Transcendental number

Pi (π) is the best-known transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The best-known transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part 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 are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x² − 2 = 0.

History

The name "transcendental" comes from Leibniz in his 1682 paper where he proved that sin(x) is not an algebraic function of x.^[1][2] Euler was probably the first person to define transcendental numbers in the modern sense.^[3]

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers in his 1768 paper proving the number π is irrational, and even proposed a tentative sketch of a proof of π's transcendence.^[4]

Joseph Liouville first proved the existence of transcendental numbers in 1844,^[5] and in 1851 gave the first decimal examples such as the Liouville constant

$\sum_{k=1}^\infty 10^{-k!} = 0.1100010000000000000000010000\ldots$

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.^[6] 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 is what we now call a Liouville number; this essentially means that it can be more closely approximated by rational numbers than can any irrational algebraic number. Liouville showed that all Liouville numbers are transcendental.^[7]

The first number to be proven transcendental without having been specifically constructed for the purpose 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.^[8][9] In 1878, Cantor published a construction that proves there are as many transcendental numbers as there are real numbers.^[10] Cantor's work established the ubiquity of transcendental numbers.

In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that e^a is transcendental when a is algebraic and not zero. Then, since e^iπ = −1 is algebraic (see Euler's identity), iπ and therefore π must be transcendental. This approach was generalized by Karl Weierstrass to 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).^[11]

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, we can immediately deduce that numbers such as 5π, (π − 3)/√2, (√π − √3)⁸ and (π⁵ + 7)^1/7 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 π + e, for example, is transcendental, though at least one of π + e and πe 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).^[12]

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=i ⁱ (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:

$2^\sqrt{2},$ the Gelfond–Schneider constant (or Hilbert number).

• The Continued Fraction Constant, Carl Ludwig Siegel (1929)

${1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}}$

• sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem).
• 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).
• W(a) if a is algebraic and nonzero, for any branch of the Lambert W Function (by the Lindemann–Weierstrass theorem).
• Γ(1/3),^[13] Γ(1/4),^[14] and Γ(1/6).^[14]
• 0.64341054629..., Cahen's constant.^[15]
• 0.12345678910111213141516..., the Champernowne constant.^[16][17]
• Ω, Chaitin's constant (since it is a non-computable number).^[18]
• The Fredholm number^[19][20]

$\sum_{n=0}^\infty 2^{-2^n}$

more generally, any number of the form

$\sum_{n=0}^\infty \beta^{2^n}$

with 0 < |β| < 1 and β algebraic.^[21]

• The aforementioned Liouville constant

$\sum_{n=1}^\infty 10^{-n!};$

more generally any number of the form

$\sum_{n=1}^\infty \beta^{n!}$

with 0 < |β| < 1 and β algebraic.

• The Prouhet–Thue–Morse constant.^[22][23]
• Any number for which the digits with respect to some fixed base form a Sturmian word.^[24]
• For β > 1

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

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

Possibly 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. π + e, π − e, πe, π/e, π^π, e^e, π^e, π^√2, e^π2 are not known to be rational, algebraic irrational or transcendental. Notable exceptions are π + e^π, πe^π and e^π√n (for any positive integer n) which have been proven to be transcendental.^[25][26]
• The Euler–Mascheroni constant γ (which has not even been proven to be irrational).
• Catalan's constant, also not known 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 known to be irrational.)
• The Feigenbaum constants, δ and α.
• Mills' constant.

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:

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

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

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

and multiply both sides of the above equation by

$\int^{\infty}_{0} f_k e^{-x}\,dx,$

to arrive at the equation:

$c_{0} \left (\int^{\infty}_{0} f_k e^{-x}\,dx\right )+ c_1e\left ( \int^{\infty}_{0}f_k e^{-x}\,dx\right )+\cdots+ c_{n}e^{n} \left (\int^{\infty}_{0}f_k e^{-x}\,dx\right ) = 0.$

This equation can be written in the form

$P+Q=0$

where

$P =c_{0}\left ( \int^{\infty}_{0}f_k e^{-x}\,dx\right )+ c_{1}e\left (\int^{\infty}_{1}f_k e^{-x}\,dx\right )+ c_{2}e^{2}\left (\int^{\infty}_{2}f_k e^{-x}\,dx\right ) +\cdots+ c_{n}e^{n}\left (\int^{\infty}_{n}f_k e^{-x}\,dx\right )$

$Q=c_{1}e\left (\int^{1}_{0} f_k e^{-x}\,dx\right )+c_{2}e^{2} \left (\int^{2}_{0} f_k e^{-x}\,dx\right )+\cdots+c_{n}e^{n}\left (\int^{n}_{0} f_k e^{-x}\,dx \right )$

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

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

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

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

$c_{a}e^{a}\int^{\infty}_{a} f_k e^{-x}\,dx$

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

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

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:

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

and thus

${\frac{1}{k!}}c_{0}\int^{\infty}_{0} f_k e^{-x}\,dx = c_{0}[(-1)^{n}(n!)]^{k+1} \qquad \mod (k+1).$

By choosing k so that k+1 is prime and larger than n and |c₀|, we get that $\tfrac{P}{k!}$ is non-zero modulo (k+1) and is thus non-zero.

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

Proof. Note that

$f_k e^{-x} = x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x} = \left ([x(x-1)\cdots(x-n)]^k \right ) \left ((x-1)\cdots(x-n)e^{-x}\right )$

Using upper bounds G and H for $|x(x-1)\cdots(x-n)|$ and $|(x-1)\cdots(x-n)e^{-x}|$ on the interval [0,n] we can infer that

$|Q|<G^{k}H(|c_{1}|e+2|c_{2}|e^{2}+\cdots+n|c_{n}|e^{n})$

and since

$\lim_{k\to\infty}\frac{G^k}{k!}=0$

it follows that

$\lim_{k\to\infty}\frac{Q}{k!}=0$

which is sufficient to finish the proof of this lemma.

Noting that one can choose k so that both Lemmas hold we get the contradiction we needed to prove the transcendence of e.

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.

Mahler's classification

Kurt Mahler in 1932 partitioned the transcendental numbers into 3 classes, called S, T, and U.^[27] Definition of these classes draws on an extension of the idea of a Liouville number (cited above).

Measure of irrationality of a real number

One way to define a Liouville number is to consider how small a given real number x makes linear polynomials |qx − p| without making them exactly 0. Here p, q are integers with |p|, |q| bounded by a positive integer H.

Let m(x, 1, H) be the minimum non-zero absolute value these polynomials take and take:

$\omega(x, 1, H) = - \frac{\log m(x, 1, H)}{\log H}$

$\omega(x, 1)= \limsup_{H\to\infty} \omega(x,1,H).$

ω(x, 1) is often called the measure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality. Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.

Measure of transcendence of a complex number

Next consider the values of polynomials at a complex number x, when these polynomials have integer coefficients, degree at most n, and height at most H, with n, H being positive integers.

Let m(x,n,H) be the minimum non-zero absolute value such polynomials take at x and take:

$\omega(x, n, H) = - \frac{\log m(x, n, H)}{n\log H}$

$\omega(x, n)= \limsup_{H\to\infty} \omega(x,n,H).$

Suppose this is infinite for some minimum positive integer n. A complex number x in this case is called a U number of degree n.

Now we can define

$\omega (x) =\limsup_{n\to\infty}\omega(x,n).$

ω(x) is often called the measure of transcendence of x. If the ω(x,n) are bounded, then ω(x) is finite, and x is called an S number. If the ω(x,n) are finite but unbounded, x is called a T number. x is algebraic if and only if ω(x) = 0.

Clearly the Liouville numbers are a subset of the U numbers. William LeVeque in 1953 constructed U numbers of any desired degree.^[28][29] The Liouville numbers and hence the U numbers are uncountable sets. They are sets of measure 0.^[30]

T numbers also comprise a set of measure 0.^[31] It took about 35 years to show their existence. Wolfgang M. Schmidt in 1968 showed that examples exist. However, almost all complex numbers are S numbers.^[32] Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:^[33][34] this shows that e is an S number and gives a proof of the transcendence of π. The most that is known about π is that it is not a U number. Many other transcendental numbers remain unclassified.

Two numbers x, y are called algebraically dependent if there is a non-zero polynomial P in 2 indeterminates with integer coefficients such that P(x, y) = 0. There is a powerful theorem that 2 complex numbers that are algebraically dependent belong to the same Mahler class.^[29][35] This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or π.

It is often speculated that S stood for the name of Mahler's teacher Carl Ludwig Siegel and that T and U are just the next two letters.

Koksma's equivalent classification

Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers.^[27][36]

Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of this finite set such that |x − α| has the minimum positive value. Define ω*(x,H,n) and ω*(x,n) by:

$|x-\alpha| = H^{-n\omega^*(x,H,n)-1}.$

$\omega^*(x,n) = \limsup_{H\to\infty} \omega^*(x,n,H).$

If for a smallest positive integer n, ω*(x,n) is infinite, x is called a U*-number of degree n.

If the ω*(x,n) are bounded and do not converge to 0, x is called an S*-number,

A number x is called an A*-number if the ω*(x,n) converge to 0.

If the ω*(x,n) are all finite but unbounded, x is called a T*-number,

Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.^[36] The A*-numbers are the algebraic numbers.^[32]

LeVeque's construction

Let

$\lambda= \tfrac{1}{3} + \sum_{k=1}^\infty 10^{-k!}$

It can be shown that the nth root of λ (a Liouville number) is a U-number of degree n.^[37]

This construction can be improved to create an uncountable family of U-numbers of degree n. Let Z be the set consisting of every other power of 10 in the series above for λ. The set of all subsets of Z is uncountable. Deleting any of the subsets of Z from the series for λ creates uncountably many distinct Liouville numbers, whose nth roots are U-numbers of degree n.

Type

The supremum of the sequence {ω(x, n)} is called the type. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.^[28]

See also

• Transcendental number theory, the study of questions related to transcendental numbers
• Diophantine approximation

Notes

1. Gottfried Wilhelm Leibniz, Karl Immanuel Gerhardt, Georg Heinrich Pertz (1858). Leibnizens mathematische Schriften 5. A. Asher & Co. pp. 97–98.
2. Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. p. 74.
3. Paul Erdős; Dudley, Underwood (December 1943). "Some Remarks and Problems in Number Theory Related to the Work of Euler". Mathematics Magazine 76 (5): 292–299. doi:10.2307/2690369. JSTOR 2690369.
4. 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.
5. Aubrey J. Kempner (October 1916). "On Transcendental Numbers". Transactions of the American Mathematical Society (American Mathematical Society) 17 (4): 476–482. doi:10.2307/1988833. JSTOR 1988833.
6. Weisstein, Eric W. "Liouville's Constant", MathWorld
7. J. Liouville (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 et Appl. 16: 133–142.
8. Georg Cantor (1874). "Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen". J. Reine Angew. Math. 77: 258–262.
9. Gray, Robert (1994). "Georg Cantor and transcendental numbers". Amer. Math. Monthly 101: 819–832. doi:10.2307/2975129.
10. Georg Cantor (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". J. Reine Angew. Math. 84: 242–258. (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. See p. 254.)
11. J J O'Connor and E F Robertson: Alan Baker. The MacTutor History of Mathematics archive 1998.
12. Boris Adamczewski and Yann Bugeaud (March 2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica 195 (1): 1–20. doi:10.1007/BF02588048.
13. Le Lionnais, F. Les nombres remarquables (ISBN 2-7056-1407-9). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, Transcendental Number
14. Chudnovsky, G. V. (1984). Contributions to the Theory of Transcendental Numbers. Providence, RI: American Mathematical Society. ISBN 0-8218-1500-8. via Wolfram Mathworld, Transcendental Number
15. Davison & Shallit 1991
16. K. Mahler (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A. (40): 421–428.
17. Mahler (1976) p.12
18. Calude, Cristian S. (2002). Information and Randomness: An Algorithmic Perspective. Texts in Theoretical Computer Science (2nd rev. and ext. ed.). Springer-Verlag. p. 239. ISBN 3-540-43466-6. Zbl 1055.68058.
19. Allouche & Shallit (2003) pp.385,403
20. Shallit, Jeffrey (1999). "Number theory and formal languages". In Hejhal, Dennis A.; Friedman, Joel; Gutzwiller, Martin C.; Odlyzko, Andrew M. 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-Verlag. pp. 547–570. ISBN 0-387-98824-6.
21. Loxton, J. H. (1988). "13. Automata and transcendence". In Baker, A. New Advances in Transcendence Theory. Cambridge University Press. pp. 215–228. ISBN 0-521-33545-0. Zbl 0656.10032.
22. Mahler, Kurt (1929). "Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen". Math. Annalen 101: 342–366. doi:10.1007/bf01454845. JFM 55.0115.01.
23. Allouche & Shallit (2003) p.387
24. Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.
25. Weisstein, Eric W., "Irrational Number", MathWorld.
26. Modular functions and transcendence questions, Yu. V. Nesterenko, Sbornik: Mathematics(1996), 187(9):1319
27. Bugeaud (2012) p.250
28. Baker (1975) p. 86.
29. LeVeque (2002) p.II:172
30. Burger and Tubbs, p. 170.
31. Burger and Tubbs, p. 172.
32. Bugeaud (2012) p.251
33. LeVeque (2002) pp.II:174–186
34. Burger and Tubbs, p. 182.
35. Burger and Tubbs, p. 163.
36. Baker (1975) p.87
37. Baker(1979), p. 90.

References

• David Hilbert, "Über die Transcendenz der Zahlen e und $\pi$", Mathematische Annalen 43:216–219 (1893).
• A. O. Gelfond, Transcendental and Algebraic Numbers, Dover reprint (1960).
• Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. ISBN 0-521-20461-5. Zbl 0297.10013.
• Mahler, Kurt (1976). Lectures on Transcendental Numbers. Lecture Notes in Mathematics 546. Springer-Verlag. ISBN 3-540-07986-6. Zbl 0332.10019.
• Sprindzhuk, Vladimir G. (1979). Metric theory of Diophantine approximations. Scripta Series in Mathematics. Transl. from the Russian and ed. by Richard A. Silverman. With a foreword by Donald J. Newman. John Wiley & Sons. Zbl 0482.10047.
• LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. ISBN 978-0-486-42539-9.
• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Burger, Edward B.; Tubbs, Robert (2004). Making transcendence transparent. An intuitive approach to classical transcendental number theory. New York, NY: Springer-Verlag. ISBN 0-387-21444-5. Zbl 1092.11031.
• Peter M Higgins, "Number Story" Copernicus Books, 2008, ISBN 978-1-84800-001-8.
• Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics 193. Cambridge: Cambridge University Press. ISBN 978-0-521-11169-0. Zbl pre06066616.
• Davison, J. Les; Shallit, Jeffrey O. (1991), "Continued fractions for some alternating series", Monatshefte für Mathematik 111 (2): 119–126, doi:10.1007/BF01332350

External links

• Weisstein, Eric W., "Transcendental Number", MathWorld.
• Weisstein, Eric W., "Liouville Number", MathWorld.
• Weisstein, Eric W., "Liouville's Constant", MathWorld.
• (English) Proof that e is transcendental
• (English) Proof that the Liouville Constant is transcendental
• (German) Proof that e is transcendental (PDF)
• (German) $\pi$