
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are π and e. 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 of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set.
All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping sets of rational, algebraic irrational, and transcendental real numbers. For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0. The golden ratio (denoted or ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2 − x − 1 = 0.
History
The name "transcendental" comes from Latin trānscendere 'to climb over or beyond, surmount', and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that sin x is not an algebraic function of x.Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the modern sense.
Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and proposed a tentative sketch proof that π is transcendental.
Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant
in which the nth digit after the decimal point is 1 if n = k! (k factorial) for some k and 0 otherwise. In other words, the nth digit of this number is 1 only if n is one of 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.
The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was 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. 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. Cantor's work established the ubiquity of transcendental numbers.
In 1882 Ferdinand von Lindemann published the first complete proof that π is transcendental. He first proved that ea is transcendental if a is a non-zero algebraic number. Then, since eiπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, 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 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 a is an algebraic number that is not 0 or 1, and b is an irrational algebraic number, is ab 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).
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, since a rational number is the root of an integer polynomial of degree one. 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 π is transcendental, it can be immediately deduced that numbers such as ,
,
, and
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, π and (1 − π) are both transcendental, but π + (1 − π) = 1 is obviously not. It is unknown whether e + π, for example, is transcendental, though at least one of e + π and eπ 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) = x2 − (a + b) x + a b . If (a + b) and 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, 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.
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 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 have a "simple" structure, and that are not eventually periodic are transcendental (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:
- π (by the Lindemann–Weierstrass theorem).
if
is algebraic and nonzero (by the Lindemann–Weierstrass theorem), in particular Euler's number e.
where
is a positive integer; in particular Gelfond's constant
(by the Gelfond–Schneider theorem).
- Algebraic combinations of
and
such as
and
(following from their algebraic independence).
where
is algebraic but not 0 or 1, and
is irrational algebraic, in particular the Gelfond–Schneider constant
(by the Gelfond–Schneider theorem).
- The natural logarithm
if
is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
if
and
are positive integers not both powers of the same integer, and
is not equal to 1 (by the Gelfond–Schneider theorem).
- All numbers of the form
are transcendental, where
are algebraic for all
and
are non-zero algebraic for all
(by Baker's theorem).
- The trigonometric functions
and their hyperbolic counterparts, for any nonzero algebraic number
, expressed in radians (by the Lindemann–Weierstrass theorem).
- Non-zero results of the inverse trigonometric functions
and their hyperbolic counterparts, for any algebraic number
(by the Lindemann–Weierstrass theorem).
, for rational
such that
.
- The fixed point of the cosine function (also referred to as the Dottie number
) – the unique real solution to the equation
, where
is in radians (by the Lindemann–Weierstrass theorem).
if
is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular the omega constant Ω.
if both
and the order
are algebraic such that
, for any branch of the generalized Lambert W function.
, 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
or
.
- Algebraic combinations of
and
or of
and
such as the lemniscate constant
(following from their respective algebraic independences).
- The values of Beta function
if
and
are non-integer rational numbers.
- The Bessel function of the first kind
, its first derivative, and the quotient
are transcendental when
is rational and
is algebraic and nonzero, and all nonzero roots of
and
are transcendental when
is rational.
- The number
, where
and
are Bessel functions and
is the Euler–Mascheroni constant.
- Any Liouville number, in particular: Liouville's constant.
- Numbers with large irrationality measure, such as the Champernowne constant
(by Roth's theorem).
- Numbers artificially constructed not to be algebraic periods.
- Any non-computable number, in particular: Chaitin's constant.
- Constructed irrational numbers which are not simply normal in any base.
- Any number for which the digits with respect to some fixed base form a Sturmian word.
- The Prouhet–Thue–Morse constant and the related rabbit constant.
- The Komornik–Loreti constant.
- The paperfolding constant (also named as "Gaussian Liouville number").
- The values of the infinite series with fast convergence rate as defined by Y. Gao and J. Gao, such as
.
- Numbers of the form
and
For b > 1 where
is the floor function.
- Any number of the form
(where
,
are polynomials in variables
and
,
is algebraic and
,
is any integer greater than 1).
- The numbers
and
with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.
- The values of the Rogers-Ramanujan continued fraction
where
is algebraic and
. The lemniscatic values of theta function
(under the same conditions for
) are also transcendental.
- j(q) where
is algebraic but not imaginary quadratic (i.e, the exceptional set of this function is the number field whose degree of extension over
is 2).
- The constants
and
in the formula for first index of occurrence of Gijswijt's sequence, where k is any integer greater than 1.
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: eπ, e + π, ππ, ee, πe, π√2, eπ2. It has been shown that both e + π and π/e do not satisfy any polynomial equation of degree
and integer coefficients of average size 109. At least one of the numbers ee and ee2 is transcendental.Schanuel's conjecture would imply that all of the above numbers are transcendental and algebraically independent.
- The Euler–Mascheroni constant γ: In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes γ/4) contains at most one algebraic number. In 2012 it was shown that at least one of γ and the Gompertz constant δ is transcendental.
- The values of the Riemann zeta function ζ(n) at odd positive integers
; in particular Apéry's constant ζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.
- The values of the Dirichlet beta function β(n) at even positive integers
; in particular Catalan's Constant β(2). (none of them are known to be irrational).
- Values of the Gamma Function Γ(1/n) for positive integers
and
are not known to be irrational, let alone transcendental. For
at least one the numbers Γ(1/n) and Γ(2/n) is transcendental.
- Any number given by some kind of limit that is not obviously algebraic.
Proofs for specific numbers
A proof that e is transcendental
The first proof that 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:
Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients c0, c1, ..., cn satisfying the equation: It is difficult to make use of the integer status of these coefficients when multiplied by a power of the irrational e, but we can absorb those powers into an integral which “mostly” will assume integer values. For a positive integer k, define the polynomial
and multiply both sides of the above equation by
to arrive at the equation:
By splitting respective domains of integration, this equation can be written in the form where
Here P will turn out to be an integer, but more importantly it grows quickly with k.
Lemma 1
There are arbitrarily large k such that is a non-zero integer.
Proof. Recall the standard integral (case of the Gamma function) valid for any natural number
. More generally,
- if
then
.
This would allow us to compute exactly, because any term of
can be rewritten as
through a change of variables. Hence
That latter sum is a polynomial in
with integer coefficients, i.e., it is a linear combination of powers
with integer coefficients. Hence the number
is a linear combination (with those same integer coefficients) of factorials
; in particular
is an integer.
Smaller factorials divide larger factorials, so the smallest occurring in that linear combination will also divide the whole of
. We get that
from the lowest power
term appearing with a nonzero coefficient in
, but this smallest exponent
is also the multiplicity of
as a root of this polynomial.
is chosen to have multiplicity
of the root
and multiplicity
of the roots
for
, so that smallest exponent is
for
and
for
with
. Therefore
divides
.
To establish the last claim in the lemma, that is nonzero, it is sufficient to prove that
does not divide
. To that end, let
be any prime larger than
and
. We know from the above that
divides each of
for
, so in particular all of those are divisible by
. It comes down to the first term
. We have (see falling and rising factorials)
and those higher degree terms all give rise to factorials
or larger. Hence
That right hand side is a product of nonzero integer factors less than the prime
, therefore that product is not divisible by
, and the same holds for
; in particular
cannot be zero.
Lemma 2
For sufficiently large k, .
Proof. Note that
where u(x), v(x) are continuous functions of x for all x, so are bounded on the interval [0, n]. That is, there are constants G, H > 0 such that
So each of those integrals composing Q is bounded, the worst case being
It is now possible to bound the sum Q as well:
where M is a constant not depending on k. It follows that
finishing the proof of this lemma.
Conclusion
Choosing a value of k that satisfies both lemmas leads to a non-zero integer added to a vanishingly small quantity
being equal to zero: an impossibility. It follows that the original assumption, that e can satisfy a polynomial equation with integer coefficients, is also impossible; that is, e is transcendental.
The transcendence of π
A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof.
For detailed information concerning the proofs of the transcendence of π and e, see the references and external links.
See also
- 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.
|
Notes
- 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.
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- Natarajan, Saradha [in French]; Thangadurai, Ravindranathan (2020). Pillars of Transcendental Number Theory. Springer Verlag. ISBN 978-981-15-4154-4.
- Pytheas Fogg, N. (2002). Berthé, V.; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Springer. ISBN 978-3-540-44141-0. Zbl 1014.11015.
- Shallit, J. (15–26 July 1996). "Number theory and formal languages". In Hejhal, D.A.; Friedman, Joel; Gutzwiller, M.C.; Odlyzko, A.M. (eds.). Emerging Applications of Number Theory. IMA Summer Program. The IMA Volumes in Mathematics and its Applications. Vol. 109. Minneapolis, MN: Springer (published 1999). pp. 547–570. ISBN 978-0-387-98824-5.
External links
- Weisstein, Eric W. "Transcendental Number". MathWorld.
- Weisstein, Eric W. "Liouville Number". MathWorld.
- Weisstein, Eric W. "Liouville's Constant". MathWorld.
- "Proof that e is transcendental". planetmath.org.
- "Proof that the Liouville constant is transcendental". deanlmoore.com. Retrieved 2018-11-12.
- Fritsch, R. (29 March 1988). Transzendenz von e im Leistungskurs? [Transcendence of e in advanced courses?] (PDF). 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]. Der mathematische und naturwissenschaftliche Unterricht (in German). Vol. 42. Kiel, DE (published 1989). pp. 75–80 (presentation), 375–376 (responses). Archived from the original (PDF) on 2011-07-16 – via University of Munich (mathematik.uni-muenchen.de ). — Proof that e is transcendental, in German.
- Fritsch, R. (2003). "Hilberts Beweis der Transzendenz der Ludolphschen Zahl π" (PDF). Дифференциальная геометрия многообразий фигур (in German). 34: 144–148. Archived from the original (PDF) on 2011-07-16 – via University of Munich (mathematik.uni-muenchen.de/~fritsch).
In mathematics a transcendental number is a real or complex number that is not algebraic that is not the root of a non zero polynomial with integer or equivalently rational coefficients The best known transcendental numbers are p and e 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 of real numbers R displaystyle mathbb R and the set of complex numbers C displaystyle mathbb C are both uncountable sets and therefore larger than any countable set All transcendental real numbers also known as real transcendental numbers or transcendental irrational numbers are irrational numbers since all rational numbers are algebraic The converse is not true Not all irrational numbers are transcendental Hence the set of real numbers consists of non overlapping sets of rational algebraic irrational and transcendental real numbers For example the square root of 2 is an irrational number but it is not a transcendental number as it is a root of the polynomial equation x2 2 0 The golden ratio denoted f displaystyle varphi or ϕ displaystyle phi is another irrational number that is not transcendental as it is a root of the polynomial equation x2 x 1 0 HistoryThe name transcendental comes from Latin transcendere to climb over or beyond surmount and was first used for the mathematical concept in Leibniz s 1682 paper in which he proved that sin x is not an algebraic function of x Euler in the eighteenth century was probably the first person to define transcendental numbers in the modern sense Johann Heinrich Lambert conjectured that e and p were both transcendental numbers in his 1768 paper proving the number p is irrational and proposed a tentative sketch proof that p is transcendental Joseph Liouville first proved the existence of transcendental numbers in 1844 and in 1851 gave the first decimal examples such as the Liouville constant Lb n 1 10 n 10 1 10 2 10 6 10 24 10 120 10 720 10 5040 10 40320 0 11000100000000000000000100000000000000000000000000000000000000000000000000000 displaystyle begin aligned L b amp sum n 1 infty 10 n 2pt amp 10 1 10 2 10 6 10 24 10 120 10 720 10 5040 10 40320 ldots 4pt amp 0 textbf 1 textbf 1 000 textbf 1 00000000000000000 textbf 1 00000000000000000000000000000000000000000000000000000 ldots end aligned in which the n th digit after the decimal point is 1 if n k k factorial for some k and 0 otherwise In other words the n th digit of this number is 1 only if n is one of 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 The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers existence was 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 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 Cantor s work established the ubiquity of transcendental numbers In 1882 Ferdinand von Lindemann published the first complete proof that p is transcendental He first proved that ea is transcendental if a is a non zero algebraic number Then since eip 1 is algebraic see Euler s identity ip must be transcendental But since i is algebraic p must therefore be transcendental This approach was generalized by Karl Weierstrass to what is now known as the Lindemann Weierstrass theorem The transcendence of p 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 a is an algebraic number that is not 0 or 1 and b is an irrational algebraic number is ab 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 PropertiesA 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 a rational number is the root of an integer polynomial of degree one 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 p is transcendental it can be immediately deduced that numbers such as 5p displaystyle 5 pi p 32 displaystyle tfrac pi 3 sqrt 2 p 3 8 displaystyle sqrt pi sqrt 3 8 and p5 74 displaystyle sqrt 4 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 p and 1 p are both transcendental but p 1 p 1 is obviously not It is unknown whether e p for example is transcendental though at least one of e p and ep 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 x2 a b x a b If a b and 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 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 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 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 p 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 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 transcendentalNumbers proven to be transcendental p by the Lindemann Weierstrass theorem ea displaystyle e a if a displaystyle a is algebraic and nonzero by the Lindemann Weierstrass theorem in particular Euler s number e epn displaystyle e pi sqrt n where n displaystyle n is a positive integer in particular Gelfond s constant ep displaystyle e pi by the Gelfond Schneider theorem Algebraic combinations of p displaystyle pi and epn n Z displaystyle e pi sqrt n n in mathbb Z such as p ep displaystyle pi e pi and pep displaystyle pi e pi following from their algebraic independence ab displaystyle a b where a displaystyle a is algebraic but not 0 or 1 and b displaystyle b is irrational algebraic in particular the Gelfond Schneider constant 22 displaystyle 2 sqrt 2 by the Gelfond Schneider theorem The natural logarithm ln a displaystyle ln a if a displaystyle a is algebraic and not equal to 0 or 1 for any branch of the logarithm function by the Lindemann Weierstrass theorem logb a displaystyle log b a if a displaystyle a and b displaystyle b are positive integers not both powers of the same integer and a displaystyle a is not equal to 1 by the Gelfond Schneider theorem All numbers of the form p b1ln a1 bnln an displaystyle pi beta 1 ln a 1 cdots beta n ln a n are transcendental where bj displaystyle beta j are algebraic for all 1 j n displaystyle 1 leq j leq n and aj displaystyle a j are non zero algebraic for all 1 j n displaystyle 1 leq j leq n by Baker s theorem The trigonometric functions sin x cos x displaystyle sin x cos x and their hyperbolic counterparts for any nonzero algebraic number x displaystyle x expressed in radians by the Lindemann Weierstrass theorem Non zero results of the inverse trigonometric functions arcsin x arccos x displaystyle arcsin x arccos x and their hyperbolic counterparts for any algebraic number x displaystyle x by the Lindemann Weierstrass theorem p 1arctan x displaystyle pi 1 arctan x for rational x displaystyle x such that x 0 1 displaystyle x notin 0 pm 1 The fixed point of the cosine function also referred to as the Dottie number d displaystyle d the unique real solution to the equation cos x x displaystyle cos x x where x displaystyle x is in radians by the Lindemann Weierstrass theorem W a displaystyle W a if a displaystyle a is algebraic and nonzero for any branch of the Lambert W Function by the Lindemann Weierstrass theorem in particular the omega constant W W r a displaystyle W r a if both a displaystyle a and the order r displaystyle r are algebraic such that a 0 displaystyle a neq 0 for any branch of the generalized Lambert W function xs displaystyle 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 G n 2 G n 3 G n 4 displaystyle Gamma n 2 Gamma n 3 Gamma n 4 or G n 6 displaystyle Gamma n 6 Algebraic combinations of p displaystyle pi and G 1 3 displaystyle Gamma 1 3 or of p displaystyle pi and G 1 4 displaystyle Gamma 1 4 such as the lemniscate constant ϖ displaystyle varpi following from their respective algebraic independences The values of Beta function B a b displaystyle mathrm B a b if a b displaystyle a b and a b displaystyle a b are non integer rational numbers The Bessel function of the first kind Jn x displaystyle J nu x its first derivative and the quotient Jn x Jn x displaystyle tfrac J nu x J nu x are transcendental when n displaystyle nu is rational and x displaystyle x is algebraic and nonzero and all nonzero roots of Jn x displaystyle J nu x and Jn x displaystyle J nu x are transcendental when n displaystyle nu is rational The number p2Y0 2 J0 2 g displaystyle tfrac pi 2 tfrac Y 0 2 J 0 2 gamma where Ya x displaystyle Y alpha x and Ja x displaystyle J alpha x are Bessel functions and g displaystyle gamma is the Euler Mascheroni constant Any Liouville number in particular Liouville s constant Numbers with large irrationality measure such as the Champernowne constant C10 displaystyle C 10 by Roth s theorem Numbers artificially constructed not to be algebraic periods Any non computable number in particular Chaitin s constant Constructed irrational numbers which are not simply normal in any base Any number for which the digits with respect to some fixed base form a Sturmian word The Prouhet Thue Morse constant and the related rabbit constant The Komornik Loreti constant The paperfolding constant also named as Gaussian Liouville number The values of the infinite series with fast convergence rate as defined by Y Gao and J Gao such as n 1 3n23n displaystyle sum n 1 infty frac 3 n 2 3 n Numbers of the form k 0 10 bk displaystyle sum k 0 infty 10 b k and k 0 10 bk displaystyle sum k 0 infty 10 left lfloor b k right rfloor For b gt 1 where b b displaystyle b mapsto lfloor b rfloor is the floor function Any number of the form n 0 En brn Fn brn displaystyle sum n 0 infty frac E n beta r n F n beta r n where En z displaystyle E n z Fn z displaystyle F n z are polynomials in variables n displaystyle n and z displaystyle z b displaystyle beta is algebraic and b 0 displaystyle beta neq 0 r displaystyle r is any integer greater than 1 The numbers a 3 3003300000 displaystyle alpha 3 3003300000 and a 1 0 3030000030 displaystyle 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 The values of the Rogers Ramanujan continued fraction R q displaystyle R q where q C displaystyle q in mathbb C is algebraic and 0 lt q lt 1 displaystyle 0 lt q lt 1 The lemniscatic values of theta function n qn2 displaystyle sum n infty infty q n 2 under the same conditions for q displaystyle q are also transcendental j q where q C displaystyle q in 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 Q displaystyle mathbb Q is 2 The constants ϵk displaystyle epsilon k and nk displaystyle nu k in the formula for first index of occurrence of Gijswijt s sequence where k is any integer greater than 1 Conjectured transcendental numbersNumbers 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 ep e p p p ee pe p 2 ep2 It has been shown that both e p and p e do not satisfy any polynomial equation of degree 8 displaystyle leq 8 and integer coefficients of average size 109 At least one of the numbers ee and ee2 is transcendental Schanuel s conjecture would imply that all of the above numbers are transcendental and algebraically independent The Euler Mascheroni constant g In 2010 it has been shown that an infinite list of Euler Lehmer constants which includes g 4 contains at most one algebraic number In 2012 it was shown that at least one of g and the Gompertz constant d is transcendental The values of the Riemann zeta function z n at odd positive integers n 3 displaystyle n geq 3 in particular Apery s constant z 3 which is known to be irrational For the other numbers z 5 z 7 z 9 even this is not known The values of the Dirichlet beta function b n at even positive integers n 2 displaystyle n geq 2 in particular Catalan s Constant b 2 none of them are known to be irrational Values of the Gamma Function G 1 n for positive integers n 5 displaystyle n 5 and n 7 displaystyle n geq 7 are not known to be irrational let alone transcendental For n 2 displaystyle n geq 2 at least one the numbers G 1 n and G 2 n is transcendental Any number given by some kind of limit that is not obviously algebraic Proofs for specific numbersA proof that e is transcendental The first proof that 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 Assume for purpose of finding a contradiction that e is algebraic Then there exists a finite set of integer coefficients c0 c1 cn satisfying the equation c0 c1e c2e2 cnen 0 c0 cn 0 displaystyle 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 e but we can absorb those powers into an integral which mostly will assume integer values For a positive integer k define the polynomial fk x xk x 1 x n k 1 displaystyle f k x x k left x 1 cdots x n right k 1 and multiply both sides of the above equation by 0 fk x e xdx displaystyle int 0 infty f k x e x mathrm d x to arrive at the equation c0 0 fk x e xdx c1e 0 fk x e xdx cnen 0 fk x e xdx 0 displaystyle c 0 left int 0 infty f k x e x mathrm d x right c 1 e left int 0 infty f k x e x mathrm d x right cdots c n e n left int 0 infty 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 displaystyle P Q 0 where P c0 0 fk x e xdx c1e 1 fk x e xdx c2e2 2 fk x e xdx cnen n fk x e xdx Q c1e 01fk x e xdx c2e2 02fk x e xdx cnen 0nfk x e xdx displaystyle begin aligned P amp c 0 left int 0 infty f k x e x mathrm d x right c 1 e left int 1 infty f k x e x mathrm d x right c 2 e 2 left int 2 infty f k x e x mathrm d x right cdots c n e n left int n infty f k x e x mathrm d x right Q amp c 1 e left int 0 1 f k x e x mathrm d x right c 2 e 2 left int 0 2 f k x e x mathrm d x right cdots c n e n left int 0 n f k x e x mathrm d x right end aligned Here P will turn out to be an integer but more importantly it grows quickly with k Lemma 1 There are arbitrarily large k such that Pk displaystyle tfrac P k is a non zero integer Proof Recall the standard integral case of the Gamma function 0 tje tdt j displaystyle int 0 infty t j e t mathrm d t j valid for any natural number j displaystyle j More generally if g t j 0mbjtj displaystyle g t sum j 0 m b j t j then 0 g t e tdt j 0mbjj displaystyle int 0 infty g t e t mathrm d t sum j 0 m b j j This would allow us to compute P displaystyle P exactly because any term of P displaystyle P can be rewritten as caea a fk x e xdx ca a fk x e x a dx t x ax t adx dt ca 0 fk t a e tdt displaystyle c a e a int a infty f k x e x mathrm d x c a int a infty f k x e x a mathrm d x left begin aligned t amp x a x amp t a mathrm d x amp 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 a 0nca 0 fk t a e tdt 0 a 0ncafk t a e tdt displaystyle 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 displaystyle t with integer coefficients i e it is a linear combination of powers tj displaystyle t j with integer coefficients Hence the number P displaystyle P is a linear combination with those same integer coefficients of factorials j displaystyle j in particular P displaystyle P is an integer Smaller factorials divide larger factorials so the smallest j displaystyle j occurring in that linear combination will also divide the whole of P displaystyle P We get that j displaystyle j from the lowest power tj displaystyle t j term appearing with a nonzero coefficient in a 0ncafk t a displaystyle textstyle sum a 0 n c a f k t a but this smallest exponent j displaystyle j is also the multiplicity of t 0 displaystyle t 0 as a root of this polynomial fk x displaystyle f k x is chosen to have multiplicity k displaystyle k of the root x 0 displaystyle x 0 and multiplicity k 1 displaystyle k 1 of the roots x a displaystyle x a for a 1 n displaystyle a 1 dots n so that smallest exponent is tk displaystyle t k for fk t displaystyle f k t and tk 1 displaystyle t k 1 for fk t a displaystyle f k t a with a gt 0 displaystyle a gt 0 Therefore k displaystyle k divides P displaystyle P To establish the last claim in the lemma that P displaystyle P is nonzero it is sufficient to prove that k 1 displaystyle k 1 does not divide P displaystyle P To that end let k 1 displaystyle k 1 be any prime larger than n displaystyle n and c0 displaystyle c 0 We know from the above that k 1 displaystyle k 1 divides each of ca 0 fk t a e tdt displaystyle textstyle c a int 0 infty f k t a e t mathrm d t for 1 a n displaystyle 1 leqslant a leqslant n so in particular all of those are divisible by k 1 displaystyle k 1 It comes down to the first term c0 0 fk t e tdt displaystyle textstyle c 0 int 0 infty f k t e t mathrm d t We have see falling and rising factorials fk t tk t 1 t n k 1 1 n n k 1tk higher degree terms displaystyle 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 displaystyle k 1 or larger Hence P c0 0 fk t e tdt c0 1 n n k 1k mod k 1 displaystyle 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 displaystyle k 1 therefore that product is not divisible by k 1 displaystyle k 1 and the same holds for P displaystyle P in particular P displaystyle P cannot be zero Lemma 2 For sufficiently large k Qk lt 1 displaystyle left tfrac Q k right lt 1 Proof Note that fke x xk x 1 x 2 x n k 1e x x x 1 x n k x 1 x n e x u x k v x displaystyle begin aligned f k e x amp x k left x 1 x 2 cdots x n right k 1 e x amp left x x 1 cdots x n right k cdot left x 1 cdots x n e x right amp u x k cdot v x end aligned where u x v x are continuous functions of x for all x so are bounded on the interval 0 n That is there are constants G H gt 0 such that fke x u x k v x lt GkH for 0 x n displaystyle left f k e x right leq u x k cdot v x lt G k H quad text for 0 leq x leq n So each of those integrals composing Q is bounded the worst case being 0nfke x d x 0n fke x d x 0nGkH d x nGkH displaystyle 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 nG k H It is now possible to bound the sum Q as well Q lt Gk nH c1 e c2 e2 cn en Gk M displaystyle Q lt G k cdot nH left c 1 e c 2 e 2 cdots c n e n right G k cdot M where M is a constant not depending on k It follows that Qk lt M Gkk 0 as k displaystyle left frac Q k right lt M cdot frac G k k to 0 quad text as k to infty finishing the proof of this lemma Conclusion Choosing a value of k that satisfies both lemmas leads to a non zero integer Pk displaystyle left tfrac P k right added to a vanishingly small quantity Qk displaystyle left tfrac Q k right being equal to zero an impossibility It follows that the original assumption that e can satisfy a polynomial equation with integer coefficients is also impossible that is e is transcendental The transcendence of p A similar strategy different from Lindemann s original approach can be used to show that the number p is transcendental Besides the gamma function and some estimates as in the proof for e facts about symmetric polynomials play a vital role in the proof For detailed information concerning the proofs of the transcendence of p and e see the references and external links See alsoMathematics portalTranscendental 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 Number systems Complex C displaystyle mathbb C Real R displaystyle mathbb R Rational Q displaystyle mathbb Q Integer Z displaystyle mathbb Z Natural N displaystyle mathbb N Zero 0One 1Prime numbersComposite numbersNegative integersFraction Finite decimalDyadic finite binary Repeating decimalIrrational Algebraic irrationalIrrational periodTranscendentalImaginaryNotesCantor 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 ReferencesPickover Cliff The 15 most famous transcendental numbers sprott physics wisc edu Retrieved 2020 01 23 Shidlovskii Andrei B June 2011 Transcendental Numbers Walter de Gruyter p 1 ISBN 9783110889055 Bunday B D Mulholland H 20 May 2014 Pure Mathematics for Advanced Level Butterworth Heinemann ISBN 978 1 4831 0613 7 Retrieved 21 March 2021 Baker A 1964 On Mahler s classification of transcendental numbers Acta Mathematica 111 97 120 doi 10 1007 bf02391010 S2CID 122023355 Heuer Nicolaus Loeh Clara 1 November 2019 Transcendental simplicial volumes arXiv 1911 06386 math GT Real number Encyclopaedia Britannica mathematics Retrieved 2020 08 11 transcendental Oxford English Dictionary s v Leibniz 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published 1999 pp 547 570 ISBN 978 0 387 98824 5 External linksWikisource has original text related to this article Uber die Transzendenz der Zahlen e und p in German Weisstein Eric W Transcendental Number MathWorld Weisstein Eric W Liouville Number MathWorld Weisstein Eric W Liouville s Constant MathWorld Proof that e is transcendental planetmath org Proof that the Liouville constant is transcendental deanlmoore com Retrieved 2018 11 12 Fritsch R 29 March 1988 Transzendenz von e im Leistungskurs Transcendence of e in advanced courses PDF Rahmen der 79 Hauptversammlung des Deutschen Vereins zur Forderung des mathematischen und naturwissenschaftlichen Unterrichts 79th Annual General Meeting of the German Association for the Promotion of Mathematics and Science Education Der mathematische und naturwissenschaftliche Unterricht in German Vol 42 Kiel DE published 1989 pp 75 80 presentation 375 376 responses Archived from the original PDF on 2011 07 16 via University of Munich mathematik uni 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