Transcendental number

In mathematics, a transcendental number is a number that is not algebraic: it is not the root of a non-zero polynomial with rational coefficients. The best known transcendental numbers are π and e.[1][2]

Pi (π) is a well known transcendental number

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 common. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers compose a countable set, but the set of real numbers and the set of complex numbers are both uncountable sets and so are larger than any countable set. All real transcendental numbers are irrational numbers since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. 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 ${\displaystyle \varphi }$ or ${\displaystyle \phi }$) 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 the Latin transcendĕre 'to climb over or beyond, surmount',[3] 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.[4][5] Euler, in the 18th century, was probably the first person to define transcendental numbers in the modern sense.[6]

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 of a proof of π's transcendence.[7]

Joseph Liouville first proved the existence of transcendental numbers in 1844,[8] and in 1851 gave the first decimal examples such as the Liouville constant

${\displaystyle {\begin{aligned}L_{b}&=\sum _{n=1}^{\infty }10^{-n!}\\&=10^{-1}+10^{-2}+10^{-6}+10^{-24}+10^{-120}+10^{-720}+10^{-5040}+10^{-40320}+\ldots \\&=0.{\textbf {1}}{\textbf {1}}000{\textbf {1}}00000000000000000{\textbf {1}}00000000000000000000000000000000000000000000000000000\ldots \\\end{aligned}}}$

in which the nth digit after the decimal point is 1 if n is equal to k! (k factorial) for some k and 0 otherwise.[9] 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 are called Liouville numbers, named in honour of him. Liouville showed that all Liouville numbers are transcendental.[10]

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.[11][12] 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.[13] Cantor's work established the ubiquity of transcendental numbers.

In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of π. He first proved that e^a is transcendental when a is any non-zero algebraic number. Then, since e^iπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, π therefore must be transcendental. This approach was generalized by Karl Weierstrass to what is now known as the Lindemann–Weierstrass theorem. The transcendence of π allowed the proof of the impossibility of several ancient geometric constructions involving compass and straightedge, including the most famous one, squaring the circle.

In 1900, David Hilbert posed an influential 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 a^b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).[14]

Properties

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, they cannot both 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.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. For example, from knowing that π is transcendental, it can be immediately deduced that numbers such as 5π, π-3/√2, (√π-√3)⁸, and ⁴√π⁵+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, 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 + ab. If (a + b) and ab 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 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 are not eventually periodic are transcendental (eventually periodic continued fractions correspond to quadratic irrationals).[15]

Numbers proven to be transcendental

Numbers proven to be transcendental:

• e^a if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem).
• π (by the Lindemann–Weierstrass theorem).
• e^π, Gelfond's constant, as well as e^−π/2 = iⁱ (by the Gelfond–Schneider theorem).
• a^b where a is algebraic but not 0 or 1, and b is irrational algebraic (by the Gelfond–Schneider theorem), in particular:

2^√2, the Gelfond–Schneider constant (or Hilbert number)

• sin a, cos a, tan a, csc a, sec a, and cot a, and their hyperbolic counterparts, for any nonzero algebraic number a, expressed in radians (by the Lindemann–Weierstrass theorem).
• The fixed point of the cosine function (also referred to as the dottie number d) – the unique real solution to the equation cos x = x, where x is in radians (by the Lindemann–Weierstrass theorem).[16]
• ln a if a is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
• log_b a if a and b are positive integers not both powers of the same integer (by the Gelfond–Schneider theorem).
• W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem), in particular: Ω the omega constant
• √x_s, the square super-root of any natural number is either an integer or transcendental (by the Gelfond-Schneider theorem)
• Γ(1/3),[17] Γ(1/4),[18] and Γ(1/6).[18]
• 0.64341054629..., Cahen's constant.[19]
• The Champernowne constants, the irrational numbers formed by concatenating representations of all positive integers.[20][21]
• Ω, Chaitin's constant (since it is a non-computable number).[22]
• The so-called Fredholm constants, such as[8][23][24]

  ${\displaystyle \sum _{n=0}^{\infty }10^{-2^{n}}=0.{\textbf {1}}{\textbf {1}}0{\textbf {1}}000{\textbf {1}}0000000{\textbf {1}}\ldots }$

which also holds by replacing 10 with any algebraic b > 1.[25]

• The Gauss constant.
• The two lemniscate constants L₁ (sometimes denoted as ϖ) and L₂.
• The aforementioned Liouville constant for any algebraic b ∈ (0, 1).
• The Prouhet–Thue–Morse constant.[26][27]
• The Komornik-Loreti constant.
• Any number for which the digits with respect to some fixed base form a Sturmian word.[28]
• For β > 1

${\displaystyle \sum _{k=0}^{\infty }10^{-\left\lfloor \beta ^{k}\right\rfloor };}$

where ${\displaystyle \beta \mapsto \lfloor \beta \rfloor }$ is the floor function.

• 3.300330000000000330033... and its reciprocal 0.30300000303..., two numbers with only two different decimal digits whose nonzero digit positions are given by the Moser–de Bruijn sequence and its double.[29]
• The number π/2Y₀(2)/J₀(2)-γ, where Y_α(x) and J_α(x) are Bessel functions and γ is the Euler-Mascheroni constant.[30][31]

Possible transcendental numbers

Numbers which have yet to be proven to be either transcendental or algebraic:

• Most sums, products, powers, etc. of the number π and the number e, e.g. eπ, e + π, π − e, π/e, π^π, e^e, π^e, π^√2, e^π2 are not known to be rational, algebraic, irrational or transcendental. A notable exception is e^π√n (for any positive integer n) which has been proven transcendental.[32]
• The Euler–Mascheroni constant γ: In 2010 M. Ram Murty and N. Saradha considered an infinite list of numbers also containing γ/4 and showed that all but at most one of them have to be transcendental.[33][34] In 2012 it was shown that at least one of γ and the Euler-Gompertz constant δ is transcendental.[35]
• Catalan's constant, not even proven to be irrational.
• Khinchin's constant, also not proven to be irrational.
• Apéry's constant ζ(3) (which Apéry proved is irrational).
• The Riemann zeta function at other odd integers, ζ(5), ζ(7), ... (not proven to be irrational).
• The Feigenbaum constants δ and α, also not proven to be irrational.
• Mills' constant, also not proven to be irrational.
• The Copeland–Erdős constant, formed by concatenating the decimal representations of prime numbers.

Conjectures:

• Schanuel's conjecture,
• Four exponentials conjecture.

Sketch of 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, ..., c_n satisfying the equation:

${\displaystyle c_{0}+c_{1}e+c_{2}e^{2}+\cdots +c_{n}e^{n}=0,\qquad c_{0},c_{n}\neq 0.}$

Now for a positive integer k, we define the following polynomial:

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

and multiply both sides of the above equation by

${\displaystyle \int _{0}^{\infty }f_{k}e^{-x}\,dx,}$

to arrive at the equation:

${\displaystyle c_{0}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)=0.}$

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

${\displaystyle P+Q=0}$

where

${\displaystyle {\begin{aligned}P&=c_{0}\left(\int _{0}^{\infty }f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int _{1}^{\infty }f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int _{2}^{\infty }f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{n}^{\infty }f_{k}e^{-x}\,dx\right)\\Q&=c_{1}e\left(\int _{0}^{1}f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int _{0}^{2}f_{k}e^{-x}\,dx\right)+\cdots +c_{n}e^{n}\left(\int _{0}^{n}f_{k}e^{-x}\,dx\right)\end{aligned}}}$

Lemma 1. For an appropriate choice of k, ${\displaystyle {\tfrac {P}{k!}}}$ is a non-zero integer.

Proof. Each term in P is an integer times a sum of factorials, which results from the relation

${\displaystyle \int _{0}^{\infty }x^{j}e^{-x}\,dx=j!}$

which is valid for any positive integer j (consider the Gamma function).

It is non-zero because for every a satisfying 0< a ≤ n, the integrand in

${\displaystyle c_{a}e^{a}\int _{a}^{\infty }f_{k}e^{-x}\,dx}$

is e^−x times a sum of terms whose lowest power of x is k+1 after substituting x for x+a in the integral. Then this becomes a sum of integrals of the form

${\displaystyle A_{j-k}\int _{0}^{\infty }x^{j}e^{-x}\,dx}$ Where A_j-k is integer.

with k+1 ≤ j, and it is therefore an integer divisible by (k+1)!. After dividing by k!, we get zero modulo (k+1). However, we can write:

${\displaystyle \int _{0}^{\infty }f_{k}e^{-x}\,dx=\int _{0}^{\infty }\left(\left[(-1)^{n}(n!)\right]^{k+1}e^{-x}x^{k}+\cdots \right)dx}$

and thus

${\displaystyle {\frac {1}{k!}}c_{0}\int _{0}^{\infty }f_{k}e^{-x}\,dx\equiv c_{0}[(-1)^{n}(n!)]^{k+1}\not \equiv 0{\pmod {k+1}}.}$

So when dividing each integral in P by k!, the initial one is not divisible by k+1, but all the others are, as long as k+1 is prime and larger than n and |c₀|. It follows that ${\displaystyle {\tfrac {P}{k!}}}$ itself is not divisible by the prime k+1 and therefore cannot be zero.

Lemma 2. ${\displaystyle \left|{\tfrac {Q}{k!}}\right|<1}$ for sufficiently large ${\displaystyle k}$.

Proof. Note that

${\displaystyle {\begin{aligned}f_{k}e^{-x}&=x^{k}[(x-1)(x-2)\cdots (x-n)]^{k+1}e^{-x}\\&=\left(x(x-1)\cdots (x-n)\right)^{k}\cdot \left((x-1)\cdots (x-n)e^{-x}\right)\\&=u(x)^{k}\cdot v(x)\end{aligned}}}$

where ${\displaystyle u(x)}$ and ${\displaystyle v(x)}$ are continuous functions of ${\displaystyle x}$ for all ${\displaystyle x}$, so are bounded on the interval ${\displaystyle [0,n]}$. That is, there are constants ${\displaystyle G,H>0}$ such that

${\displaystyle \left|f_{k}e^{-x}\right|\leq |u(x)|^{k}\cdot |v(x)|<G^{k}H\quad {\text{ for }}0\leq x\leq n.}$

So each of those integrals composing ${\displaystyle Q}$ is bounded, the worst case being

${\displaystyle \left|\int _{0}^{n}f_{k}e^{-x}\,dx\right|\leq \int _{0}^{n}\left|f_{k}e^{-x}\right|\,dx\leq \int _{0}^{n}G^{k}H\,dx=nG^{k}H.}$

It is now possible to bound the sum ${\displaystyle Q}$ as well:

${\displaystyle |Q|<G^{k}\cdot nH\left(|c_{1}|e+|c_{2}|e^{2}+\cdots +|c_{n}|e^{n}\right)=G^{k}\cdot M,}$

where ${\displaystyle M}$ is a constant not depending on ${\displaystyle k}$. It follows that

${\displaystyle \left|{\frac {Q}{k!}}\right|<M\cdot {\frac {G^{k}}{k!}}\to 0\quad {\text{ as }}k\to \infty ,}$

finishing the proof of this lemma.

Choosing a value of ${\displaystyle k}$ satisfying both lemmas leads to a non-zero integer (${\displaystyle P/k!}$) added to a vanishingly small quantity (${\displaystyle Q/k!}$) being equal to zero, is 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
• Gelfond-Schneider theorem
• Diophantine approximation
• Periods, a set of numbers (including both transcendental and algebraic numbers) which may be defined by integral equations.

Notes

1. "The 15 Most Famous Transcendental Numbers - Cliff Pickover". sprott.physics.wisc.edu. Retrieved 2020-01-23.
2. Shidlovskii, Andrei B. (June 2011). Transcendental numbers. Walter de Gruyter. p. 1. ISBN 9783110889055.
3. Oxford English Dictionary, s.v.
4. Leibniz, Gerhardt & Pertz 1858, pp. 97–98.
5. Bourbaki 1994, p. 74.
6. Erdős & Dudley 1983.
7. Lambert 1768.
8. Kempner 1916.
9. Weisstein, Eric W. "Liouville's Constant", MathWorld
10. Liouville 1851.
11. Cantor 1874.
12. Gray 1994.
13. Cantor 1878, p. 254. 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.
14. J J O'Connor and E F Robertson: Alan Baker. The MacTutor History of Mathematics archive 1998.
15. Adamczewski & Bugeaud 2005.
16. Weisstein, Eric W. "Dottie Number". Wolfram MathWorld. Wolfram Research, Inc. Retrieved 23 July 2016.
17. Le Lionnais 1979, p. 46 via Wolfram Mathworld, Transcendental Number
18. Chudnovsky 1984 via Wolfram Mathworld, Transcendental Number
19. Davison & Shallit 1991.
20. Mahler 1937.
21. Mahler 1976, p. 12.
22. Calude 2002, p. 239.
23. Allouche & Shallit 2003, pp. 385,403. The name 'Fredholm number' is misplaced: Kempner first proved this number is transcendental, and the note on page 403 states that Fredholm never studied this number.
24. Shallit 1999.
25. Loxton 1988.
26. Mahler 1929.
27. Allouche & Shallit 2003, p. 387.
28. Pytheas Fogg 2002.
29. Blanchard & Mendès France 1982.
30. Mahler, Kurt; Mordell, Louis Joel (1968-06-04). "Applications of a theorem by A. B. Shidlovski". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 305 (1481): 149–173. Bibcode:1968RSPSA.305..149M. doi:10.1098/rspa.1968.0111. S2CID 123486171.
31. Lagarias, Jeffrey C. (2013-07-19). "Euler's constant: Euler's work and modern developments". Bulletin of the American Mathematical Society. 50 (4): 527–628. arXiv:1303.1856. doi:10.1090/S0273-0979-2013-01423-X. ISSN 0273-0979.
32. Weisstein, Eric W. "Irrational Number". MathWorld.
33. Murty, M. Ram; Saradha, N. (2010-12-01). "Euler–Lehmer constants and a conjecture of Erdös". Journal of Number Theory. 130 (12): 2671–2682. doi:10.1016/j.jnt.2010.07.004. ISSN 0022-314X.
34. Murty, M. Ram; Zaytseva, Anastasia (2013-01-01). "Transcendence of Generalized Euler Constants". The American Mathematical Monthly. 120 (1): 48–54. doi:10.4169/amer.math.monthly.120.01.048. ISSN 0002-9890. S2CID 20495981.
35. Rivoal, Tanguy (2012). "On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant". Michigan Mathematical Journal. 61 (2): 239–254. doi:10.1307/mmj/1339011525. ISSN 0026-2285.

References

• Adamczewski, Boris; Bugeaud, Yann (2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica. 195 (1): 1–20. arXiv:math/0511677. Bibcode:2005math.....11677A. doi:10.1007/BF02588048. S2CID 15521751.
• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Baker, Alan (1990). Transcendental Number Theory (paperback ed.). Cambridge University Press. ISBN 978-0-521-20461-3. Zbl 0297.10013.
• Blanchard, André; Mendès France, Michel (1982). "Symétrie et transcendance". Bulletin des Sciences Mathématiques. 106 (3): 325–335. MR 0680277.
• Bourbaki, Nicolas (1994). Elements of the History of Mathematics. Springer.
• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. 193. Cambridge University Press. ISBN 978-0-521-11169-0. Zbl 1260.11001.
• Burger, Edward B.; Tubbs, Robert (2004). Making transcendence transparent. An intuitive approach to classical transcendental number theory. Springer. ISBN 978-0-387-21444-3. Zbl 1092.11031.
• Calude, Cristian S. (2002). Information and Randomness: An Algorithmic Perspective. Texts in Theoretical Computer Science (2nd rev. and ext. ed.). Springer. ISBN 978-3-540-43466-5. Zbl 1055.68058.
• Cantor, Georg (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262.
• Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". J. Reine Angew. Math. 84: 242–258.
• Chudnovsky, G. V. (1984). Contributions to the Theory of Transcendental Numbers. American Mathematical Society. ISBN 978-0-8218-1500-7.
• Davison, J. Les; Shallit, Jeffrey O. (1991). "Continued fractions for some alternating series". Monatshefte für Mathematik. 111 (2): 119–126. doi:10.1007/BF01332350. S2CID 120003890.
• Erdős, Paul; Dudley, Underwood (1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler" (PDF). Mathematics Magazine. 56 (5): 292–298. CiteSeerX 10.1.1.210.6272. doi:10.2307/2690369. JSTOR 2690369.
• Gelfond, Alexander (1960) [1956]. Transcendental and Algebraic Numbers. Dover.
• Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly. 101 (9): 819–832. doi:10.2307/2975129. JSTOR 2975129. Zbl 0827.01004.
• Higgins, Peter M. (2008). Number Story. Copernicus Books. ISBN 978-1-84800-001-8.
• Hilbert, David (1893). "Über die Transcendenz der Zahlen e und ${\displaystyle \pi }$". Mathematische Annalen. 43: 216–219. doi:10.1007/BF01443645. S2CID 179177945.
• Kempner, Aubrey J. (1916). "On Transcendental Numbers". Transactions of the American Mathematical Society. 17 (4): 476–482. doi:10.2307/1988833. JSTOR 1988833.
• Lambert, Johann Heinrich (1768). "Mémoire sur quelques propriétés remarquables des quantités transcendantes, circulaires et logarithmiques". Mémoires de l'Académie Royale des Sciences de Berlin: 265–322.
• Leibniz, Gottfried Wilhelm; Gerhardt, Karl Immanuel; Pertz, Georg Heinrich (1858). Leibnizens mathematische Schriften. 5. A. Asher & Co. pp. 97–98.
• Le Lionnais, François (1979). Les nombres remarquables. Hermann. ISBN 2-7056-1407-9.
• LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. Dover. ISBN 978-0-486-42539-9.
• Liouville, Joseph (1851). "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques" (PDF). J. Math. Pures Appl. 16: 133–142.
• Loxton, J. H. (1988). "13. Automata and transcendence". In Baker, A. (ed.). New Advances in Transcendence Theory. Cambridge University Press. pp. 215–228. ISBN 978-0-521-33545-4. Zbl 0656.10032.
• Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen. 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01. S2CID 120549929.
• Mahler, Kurt (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A. (40): 421–428.
• Mahler, Kurt (1976). Lectures on Transcendental Numbers. Lecture Notes in Mathematics. 546. Springer. ISBN 978-3-540-07986-6. Zbl 0332.10019.
• Natarajan, Saradha; Thangadurai, Ravindranathan (2020). Pillars of Transcendental Number Theory. Springer Verlag. ISBN 978-981-15-4154-4.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Springer. ISBN 978-3-540-44141-0. Zbl 1014.11015.
• Shallit, Jeffrey (1999). "Number theory and formal languages". In Hejhal, Dennis A.; Friedman, Joel; Gutzwiller, Martin C.; Odlyzko, Andrew M. (eds.). Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15-26, 1996. The IMA volumes in mathematics and its applications. 109. Springer. pp. 547–570. ISBN 978-0-387-98824-5.

External links

• Transcendental number (mathematics) at the Encyclopædia Britannica
• Weisstein, Eric W. "Transcendental Number". MathWorld.
• Weisstein, Eric W. "Liouville Number". MathWorld.
• Weisstein, Eric W. "Liouville's Constant". MathWorld.
• (in English) Proof that e is transcendental
• (in English) Proof that the Liouville Constant is transcendental
• (in German) Proof that e is transcendental (PDF)
• (in German) Proof that π is transcendental (PDF)