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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a transcendental number is a real or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
that is not algebraic: that is, not the
root In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
of a non-zero
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
with
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
(or, equivalently,
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
)
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s. The best-known transcendental numbers are and . 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 In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...
real and complex numbers are transcendental, since the algebraic numbers form a
countable set In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
, while the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
real numbers In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
and the set of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s are both
uncountable set In mathematics, an uncountable set, informally, is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger t ...
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 In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
s, since all
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for examp ...
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 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation . The
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
(denoted \varphi or \phi) is another irrational number that is not transcendental, as it is a root of the polynomial equation .


History

The name "transcendental" comes , and was first used for the mathematical concept in Leibniz's 1682 paper in which he proved that is not an algebraic function of .
Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, in the eighteenth century, was probably the first person to define transcendental ''numbers'' in the modern sense.
Johann Heinrich Lambert Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...
conjectured that and were both transcendental numbers in his 1768 paper proving the number is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, and proposed a tentative sketch proof that is transcendental.
Joseph Liouville Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérès ...
first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant \begin L_b &= \sum_^\infty 10^ \\ pt &= 10^ + 10^ + 10^ + 10^ + 10^ + 10^ + 10^ + 10^ + \ldots \\ pt &= 0.\textbf\textbf000\textbf00000000000000000\textbf00000000000000000000000000000000000000000000000000000\ \ldots \end in which the th digit after the decimal point is if = (
factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...
) for some and otherwise. In other words, the th digit of this number is 1 only if is one of , etc. Liouville showed that this number belongs to a class of transcendental numbers that can be more closely approximated by
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s than can any irrational algebraic number, and this class of numbers is called the
Liouville number In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that :0<\left, x-\frac\<\frac. The inequality implies that Liouville numbers po ...
s. 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 , by Charles Hermite in 1873. In 1874
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
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 is transcendental. He first proved that is transcendental if is a non-zero algebraic number. Then, since is algebraic (see
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definit ...
), must be transcendental. But since is algebraic, must therefore be transcendental. This approach was generalized by
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
to what is now known as the
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transc ...
. The transcendence of implies that geometric constructions involving
compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an Idealiz ...
only cannot produce certain results, for example
squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...
. In 1900
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
posed a question about transcendental numbers, Hilbert's seventh problem: If is an
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
that is not 0 or 1, and is an irrational algebraic number, is 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 Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
, since every
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
is the root of some integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, and since each such polynomial has a finite number of zeroes, the
algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s must also be countable. However,
Cantor's diagonal argument Cantor's diagonal argument (among various similar namesthe diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof) is a mathematical proof that there are infin ...
proves that the real numbers (and therefore also the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s to be countable. This makes the transcendental numbers uncountable. No
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
is transcendental and all real transcendental numbers are irrational. The
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
s contain all the real transcendental numbers and a subset of the algebraic numbers, including the
quadratic irrational In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numb ...
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 is transcendental, it can be immediately deduced that numbers such as 5\pi, \tfrac, (\sqrt-\sqrt)^8, and \sqrt /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 In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically i ...
. For example, and are both transcendental, but is obviously not. It is unknown whether , for example, is transcendental, though at least one of and must be transcendental. More generally, for any two transcendental numbers and , at least one of and must be transcendental. To see this, consider the polynomial  . If and were both algebraic, then this would be a polynomial with algebraic coefficients. Because algebraic numbers form an
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
, this would imply that the roots of the polynomial, and , 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 In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that :0<\left, x-\frac\<\frac. The inequality implies that Liouville numbers po ...
s are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its
simple continued fraction A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence \ of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fr ...
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 , one can show that is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that 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: * (by the
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transc ...
). * if is algebraic and nonzero (by the Lindemann–Weierstrass theorem), in particular Euler's number . * 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 2^ (by the Gelfond–Schneider theorem). * The
natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
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 \pi + \beta_1 \ln (a_1) + \cdots + \beta_n \ln (a_n) are transcendental, where \beta_j are algebraic for all 1 \leq j \leq n and a_j are non-zero algebraic for all 1 \leq j \leq n (by Baker's theorem). *The
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and their hyperbolic counterparts, for any nonzero algebraic number , expressed in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s (by the Lindemann–Weierstrass theorem). *Non-zero results of the
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a functi ...
and their hyperbolic counterparts, for any algebraic number (by the Lindemann–Weierstrass theorem). *\pi^, for rational such that x \notin \. *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 In mathematics, the Lambert function, also called the omega function or product logarithm, is a multivalued function, namely the Branch point, branches of the converse relation of the function , where is any complex number and is the expone ...
(by the Lindemann–Weierstrass theorem), in particular the omega constant . * if both and the order are algebraic such that a \neq 0, 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 In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
of rational numbers that are of the form \Gamma(n/2),\Gamma(n/3),\Gamma(n/4) or \Gamma(n/6). * Algebraic combinations of and or of and such as the lemniscate constant \varpi (following from their respective algebraic independences). * The values of
Beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^ ...
\Beta(a,b) if a, b and a+b are non-integer rational numbers. * The Bessel function of the first kind , its first derivative, and the quotient \tfrac are transcendental when ' is rational and ' is algebraic and nonzero, and all nonzero roots of and are transcendental when ' is rational. * The number \tfrac \tfrac - \gamma, where and are Bessel functions and is the Euler–Mascheroni constant. * Values of the Fibonacci zeta function at the positive even argument. * Any
Liouville number In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that :0<\left, x-\frac\<\frac. The inequality implies that Liouville numbers po ...
, in particular: Liouville's constant \sum_^\infty\frac1. * Numbers with irrationality measure larger than 2, such as the Champernowne constant C_ (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 \sum_^\infty \frac. * Any number of the form \sum_^\infty \frac (where E_n(z), F_n(z) are polynomials in variables n and z, \beta is algebraic and \beta \neq 0, r is any integer greater than 1). * Numbers of the form \sum_^\infty 10^ and \sum_^\infty 10^ For where b \mapsto\lfloor b \rfloor is the
floor function In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...
. * The numbers \alpha = 3.3003300000... and \alpha^ = 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) where \in \mathbb C is algebraic and 0 < , q, < 1. The lemniscatic values of theta function \sum_^\infty q^ (under the same conditions for ) are also transcendental. * where \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 \mathbb Q is 2). * The constants \epsilon_k and \nu_k 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: , , , , , , . It has been shown that both and do not satisfy any
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For example, x^5-3x+1=0 is a ...
of degree and integer coefficients of average size 109. At least one of the numbers and is transcendental. Schanuel's conjecture would imply that all of the above numbers are transcendental and
algebraically independent In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K. In particular, a one element set \ is algebraically i ...
. * The Euler–Mascheroni constant '':'' In 2010 it has been shown that an infinite list of Euler-Lehmer constants (which includes ) contains at most one algebraic number. In 2012 it was shown that at least one of and the Gompertz constant is transcendental. * The values of the Riemann zeta function at odd positive integers n\geq3; in particular Apéry's constant , which is known to be irrational. For the other numbers even this is not known. * The values of the Dirichlet beta function at even positive integers n\geq2; in particular Catalan's Constant . (none of them are known to be irrational). * Values of the
Gamma Function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
for positive integers n=5 and n\geq7 are not known to be irrational, let alone transcendental. For n\geq2 at least one the numbers and is transcendental. * Any number given by some kind of limit that is not obviously algebraic.


Proofs for specific numbers


A proof that is transcendental

The first proof that the base of the natural logarithms, , is transcendental dates from 1873. We will now follow the strategy of
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
(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 is algebraic. Then there exists a finite set of integer coefficients satisfying the equation: c_ + c_e + c_ e^ + \cdots + c_ e^ = 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 , but we can absorb those powers into an integral which “mostly” will assume integer values. For a positive integer , define the polynomial f_k(x) = x^ \left x-1)\cdots(x-n) \right , and multiply both sides of the above equation by \int^_ f_k(x) \, e^\, \mathrmx\ , to arrive at the equation: c_0 \left (\int^_ f_k(x) e^ \,\mathrmx \right) + c_1 e \left( \int^_ f_k(x) e^ \,\mathrmx \right ) + \cdots + c_e^ \left( \int^_ f_k(x) e^ \,\mathrmx \right) = 0 ~. By splitting respective domains of integration, this equation can be written in the form P + Q = 0 where \begin P &= c_ \left( \int^_ f_k(x) e^ \,\mathrmx \right) + c_ e \left( \int^_ f_k(x) e^ \,\mathrmx \right) + c_ e^ \left( \int^_ f_k(x) e^ \,\mathrmx \right) + \cdots + c_ e^ \left( \int^_ f_k(x) e^ \,\mathrmx \right) \\ Q &= c_ e \left(\int^_ f_k(x) e^ \,\mathrmx \right) + c_e^ \left( \int^_ f_k(x) e^ \,\mathrmx \right) + \cdots+c_ e^ \left( \int^_ f_k(x) e^ \,\mathrmx \right) \end Here will turn out to be an integer, but more importantly it grows quickly with .


Lemma 1

''There are arbitrarily large such that \ \tfrac\ is a non-zero integer.'' Proof. Recall the standard integral (case of the
Gamma function In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...
) \int^_ t^ e^ \,\mathrmt = j! valid for any
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
j. More generally, : if g(t) = \sum_^m b_j t^j then \int^_ g(t) e^ \,\mathrmt = \sum_^m b_j j! . This would allow us to compute P exactly, because any term of P can be rewritten as c_ e^ \int^_ f_k(x) e^ \,\mathrmx = c_ \int^_ f_k(x) e^ \,\mathrmx = \left\ = c_a \int_0^\infty f_k(t+a) e^ \,\mathrmt through a change of variables. Hence P = \sum_^n c_a \int_0^\infty f_k(t+a) e^ \,\mathrmt = \int_0^\infty \biggl( \sum_^n c_a f_k(t+a) \biggr) e^ \,\mathrmt 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 \textstyle \sum_^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,\dots,n, so that smallest exponent is t^k for f_k(t) and t^ 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
larger than n and , c_0, . We know from the above that (k+1)! divides each of \textstyle c_a \int_0^\infty f_k(t+a) e^ \,\mathrmt for 1 \leqslant a \leqslant n , so in particular all of those ''are'' divisible by k+1. It comes down to the first term \textstyle c_0 \int_0^\infty f_k(t) e^ \,\mathrmt . We have (see falling and rising factorials) f_k(t) = t^k \bigl (t-1) \cdots (t-n) \bigr = \bigl (-1)^(n!) \bigr t^k + \text 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^ \,\mathrmt \equiv c_0 \bigl (-1)^(n!) \bigr k! \pmod 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.


Lemma 2

''For sufficiently large , \left, \tfrac \ <1.'' Proof. Note that \begin f_k e^ &= x^ \left (x-1)(x-2) \cdots (x-n) \right e^\\ &= \left (x(x-1)\cdots(x-n) \right)^k \cdot \left( (x-1) \cdots (x-n) e^ \right) \\ &= u(x)^k \cdot v(x) \end where are continuous functions of for all , so are bounded on the interval . That is, there are constants such that \ \left, f_k e^ \ \leq , u(x), ^k \cdot , v(x), < G^k H \quad \text 0 \leq x \leq n ~. So each of those integrals composing is bounded, the worst case being \left, \int_^ f_ e^\ \mathrm\ x \ \leq \int_^ \left, f_ e^ \ \ \mathrm\ x \leq \int_^G^k H\ \mathrm\ x = n G^k H ~. It is now possible to bound the sum as well: , Q, < G^ \cdot n H \left( , c_1, e+, c_2, e^2 + \cdots+, c_n, e^ \right) = G^k \cdot M\ , where is a constant not depending on . It follows that \ \left, \frac \ < M \cdot \frac \to 0 \quad \text k \to \infty\ , finishing the proof of this lemma.


Conclusion

Choosing a value of that satisfies both lemmas leads to a non-zero integer \left(\tfrac\right) added to a vanishingly small quantity \left(\tfrac\right) being equal to zero: an impossibility. It follows that the original assumption, that can satisfy a polynomial equation with integer coefficients, is also impossible; that is, is transcendental.


The transcendence of

A similar strategy, different from Lindemann's original approach, can be used to show that the number is transcendental. Besides the gamma-function and some estimates as in the proof for , facts about symmetric polynomials play a vital role in the proof. For detailed information concerning the proofs of the transcendence of and , see the references and external links.


See also

*
Transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendenc ...
, the study of questions related to transcendental numbers * Transcendental element, generalization of transcendental numbers in abstract algebra * Gelfond–Schneider theorem *
Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...
* Periods, a countable set of numbers (including all algebraic and some transcendental numbers) which may be defined by integral equations.


Notes


References


Sources

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External links

* * * * * * — Proof that is transcendental, in German. * {{Authority control Articles containing proofs