Transcendental number

In mathematics, a transcendental number is a complex number which is not algebraic, that is, not a solution of a non-zero polynomial equation with rational coefficients.

The most prominent examples of transcendental numbers are π and e. Only a few classes of transcendental numbers are known, and proving that a given number is transcendental can be extremely difficult.

However, transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable, but the sets of real and complex numbers are uncountable.

History

Euler was probably the first person to define transcendental numbers in the modern sense.^[1] The name "transcendentals" comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x.^[2] Joseph Liouville first proved the existence of transcendental numbers in 1844,^[3] and in 1851 gave the first decimal examples such as the Liouville constant

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=1}^\infty 10^{-k!} = 0.110001000000000000000001000\ldots

in which the nth digit after the decimal point is 1 if n is a factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise.^[4] Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be particularly well approximated by rational numbers. Liouville showed that all Liouville numbers are transcendental.

Johann Heinrich Lambert conjectured that e and π were both transcendental numbers, in his 1761 paper proving the number π is irrational. 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 found the argument mentioned above establishing the ubiquity of transcendental numbers.

In 1882, Ferdinand von Lindemann published a proof that the number π is transcendental. He first showed that e to any nonzero algebraic power is transcendental, and 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 a non-zero algebraic number 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).

Properties

The set of transcendental numbers is uncountably infinite. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. But Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable.

Transcendental numbers are never rational, but some irrational numbers are not transcendental. For example, the square root of 2 is irrational, but it is a solution of the polynomial x² − 2 = 0, so it is algebraic, not transcendental.

Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument. So, 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.

Every non–computable number is also transcendental since all algebraic numbers are computable.

All Liouville numbers are transcendental; however, not all transcendental numbers are Liouville numbers. 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).^[5]

Known transcendental numbers and open problems

Here is a list of some numbers known to be transcendental:

• e^a if a is algebraic and nonzero (by the Lindemann–Weierstrass theorem), and in particular, e itself.
• 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 a ≠ 0) and b is algebraic but not rational (by the Gelfond–Schneider theorem), in particular :
  • Failed to parse (Missing texvc executable; please see math/README to configure.): 2^\sqrt{2}

, the Gelfond–Schneider constant (Hilbert number),

• 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).
• Γ(1/3),^[6] Γ(1/4),^[7] and Γ(1/6).^[7]
  • 0.12345678910111213141516..., the Champernowne constant.^[8]
    • Ω, Chaitin's constant (since it is a non-computable number).
    • Prouhet–Thue–Morse constant
    • Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^\infty 10^{-\lfloor \beta^{k} \rfloor};

        where Failed to parse (Missing texvc executable; please see math/README to configure.): \beta > 1
    and Failed to parse (Missing texvc executable; please see math/README to configure.): \beta\mapsto\lfloor \beta \rfloor
    is the floor function.
    

    Numbers for which it is unknown whether they are transcendental or not:

    • Sums, products, powers, etc. (except for Gelfond's constant) of the number π and the number e: π + e, π − e, π·e, π/e, π^π, e^e, π^e
    • 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), and in fact, ζ(2n + 1) for any positive integer n (see Riemann zeta function).

    Conjectures:

    • Schanuel's conjecture

    Proof sketch 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 Failed to parse (Missing texvc executable; please see math/README to configure.): c_{0},c_{1},\ldots,c_{n},

    satisfying the equation:
    

    Failed to parse (Missing texvc executable; please see math/README to configure.): c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0

    
    and such that Failed to parse (Missing texvc executable; please see math/README to configure.): c_0

    and Failed to parse (Missing texvc executable; please see math/README to configure.): c_n
    are both non-zero.
    

    Depending on the value of n, we specify a sufficiently large positive integer k (to meet our needs later), and multiply both sides of the above equation by Failed to parse (Missing texvc executable; please see math/README to configure.): \int^{\infty}_{0} , where the notation Failed to parse (Missing texvc executable; please see math/README to configure.): \int^{b}_{a}

    will be used in this proof as shorthand for the integral:
    

    Failed to parse (Missing texvc executable; please see math/README to configure.): \int^{b}_{a}:=\int^{b}_{a}x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}\,dx.

    
    We have arrived at the equation:

    Failed to parse (Missing texvc executable; please see math/README to configure.): c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{0}+\cdots+c_{n}e^{n}\int^{\infty}_{0} = 0

    
    which can now be written in the form

    Failed to parse (Missing texvc executable; please see math/README to configure.): P_{1}+P_{2}=0\;

    
    where

    Failed to parse (Missing texvc executable; please see math/README to configure.): P_{1}=c_{0}\int^{\infty}_{0}+c_{1}e\int^{\infty}_{1}+c_{2}e^{2}\int^{\infty}_{2}+\cdots+c_{n}e^{n}\int^{\infty}_{n}

    Failed to parse (Missing texvc executable; please see math/README to configure.): P_{2}=c_{1}e\int^{1}_{0}+c_{2}e^{2}\int^{2}_{0}+\cdots+c_{n}e^{n}\int^{n}_{0}

    
    The plan of attack now is to show that for k sufficiently large, the above relations are impossible to satisfy because

    Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{P_{1}}{k!}

    is a non-zero integer and Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{P_{2}}{k!}
    is not. 
    

    The fact that Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{P_{1}}{k!}

    is a nonzero integer results from the relation
    

    Failed to parse (Missing texvc executable; please see math/README to configure.): \int^{\infty}_{0}x^{j}e^{-x}\,dx=j!

    
    which is valid for any positive integer j and can be proved using integration by parts and mathematical induction.

    To show that

    Failed to parse (Missing texvc executable; please see math/README to configure.): \left|\frac{P_{2}}{k!}\right|<1

    for sufficiently large k 
    

    we first note that Failed to parse (Missing texvc executable; please see math/README to configure.): x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}

    is the product of the functions Failed to parse (Missing texvc executable; please see math/README to configure.): [x(x-1)(x-2)\cdots(x-n)]^{k}
    and Failed to parse (Missing texvc executable; please see math/README to configure.): (x-1)(x-2)\cdots(x-n)e^{-x}
    

    . Using upper bounds for Failed to parse (Missing texvc executable; please see math/README to configure.): |x(x-1)(x-2)\cdots(x-n)|

    and Failed to parse (Missing texvc executable; please see math/README to configure.): |(x-1)(x-2)\cdots(x-n)e^{-x}|
    on the interval [0,n] and employing the fact
    

    Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{k\to\infty}\frac{G^k}{k!}=0

    for every real number G
    

    is then sufficient to finish the proof.

    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

    • Transcendence theory, the study of questions related to transcendental numbers

    References

    1. ^ Paul Erdős, Underwood Dudley (November 1983). "Some Remarks and Problems in Number Theory Related to the Work of Euler". Mathematics Magazine 56 (5): 292-298. 
    2. ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. 
    3. ^ Aubrey J. Kempner (October 1916). "On Transcendental Numbers". Transactions of the American Mathematical Society 17 (4): 476-482. 
    4. ^ Weisstein, Eric W. "Liouville's Constant", MathWorld [1]
    5. ^ Boris Adamczewski and Yann Bugeaud (March 2005). "On the complexity of algebraic numbers, II. Continued fractions". Acta Mathematica 195 (1): 1-20. 
    6. ^ Le Lionnais, F. Les nombres remarquables (ISBN 2705614079). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, Transcendental Number
    7. ^ ^{a b} Chudnovsky, G. V. Contributions to the Theory of Transcendental Numbers (ISBN 0821815008). Providence, RI: Amer. Math. Soc., 1984. via Wolfram Mathworld, Transcendental Number
    8. ^ K. Mahler (1937). "Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen". Proc. Konin. Neder. Akad. Wet. Ser. A. (40): 421–428. 

    
    

    • David Hilbert, "Über die Transcendenz der Zahlen Failed to parse (Missing texvc executable; please see math/README to configure.): e

    und Failed to parse (Missing texvc executable; please see math/README to configure.): \pi
    

    ", Mathematische Annalen 43:216–219 (1893).

    • Alan Baker, Transcendental Number Theory, Cambridge University Press, 1975, ISBN 0-521-39791-X.

    External links

    • (English) Proof that Failed to parse (Missing texvc executable; please see math/README to configure.): e

    is transcendental
    

    • (German) Proof that Failed to parse (Missing texvc executable; please see math/README to configure.): e

    is transcendental (PDF)
    

    • (German) Proof that Failed to parse (Missing texvc executable; please see math/README to configure.): \pi

    is transcendental (PDF)
    

    v • d • e Number systems
    Basic		Complex extensions		Other extensions
    Natural numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{N} Negative numbers Integers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Z} Rational numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q} Irrational numbers	Real numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} Imaginary numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{I} Complex numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{C} Algebraic numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{A} Transcendental numbers	Quaternions Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{H} Octonions Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{O} Sedenions Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{S} Cayley-Dickson construction Split-complex numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}^{1,1}	Bicomplex numbers Biquaternions Coquaternions Tessarines Hypercomplex numbers	Musean hypernumbers Superreal numbers Hyperreal numbers Surreal numbers Dual numbers Transfinite numbers
    Other
    Cardinal numbers  · Computable numbers  · Constructible numbers  · ∞ (infinity)  · Integer sequences  · Large numbers  · Mathematical constants  · Nominal numbers  · Ordinal numbers  · p-adic numbers  · Prime numbers  ·

    bg:Трансцендентно число

    ca:Nombre transcendent cs:Transcendentní číslo da:Transcendente tal de:Transzendente Zahl es:Número trascendente eu:Zenbaki transzendente fr:Nombre transcendant ko:초월수 it:Numero trascendente he:מספר טרנסצנדנטי lt:Transcendentinis skaičius hu:Transzcendens szám nl:Transcendent getal ja:超越数 pl:Liczba przestępna pt:Número transcendente ro:Număr transcendent ru:Трансцендентное число sk:Transcendentné číslo sl:Transcendentno število fi:Transsendenttiluku sv:Transcendent tal ta:இயற்கணித எண்களும் விஞ்சிய எண்களும் th:จำนวนอดิศัย vi:Số siêu việt tr:Aşkın sayı uk:Трансцендентні числа