cdot M\ ,</math>

where {{mvar|M}} is a constant not depending on {{mvar|k}}. It follows that

<math display=block>\ \left| \frac{Q}{k!} \right| < M \cdot \frac{G^k}{k!} \to 0 \quad \text{ as } k \to \infty\ ,</math>

finishing the proof of this lemma.

====Conclusion====

Choosing a value of {{mvar|k}} that satisfies both lemmas leads to a non-zero integer <math>\left(\tfrac{P}{k!}\right)</math> added to a vanishingly small quantity <math>\left(\tfrac{Q}{k!}\right)</math> being equal to zero: an impossibility. It follows that the original assumption, that {{mvar|e}} can satisfy a polynomial equation with integer coefficients, is also impossible; that is, {{mvar|e}} is transcendental.

===The transcendence of {{mvar|π}}===
A similar strategy, different from [[Ferdinand von Lindemann|Lindemann]]'s original approach, can be used to show that the [[Pi|number {{mvar|π}}]] is transcendental. Besides the [[gamma-function]] and some estimates as in the proof for {{mvar|e}}, facts about [[symmetric polynomial]]s play a vital role in the proof.

For detailed information concerning the proofs of the transcendence of {{mvar|π}} and {{mvar|e}}, see the references and external links.

==See also==
{{Portal|Mathematics}}
* [[Transcendental number theory]], the study of questions related to transcendental numbers
* [[Transcendental element]], generalization of transcendental numbers in abstract algebra
* [[Gelfond–Schneider theorem]]
* [[Diophantine approximation]]
* [[Ring of periods|Periods]], a countable set of numbers (including all algebraic and some transcendental numbers) which may be defined by integral equations.
{{Classification of numbers}}

==Notes==
{{notelist}}

==References==
{{reflist|25em}}

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 | series=Lecture Notes in Mathematics   | volume=546
 | publisher=[[Springer-Verlag|Springer]]
 | isbn=978-3-540-07986-6 | zbl=0332.10019
}}
*{{cite book
 | last1=Natarajan   | first1=Saradha   | author1-link=:fr:Saradha Natarajan
 | last2=Thangadurai | first2=Ravindranathan
 | year=2020
 | publisher=[[Springer Verlag]]
 | title=Pillars of Transcendental Number Theory
 | isbn=978-981-15-4154-4
}}
*{{cite book
 | last=Pytheas Fogg | first=N.
 | year=2002
 | title=Substitutions in dynamics, arithmetics and combinatorics
 | editor1-last=Berthé   | editor1-first=V. | editor1-link=Valérie Berthé
 | editor2-last=Ferenczi | editor2-first=Sébastien
 | editor3-last=Mauduit  | editor3-first=Christian
 | editor4-last=Siegel   | editor4-first=A.
 | series=Lecture Notes in Mathematics  | volume=1794
 | publisher=[[Springer-Verlag|Springer]]
 | isbn=978-3-540-44141-0 | zbl=1014.11015
}}
*{{cite conference
 | last=Shallit | first=J. | author-link=Jeffrey Shallit
 | date=15–26 July 1996  | publication-date=1999
 | title=Number theory and formal languages
 | editor1-last=Hejhal     | editor1-first=D.A. | editor1-link=Dennis Hejhal
 | editor2-last=Friedman   | editor2-first=Joel
 | editor3-last=Gutzwiller | editor3-first=M.C. | editor3-link=Martin Gutzwiller
 | editor4-last=Odlyzko    | editor4-first=A.M. | editor4-link=Andrew Odlyzko
 | book-title=Emerging Applications of Number Theory
 | conference=IMA Summer Program
 | place=Minneapolis, MN
 | series=The IMA Volumes in Mathematics and its Applications  | volume=109  | pages=547–570
 | publisher=[[Springer-Verlag|Springer]]
 | isbn=978-0-387-98824-5
}}

{{refend}}

==External links==
{{wikisource|de:David Hilbert Gesammelte Abhandlungen Erster Band – Zahlentheorie/Kapitel 1|Über die Transzendenz der Zahlen {{mvar|e}} und {{mvar|π}}. (in German)}}
<!-- victim of a past cut-and-paste error ---
602440|Transcendental number (mathematics)} -->
* {{MathWorld |title=Transcendental Number |id=TranscendentalNumber}}
* {{MathWorld |title=Liouville Number      |id=LiouvilleNumber}}
* {{MathWorld |title=Liouville's Constant  |id=LiouvillesConstant}}
* {{cite web
 |title=Proof that {{mvar|e}} is transcendental |language=en
 |website=planetmath.org
 |url=http://planetmath.org/EIsTranscendental
}}
* {{cite web
 |title=Proof that the Liouville constant is transcendental
 |language=en
 |website=deanlmoore.com
 |url=https://deanlmoore.com/liouvilles-proof
 |access-date=2018-11-12
 }}
* {{cite conference
 |author=Fritsch, R.
 |date=29 March 1988  |publication-date=1989
 |title=Transzendenz von {{mvar|e}} im Leistungskurs?  |lang=de
 |trans-title=Transcendence of {{mvar|e}} in advanced courses?
 |conference=Rahmen der 79.&nbsp;Hauptversammlung des Deutschen Vereins zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts [79th&nbsp;Annual, General Meeting of the German Association for the Promotion of Mathematics and Science Education]
 |journal=Der mathematische und naturwissenschaftliche Unterricht
 |volume=42 |pages=75–80 (presentation), 375–376 (responses)
 |place=Kiel, DE
 |via=[[University of Munich]] (mathematik.uni-muenchen.de ) |url=http://www.mathematik.uni-muenchen.de/~fritsch/euler.pdf |archive-url=https://web.archive.org/web/20110716060646/http://www.mathematik.uni-muenchen.de/~fritsch/euler.pdf |archive-date=2011-07-16
}} — Proof that {{mvar|e}} is transcendental, in German.
* {{cite journal
 |author=Fritsch, R.
 |year=2003
 |title=Hilberts Beweis der Transzendenz der Ludolphschen Zahl {{mvar|π}} |language=de
 |journal=Дифференциальная геометрия многообразий фигур
 |volume=34 |pages=144–148
 |via=[[University of Munich]] (mathematik.uni-muenchen.de/~fritsch)
 |url=http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf
 |archive-url=https://web.archive.org/web/20110716060726/http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf
 |archive-date=2011-07-16
}}

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[[Category:Transcendental numbers| ]]
[[Category:Articles containing proofs]]