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.[3][4][5][6] 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.[3] 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',[7] 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.[8]Euler, in the eighteenth century, was probably the first person to define transcendental numbers in the modern sense.[9]
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.[10]
Joseph Liouville first proved the existence of transcendental numbers in 1844,[11] 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 is equal to k! (kfactorial) for some k and 0 otherwise.[12] In other words, the nth digit of this number is 1 only if n is one of the numbers 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, named in his honour. Liouville showed that all Liouville numbers are transcendental.[13]
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.[14] 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.[a]
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 zero or one, 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).[16]
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.[17] 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.
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[18] (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).
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).[21]
if is algebraic and nonzero, for any branch of the
Lambert W Function (by the Lindemann–Weierstrass theorem), in particular the
omega constantΩ.
if both and the order are algebraic such that , for any branch of the generalized Lambert W function.[22]
, 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 .[23]
Algebraic combinations of and or of and such as the
lemniscate constant (following from their respective algebraic independences).[19]
The values of
Beta function if and are non-integer rational numbers.[24]
The
Bessel function of the first kind, its first derivative, and the quotient are transcendental when is rational and is algebraic and nonzero,[25] and all nonzero roots of and are transcendental when is rational.[26]
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.[46]
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.[47][48] At least one of the numbers ee and ee2 is transcendental.[49]Schanuel's conjecture would imply that all of the above numbers are transcendental and
algebraically independent.[50]
The values of the
Riemann zeta functionζ(n) at odd positive integers ; in particular
Apéry's constantζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.
Values of the
Gamma FunctionΓ(1/n) for positive integers and are not known to be irrational, let alone transcendental.[55][56] For at least one the numbers Γ(1/n) and Γ(2/n) is transcendental.[24]
Any number given by some kind of
limit that is not obviously algebraic.[56]
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.
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.
^
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.[15]
^Weisstein, Eric W.
"Dottie Number". Wolfram MathWorld. Wolfram Research, Inc. Retrieved 23 July 2016.
^Mező, István; Baricz, Árpád (June 22, 2015). "On the generalization of the Lambert W function".
arXiv:1408.3999 [
math.CA].
^Chudnovsky, G. (1984). Contributions to the theory of transcendental numbers. Mathematical surveys and monographs (in English and Russian). Providence, R.I: American Mathematical Society.
ISBN978-0-8218-1500-7.
^Weisstein, Eric W.
"Rabbit Constant". mathworld.wolfram.com. Retrieved 2023-08-09.
^Allouche, Jean-Paul; Cosnard, Michel (2000), "The Komornik–Loreti constant is transcendental", American Mathematical Monthly, 107 (5): 448–449,
doi:
10.2307/2695302,
JSTOR2695302,
MR1763399
Burger, Edward B.; Tubbs, Robert (2004). Making transcendence transparent. An intuitive approach to classical transcendental number theory.
Springer.
ISBN978-0-387-21444-3.
Zbl1092.11031.
Calude, Cristian S. (2002). Information and Randomness: An algorithmic perspective. Texts in Theoretical Computer Science (2nd rev. and ext. ed.).
Springer.
ISBN978-3-540-43466-5.
Zbl1055.68058.
Lambert, J.H. (1768). "Mémoire sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques". Mémoires de l'Académie Royale des Sciences de Berlin: 265–322.
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.
ISBN978-3-540-44141-0.
Zbl1014.11015.
Shallit, J. (15–26 July 1996). "Number theory and formal languages". In
Hejhal, D.A.; Friedman, Joel;
Gutzwiller, M.C.;
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External links
Wikisource has original text related to this article:
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.