Transcendental Number
A number which is not the Root of any Polynomial equation with IntegerCoefficients, meaning that it is not an Algebraic Number of any degree, is said to betranscendental. This definition guarantees that every transcendental number must also be Irrational, since a Rational Number is, by definition, an Algebraic Number of degree one.
Transcendental numbers are important in the history of mathematics because their investigation provided the first proofthat Circle Squaring, one of the Geometric Problems of Antiquity which had baffled mathematicians for morethan 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a GeometricConstruction using the ancient Greek rules, it must be either Rational or a very special kindof Algebraic Number known as a Euclidean Number. Because the number 
is transcendental, the constructioncannot be done according to the Greek rules.
Georg Cantor 
was the first to prove the Existence of transcendental numbers. Liouville subsequently showed how to construct special cases (such as Liouville's Constant) usingLiouville's Rational Approximation Theorem. In particular, he showed that any number which has a rapidly convergingsequence of rational approximations must be transcendental. For many years, it was only known how to determine if specialclasses of numbers were transcendental. The determination of the status of more general numbers was considered animportant enough unsolved problem that it was one of Hilbert's Problems.
Great progress was subsequently made by Gelfond's Theorem, which gives a general rule for determining if specialcases of numbers of the form 
are transcendental. Baker produced a further revolution by proving thetranscendence of sums of numbers of the form 
for AlgebraicNumbers 
and 
.
The number Hermite in 1873, and Pi () by Lindemann in 1882. 
is transcendental by Gelfond's Theorem since
The Gelfond-Schneider Constant
is also transcendental. Other known transcendentals are 
where
is the Sine function, 
where 
is a Bessel Function of the First Kind (Hardy and Wright1985), 
, 
, the first zero 
of the Bessel Function 
(Le Lionnais1983, p. 46), 
(Borwein et al. 1989), the Thue-Morse Constant 
(Dekking 1977, Allouche and Shallit), the Champernowne Constant 0.1234567891011..., the Thue Constant
(Le Lionnais 1983, p. 46), 
(Davis 1959), and 
(Chudnovsky, Waldschmidt),where 
is the Gamma Function. At least one of 
and 
(and probably both) are transcendental,but transcendence has not been proven for either number on its own.It is not known if 
, 
, 
, 
(the Euler-Mascheroni Constant), 
, or 
(where
is a Modified Bessel Function of the First Kind) are transcendental.
The ``degree'' of transcendence of a number can be characterized by a so-called Liouville-Roth Constant. There arestill many fundamental and outstanding problems in transcendental number theory, including the Constant Problem andSchanuel's Conjecture.
See also Algebraic Number, Constant Problem, Gelfond's Theorem, Irrational Number,Lindemann-Weierstraß Theorem, Liouville-Roth Constant, Roth'sTheorem, Schanuel's Conjecture, Thue-Siegel-Roth Theorem
References
Allouche, J. P. and Shallit, J. In preparation. Baker, A. ``Approximations to the Logarithm of Certain Rational Numbers.'' Acta Arith. 10, 315-323, 1964. Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers I.'' Mathematika 13, 204-216, 1966. Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers II.'' Mathematika 14, 102-107, 1966. Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers III.'' Mathematika 14, 220-228, 1966. Baker, A. ``Linear Forms in the Logarithms of Algebraic Numbers IV.'' Mathematika 15, 204-216, 1966. Borwein, J. M.; Borwein, P. B.; and Bailey, D. H. ``Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi.'' Amer. Math. Monthly 96, 201-219, 1989. Chudnovsky, G. V. Contributions to the Theory of Transcendental Numbers. Providence, RI: Amer. Math. Soc., 1984. Courant, R. and Robbins, H. ``Algebraic and Transcendental Numbers.'' §2.6 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 103-107, 1996. Davis, P. J. ``Leonhard Euler's Integral: A Historical Profile of the Gamma Function.'' Amer. Math. Monthly 66, 849-869, 1959. Dekking, F. M. ``Transcendence du nombre de Thue-Morse.'' Comptes Rendus de l'Academie des Sciences de Paris 285, 157-160, 1977. Gray, R. ``Georg Cantor and Transcendental Numbers.'' Amer. Math. Monthly 101, 819-832, 1994. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University Press, 1985. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 46, 1983. Siegel, C. L. Transcendental Numbers. New York: Chelsea, 1965.
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