- Transcendental number
In

mathematics , a**transcendental number**is acomplex number that is not algebraic, that is, not a solution of a non-zeropolynomial equation with rationalcoefficient s.The most prominent examples of transcendental numbers are "π" and "e". Only a few classes of transcendental numbers are known, indicating that it can be extremely difficult to show that a given number is transcendental.

However, transcendental numbers are not rare: indeed,

almost all real and complex numbers are transcendental, since the algebraic numbers arecountable , but the sets of real and complex numbers areuncountable . All transcendental numbers are irrational, since all rational numbers are algebraic. (The converse is not true: not all irrational numbers are transcendental.)**History**Euler was probably the first person to define transcendental numbers in the modern sense. [*cite journal|title=Some Remarks and Problems in Number Theory Related to the Work of Euler|author=*] The name "transcendentals" comes fromPaul Erdős , Underwood Dudley|journal=Mathematics Magazine|volume=56|issue=5|month=November | year=1983|pages=292–298Leibniz in his 1682 paper where he proved sin "x" is not analgebraic function of "x". [*cite book|title=Elements of the History of Mathematics|author=Nicolás Bourbaki|publisher=Springer|year=1994*]Joseph Liouville first proved the existence of transcendental numbers in 1844, [*cite journal|title=On Transcendental Numbers|author=Aubrey J. Kempner|journal=Transactions of the American Mathematical Society|volume=17|issue=4|month=October | year=1916|pages=476–482|doi=10.2307/1988833*] and in 1851 gave the first decimal examples such as the Liouville constant:$sum\_\{k=1\}^infty\; 10^\{-k!\}\; =\; 0.110001000000000000000001000ldots$

in which the "n"th digit after the decimal point is 1 if "n" is a

factorial (i.e., 1, 2, 6, 24, 120, 720, ...., etc.) and 0 otherwise. [*Weisstein, Eric W. "Liouville's Constant", MathWorld [*] Liouville showed that this number is what we now call a*http://mathworld.wolfram.com/LiouvillesConstant.html*]Liouville number ; this essentially means that it can be particularly well approximated byrational number s. Liouville showed that all Liouville numbers are transcendental [*J. Liouville, "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationnelles algébriques"," J. Math. Pures et Appl.*] .**18**, 883-885, and 910-911, (1844).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", byCharles Hermite in1873 . In1874 ,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 (seeEuler's identity ), "iπ" and therefore "π" must be transcendental. This approach was generalized byKarl Weierstrass to theLindemann–Weierstrass theorem . The transcendence of "π" allowed the proof of the impossibility of several ancient geometric constructions involvingcompass 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 irrationalalgebraic number , is "a"^{"b"}necessarily transcendental? The affirmative answer was provided in 1934 by theGelfond–Schneider theorem . This work was extended byAlan 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, thealgebraic number s must also becountable . ButCantor'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 number s are not transcendental. For example, thesquare root of 2 is irrational, but it is a solution of the polynomial "x"^{2}− 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 (π^{5}+ 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 π"e" 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") = "x"^{2}− ("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 analgebraically 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 number s are a strict subset of the transcendental numbers.All

Liouville number s are transcendental; however, not all transcendental numbers are Liouville numbers. Any Liouville number must have unbounded partial quotients in itscontinued 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). [*cite journal|title=On the complexity of algebraic numbers, II. Continued fractions|author=Boris Adamczewski and Yann Bugeaud|journal=Acta Mathematica|volume=195|issue=1|month=March | year=2005|pages=1–20|doi=10.1007/BF02588048*]**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 theLindemann–Weierstrass theorem ), and in particular, "e" itself.

*π (by theLindemann–Weierstrass theorem ).

* "e"^{π},Gelfond's constant , as well as "e"^{-π/2}="i"^{i}(by theGelfond–Schneider theorem ).

* "a^{b}" where "a" is non-zero algebraic and "b" is irrational algebraic (by theGelfond–Schneider theorem ), in particular:

** $2^sqrt\{2\}$, theGelfond–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 theLindemann–Weierstrass theorem ).

*ln("a") if "a" is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by theLindemann–Weierstrass theorem ).

*Γ(1/3), [*Le Lionnais, F. Les nombres remarquables (ISBN 2705614079). Paris: Hermann, p. 46, 1979. via Wolfram Mathworld, [*] Γ(1/4),Chudnovsky, G. V. Contributions to the Theory of Transcendental Numbers (ISBN 0821815008). Providence, RI: Amer. Math. Soc., 1984. via Wolfram Mathworld, [*http://mathworld.wolfram.com/TranscendentalNumber.html Transcendental Number*]*http://mathworld.wolfram.com/TranscendentalNumber.html Transcendental Number*] ] and Γ(1/6).

*0.12345678910111213141516..., theChampernowne constant . [*cite journal|author=K. Mahler|title=Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen|journal=Proc. Konin. Neder. Akad. Wet. Ser. A.|issue=40|year=1937|pages=421–428*]

*Ω,Chaitin's constant (since it is a non-computable number).

*Prouhet–Thue–Morse constant

* $sum\_\{k=0\}^infty\; 10^\{-leftlfloor\; eta^\{k\}\; ight\; floor\};$ where $eta\; 1$ and $etamapstolfloor\; eta\; floor$ is thefloor function .Numbers for which it is unknown whether they are transcendental or not:

* Sums, products, powers, etc. (except forGelfond's constant ) of the number "π" and the number "e": "π" + "e", "π" − "e", "π"·"e", "π"/"e", "π"^{"π"}, "e"^{"e"}, "π"^{"e"}

* theEuler–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"(2"n" + 1) for any positive integer "n" (seeRiemann 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 ofCharles 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 $c\_\{0\},c\_\{1\},ldots,c\_\{n\},$ satisfying the equation:

:$c\_\{0\}+c\_\{1\}e+c\_\{2\}e^\{2\}+cdots+c\_\{n\}e^\{n\}=0$

and such that $c\_0$ and $c\_n$ 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 $int^\{infty\}\_\{0\}$, where the notation $int^\{b\}\_\{a\}$ will be used in this proof as shorthand for the integral:

:$int^\{b\}\_\{a\}:=int^\{b\}\_\{a\}x^\{k\}\; [(x-1)(x-2)cdots(x-n)]\; ^\{k+1\}e^\{-x\},dx.$

We have arrived at the equation:

:$c\_\{0\}int^\{infty\}\_\{0\}+c\_\{1\}eint^\{infty\}\_\{0\}+cdots+c\_\{n\}e^\{n\}int^\{infty\}\_\{0\}\; =\; 0$

which can now be written in the form

:$P\_\{1\}+P\_\{2\}=0;$

where

:$P\_\{1\}=c\_\{0\}int^\{infty\}\_\{0\}+c\_\{1\}eint^\{infty\}\_\{1\}+c\_\{2\}e^\{2\}int^\{infty\}\_\{2\}+cdots+c\_\{n\}e^\{n\}int^\{infty\}\_\{n\}$:$P\_\{2\}=c\_\{1\}eint^\{1\}\_\{0\}+c\_\{2\}e^\{2\}int^\{2\}\_\{0\}+cdots+c\_\{n\}e^\{n\}int^\{n\}\_\{0\}$

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

:$frac\{P\_\{1\{k!\}$ is a non-zero integer and $frac\{P\_\{2\{k!\}$ is not.

The fact that $frac\{P\_\{1\{k!\}$ is a nonzero integer results from the relation

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

which is valid for any positive integer "j" and can be proved using

integration by parts andmathematical induction .To show that

:$left|frac\{P\_\{2\{k!\}\; ight|<1$ for sufficiently large "k"

we construct an

auxiliary function $x^\{k\}\; [(x-1)(x-2)cdots(x-n)]\; ^\{k+1\}e^\{-x\}$, noting that it is the product of the functions $[x(x-1)(x-2)cdots(x-n)]\; ^\{k\}$ and $(x-1)(x-2)cdots(x-n)e^\{-x\}$.Using upper bounds for $|x(x-1)(x-2)cdots(x-n)|$ and $|(x-1)(x-2)cdots(x-n)e^\{-x\}|$ on the interval [0,"n"] and employing the fact:$lim\_\{k\; oinfty\}frac\{G^k\}\{k!\}=0$ 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 aboutsymmetric polynomial s 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***

David Hilbert , "Über die Transcendenz der Zahlen $e$ und $pi$", "Mathematische Annalen"**43**:216–219 (1893).

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

*Peter M Higgins , "Number Story" Copernicus Books, 2008, ISBN 978-84800-000-1.

** External links **

* [*http://planetmath.org/encyclopedia/EIsTranscendental.html Proof that $e$ is transcendental*]

* [*http://www.mathematik.uni-muenchen.de/~fritsch/euler.pdf Proof that $e$ is transcendental (PDF)*]

* [*http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf Proof that $pi$ is transcendental (PDF)*]

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