Transcendental number#Properties

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

Johann Heinrich Lambert conjectured that {{mvar|e}} and Pi were both transcendental numbers in his 1768 paper proving the number {{mvar|π}} is irrational, and proposed a tentative sketch proof that {{mvar|π}} is transcendental.{{harvnb|Lambert|1768}}

Joseph Liouville first proved the existence of transcendental numbers in 1844,{{harvnb|Kempner|1916}} and in 1851 gave the first decimal examples such as the Liouville constant

\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}

in which the {{mvar|n}}th digit after the decimal point is {{math|1}} if {{mvar|n}} = {{mvar|k}}{{math|!}} ({{mvar|k}} factorial) for some {{mvar|k}} and {{math|0}} otherwise.{{cite web| url = http://mathworld.wolfram.com/LiouvillesConstant.html| title = Weisstein, Eric W. "Liouville's Constant", MathWorld}} 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 numbers than can any irrational algebraic number, and this class of numbers is called the Liouville numbers. Liouville showed that all Liouville numbers are transcendental.{{harvnb|Liouville|1851}}

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 new method for constructing transcendental numbers.{{harvnb|Cantor|1874}}; {{harvnb|Gray|1994}} 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.{{harvnb|Cantor|1878|p=254}}

}}

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 in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).{{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. Andrew's, Scotland |website=www-history.mcs.st-andrews.ac.uk |url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Baker_Alan.html}}

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, since every rational number is the root of some integer polynomial of degree one.{{harvnb|Hardy|1979}} The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable.

No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals 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 $5\backslash pi$, $\backslash tfrac\{\backslash pi\; -\; 3\}\{\backslash sqrt\{2\}\}$, $(\backslash sqrt\{\backslash pi\}-\backslash sqrt\{3\})^8$, and $\backslash sqrt[4]\{\backslash pi^5+7\}$ 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) {{=}} x² − (a + b) x + a b}} . 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 numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its simple 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.

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{{harvnb|Adamczewski|Bugeaud|2005}} (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:

• pi (by the Lindemann–Weierstrass theorem).
• {{math|$e^a$}} if {{Math|$a$}} is algebraic and nonzero (by the Lindemann–Weierstrass theorem), in particular Euler's number {{mvar|e}}.
• {{math|$e^\{\backslash pi\; \backslash sqrt\; n\}$}} where {{math|$n$}} is a positive integer; in particular Gelfond's constant {{math|$e^\backslash pi$}} (by the Gelfond–Schneider theorem).
• Algebraic combinations of {{math|$\backslash pi$}} and {{math|$e^\{\backslash pi\; \backslash sqrt\; n\}\; ,\; n\backslash in\backslash mathbb\; Z^\{+\}$}} such as {{math|$\backslash pi\; +\; e^\{\backslash pi\}$}} and {{math|$\backslash pi\; e^\{\backslash pi\}$}} (following from their algebraic independence).{{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}}
• {{math|$a^b$}} where {{Math|$a$}} is algebraic but not 0 or 1, and {{Math|$b$}} is irrational algebraic, in particular the Gelfond–Schneider constant $2^\{\backslash sqrt\{2\}\}$ (by the Gelfond–Schneider theorem).
• The natural logarithm {{math|$\backslash ln(a)$}} if {{math|$a$}} is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
• {{math|$\backslash log\_b(a)$}} if {{math|$a$}} and {{math|$b$}} are positive integers not both powers of the same integer, and {{math|$a$}} is not equal to 1 (by the Gelfond–Schneider theorem).
• All numbers of the form $\backslash pi\; +\; \backslash beta\_1\; \backslash ln\; (a\_1)\; +\; \backslash cdots\; +\; \backslash beta\_n\; \backslash ln\; (a\_n)$ are transcendental, where $\backslash beta\_j$ are algebraic for all $1\; \backslash leq\; j\; \backslash leq\; n$ and $a\_j$ are non-zero algebraic for all $1\; \backslash leq\; j\; \backslash leq\; n$ (by Baker's theorem).

• The trigonometric functions {{math|$\backslash sin(x),\; \backslash cos(x),\; ...$}} and their hyperbolic counterparts, for any nonzero algebraic number {{math|$x$}}, expressed in radians (by the Lindemann–Weierstrass theorem).
• Non-zero results of the inverse trigonometric functions {{math|$\backslash arcsin(x),\; \backslash arccos(x),\; ...$}} and their hyperbolic counterparts, for any algebraic number {{math|$x$}} (by the Lindemann–Weierstrass theorem).
• $\backslash pi^\{-1\}\{\backslash arctan(x)\}$, for rational {{math|$x$}} such that $x\; \backslash notin\; \backslash \{0,\backslash pm\{1\}\backslash \}$.{{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}}
• The fixed point of the cosine function (also referred to as the Dottie number {{math|$d$}}) – the unique real solution to the equation {{math|$\backslash cos(x)=x$}}, where {{math|$x$}} is in radians (by the Lindemann–Weierstrass theorem).{{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}}
• {{math|$W(a)$}} if {{math|$a$}} is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular the omega constant {{math|Ω}}.
• {{math|$W(r,a)$}} if both {{math|$a$}} and the order {{math|$r$}} are algebraic such that $a\; \backslash neq\; 0$, for any branch of the generalized Lambert W function.{{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}}
• {{math|$\backslash sqrt\; x\; \_s$}}, the 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 $\backslash Gamma(n/2),\backslash Gamma(n/3),\backslash Gamma(n/4)$ or $\backslash Gamma(n/6)$.{{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}}
• Algebraic combinations of {{math|$\backslash pi$}} and {{math|$\backslash Gamma(1/3)$}} or of {{math|$\backslash pi$}} and {{math|$\backslash Gamma(1/4)$}} such as the lemniscate constant $\backslash varpi$ (following from their respective algebraic independences).
• The values of Beta function $\backslash Beta(a,b)$ if $a,\; b$ and $a+b$ are non-integer rational numbers.{{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}}
• The Bessel function of the first kind {{math|$J\_\backslash nu(x)$}}, its first derivative, and the quotient $\backslash tfrac\{J\text{'}\_\backslash nu\; (x)\}\{J\_\backslash nu\; (x)\}$ are transcendental when {{math|$\backslash nu$}} is rational and {{math|$x$}} is algebraic and nonzero,{{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 }} and all nonzero roots of {{math|$J\_\backslash nu(x)$}} and {{math|$J\text{'}\_\backslash nu(x)$}} are transcendental when {{math|$\backslash nu$}} is rational.{{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 }}
• The number $\backslash tfrac\{\backslash pi\}\{2\}\; \backslash tfrac\{Y\_0\; (2)\}\{J\_0\; (2)\}\; -\; \backslash gamma$, where {{math|$Y\_\backslash alpha(x)$}} and {{math|$J\_\backslash alpha(x)$}} are Bessel functions and {{math|$\backslash gamma$}} is the Euler–Mascheroni constant.{{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}}{{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}}
• Values of the Fibonacci zeta function at the positive even argument.{{citation

|last = Murty | first = M. Ram

| editor1-last = Prasad | editor1-first = D.

| editor2-last = Rajan | editor2-first = C. S.

| editor3-last = Sankaranarayanan | editor3-first = A.

| editor4-last = Sengupta | editor4-first = J.

| contribution = The Fibonacci zeta function | isbn = 978-93-80250-49-6 | mr = 3156859 | pages = 409–425

| publisher = Tata Institute of Fundamental Research

| series = Tata Institute of Fundamental Research Studies in Mathematics

| title = Automorphic representations and {{mvar|L}}-functions

| volume = 22 | year = 2013}}

• Any Liouville number, in particular: Liouville's constant $\backslash sum\_\{k=1\}^\backslash infty\backslash frac1\{10^\{k!\}\}$.
• Numbers with irrationality measure larger than 2, such as the Champernowne constant $C\_\{10\}$ (by Roth's theorem).
• Numbers artificially constructed not to be algebraic periods.{{cite arXiv |eprint=0805.0349 |class=math.AG |first=Masahiko |last=Yoshinaga |title=Periods and elementary real numbers |date=2008-05-03}}
• Any non-computable number, in particular: Chaitin's constant.
• Constructed irrational numbers which are not simply 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.{{harvnb|Pytheas Fogg|2002}}
• The Prouhet–Thue–Morse constant{{harvnb|Mahler|1929}}; {{harvnb|Allouche|Shallit|2003|p=387}} and the related rabbit constant.{{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}}
• The Komornik–Loreti constant.{{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}}
• The paperfolding constant (also named as "Gaussian Liouville number").{{Cite web |title=A143347 - OEIS |url=https://oeis.org/A143347 |access-date=2023-08-09 |website=oeis.org}}
• The values of the infinite series with fast convergence rate as defined by Y. Gao and J. Gao, such as $\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{3^n\}\{2^\{3^n\}\}$.{{Cite web |title=A140654 - OEIS |url=https://oeis.org/A140654 |access-date=2023-08-12 |website=oeis.org}}
• Any number of the form $\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{E\_n(\backslash beta^\{r^n\})\}\{F\_n(\backslash beta^\{r^n\})\}$ (where $E\_n(z)$, $F\_n(z)$ are polynomials in variables $n$ and $z$, $\backslash beta$ is algebraic and $\backslash beta\; \backslash neq\; 0$, $r$ is any integer greater than 1).{{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}}
• Numbers of the form $\backslash sum\_\{k=0\}^\backslash infty\; 10^\{-b^k\}$ and $\backslash sum\_\{k=0\}^\backslash infty\; 10^\{-\backslash left\backslash lfloor\; b^\{k\}\; \backslash right\backslash rfloor\}$ For {{math|b > 1}} where $b\; \backslash mapsto\backslash lfloor\; b\; \backslash rfloor$ is the floor function.{{Cite arXiv |eprint=1303.1685 |class=math.NT |first=Boris |last=Adamczewski |title=The Many Faces of the Kempner Number |date=March 2013}}{{harvnb|Shallit|1996}}{{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}}{{harvnb|Loxton|1988}}{{harvnb|Allouche|Shallit|2003|pp=385,403}}
• The numbers $\backslash alpha\; =\; 3.3003300000...$ and $\backslash alpha^\{-1\}\; =\; 0.3030000030...$ with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.{{harvnb|Blanchard|Mendès France|1982}}
• The values of the Rogers-Ramanujan continued fraction $R(q)$ where $\{\{q\}\}\; \backslash in\; \backslash mathbb\; C$ is algebraic and .{{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}} The lemniscatic values of theta function $\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; q^\{n^2\}$ (under the same conditions for $\{\{q\}\}$) are also transcendental.{{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}}
• {{math|j(q)}} where $\{\{q\}\}\; \backslash in\; \backslash mathbb\; C$ is algebraic but not imaginary quadratic (i.e, the exceptional set of this function is the number field whose degree of extension over $\backslash mathbb\; Q$ is 2).
• The constants $\backslash epsilon\_k$ and $\backslash nu\_k$ in the formula for first index of occurrence of Gijswijt's sequence, where k is any integer greater than 1.{{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 }}

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}}^{{mvar|π}}, {{math|e^e}}, {{math|π^e}}, {{math|π{{sup|{{sqrt|2}}}}}}, {{math|e^π2}}. It has been shown that both {{math|e + π}} and {{math|π/e}} do not satisfy any polynomial equation of degree {{math|$\backslash leq\; 8$}} and integer coefficients of average size 10⁹.{{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}}{{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}} At least one of the numbers {{math|e^e}} and {{math|e^e2}} is transcendental.{{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 }} Schanuel's conjecture would imply that all of the above numbers are transcendental and algebraically independent.{{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}}
• The Euler–Mascheroni constant {{mvar|γ}}: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes {{math|{{var|γ}}/4}}) contains at most one algebraic number.{{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}}{{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}} In 2012 it was shown that at least one of {{mvar|γ}} and the Gompertz constant {{mvar|δ}} is transcendental.{{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}}
• The values of the Riemann zeta function {{math|ζ(n)}} at odd positive integers $n\backslash geq3$; 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 $n\backslash geq2$; in particular Catalan's Constant {{math|β(2)}}. (none of them are known to be irrational).{{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}}
• Values of the Gamma Function {{math|Γ(1/n)}} for positive integers $n=5$ and $n\backslash geq7$ are not known to be irrational, let alone transcendental.{{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)}}{{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}} For $n\backslash geq2$ at least one the numbers {{math|Γ(1/n)}} and {{math|Γ(2/n)}} is transcendental.
• Any number given by some kind of limit that is not obviously algebraic.

The first proof that E (mathematical constant), 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 finding a contradiction, that {{mvar|e}} is algebraic. Then there exists a finite set of integer coefficients {{math|c₀, c₁, ..., c_n}} satisfying the equation:

c_{0} + c_{1}e + c_{2} e^{2} + \cdots + c_{n} e^{n} = 0, \qquad c_0, c_n \neq 0 ~.

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

f_k(x) = x^{k} \left [(x-1)\cdots(x-n) \right ]^{k+1},

and multiply both sides of the above equation by

\int^{\infty}_{0} f_k(x) \, e^{-x}\, \mathrm{d}x\ ,

to arrive at the equation:

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 ~.

By splitting respective domains of integration, this equation can be written in the form

P + Q = 0

where

\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}

Here {{mvar|P}} will turn out to be an integer, but more importantly it grows quickly with {{mvar|k}}.

There are arbitrarily large {{mvar|k}} such that $\backslash \; \backslash tfrac\{P\}\{k!\}\backslash$ is a non-zero integer.

Proof. Recall the standard integral (case of the Gamma function)

\int^{\infty}_{0} t^{j} e^{-t} \,\mathrm{d}t = j!

valid for any natural number $j$. More generally,

: if $g(t)\; =\; \backslash sum\_\{j=0\}^m\; b\_j\; t^j$ then $\backslash int^\{\backslash infty\}\_\{0\}\; g(t)\; e^\{-t\}\; \backslash ,\backslash mathrm\{d\}t\; =\; \backslash sum\_\{j=0\}^m\; b\_j\; j!$.

This would allow us to compute $P$ exactly, because any term of $P$ can be rewritten as

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

through a change of variables. Hence

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

That latter sum is a polynomial in $t$ with integer coefficients, i.e., it is a linear combination of powers $t^j$ with integer coefficients. Hence the number $P$ is a linear combination (with those same integer coefficients) of factorials $j!$; in particular $P$ is an integer.

Smaller factorials divide larger factorials, so the smallest $j!$ occurring in that linear combination will also divide the whole of $P$. We get that $j!$ from the lowest power $t^j$ term appearing with a nonzero coefficient in $\backslash textstyle\; \backslash sum\_\{a=0\}^n\; c\_a\; f\_k(t+a)$, but this smallest exponent $j$ is also the multiplicity of $t=0$ as a root of this polynomial. $f\_k(x)$ is chosen to have multiplicity $k$ of the root $x=0$ and multiplicity $k+1$ of the roots $x=a$ for $a=1,\backslash dots,n$, so that smallest exponent is $t^k$ for $f\_k(t)$ and $t^\{k+1\}$ for $f\_k(t+a)$ with $a>0$. Therefore $k!$ divides $P$.

To establish the last claim in the lemma, that $P$ is nonzero, it is sufficient to prove that $k+1$ does not divide $P$. To that end, let $k+1$ be any prime larger than $n$ and $|c\_0|$. We know from the above that $(k+1)!$ divides each of $\backslash textstyle\; c\_a\; \backslash int\_0^\backslash infty\; f\_k(t+a)\; e^\{-t\}\; \backslash ,\backslash mathrm\{d\}t$ for $1\; \backslash leqslant\; a\; \backslash leqslant\; n$, so in particular all of those are divisible by $k+1$. It comes down to the first term $\backslash textstyle\; c\_0\; \backslash int\_0^\backslash infty\; f\_k(t)\; e^\{-t\}\; \backslash ,\backslash mathrm\{d\}t$. We have (see falling and rising factorials)

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}

and those higher degree terms all give rise to factorials $(k+1)!$ or larger. Hence

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)}

That right hand side is a product of nonzero integer factors less than the prime $k+1$, therefore that product is not divisible by $k+1$, and the same holds for $P$; in particular $P$ cannot be zero.

For sufficiently large {{mvar|k}}, .

Proof. Note that

$$\backslash begin\{align\}$$

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}

where {{math|u(x), v(x)}} are 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

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

$$\backslash left|\; \backslash int\_\{0\}^\{n\}\; f\_\{k\}\; e^\{-x\}\backslash \; \backslash mathrm\{d\}\backslash \; x\; \backslash right|\; \backslash leq\; \backslash int\_\{0\}^\{n\}\; \backslash left|\; f\_\{k\}\; e^\{-x\}\; \backslash right|\; \backslash \; \backslash mathrm\{d\}\backslash \; x\; \backslash leq\; \backslash int\_\{0\}^\{n\}G^k\; H\backslash \; \backslash mathrm\{d\}\backslash \; x\; =\; n\; G^k\; H\; ~.$$

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

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

finishing the proof of this lemma.

Choosing a value of {{mvar|k}} that satisfies both lemmas leads to a non-zero integer $\backslash left(\backslash tfrac\{P\}\{k!\}\backslash right)$ added to a vanishingly small quantity $\backslash left(\backslash tfrac\{Q\}\{k!\}\backslash right)$ 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.

A similar strategy, different from Lindemann's original approach, can be used to show that the Pi is transcendental. Besides the gamma-function and some estimates as in the proof for {{mvar|e}}, facts about symmetric polynomials 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.

{{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
• Periods, a countable set of numbers (including all algebraic and some transcendental numbers) which may be defined by integral equations.

{{Classification of numbers}}

{{notelist}}

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{{refend}}

{{wikisource|de:David Hilbert Gesammelte Abhandlungen Erster Band – Zahlentheorie/Kapitel 1|Über die Transzendenz der Zahlen {{mvar|e}} und {{mvar|π}}. (in German)}}

• {{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. Hauptversammlung des Deutschen Vereins zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts [79th 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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