A transcendental number is a number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. This definition guarantees that every transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. A number x can then be tested to see if it is transcendental using the Mathematica command Not[Element[x, Algebraics]].
Transcendental numbers are important in the history of mathematics because their investigation provided the first proof that circle squaring, one of the geometric problems of antiquity that had baffled mathematicians for more than 2000 years was, in fact, insoluble. Specifically, in order for a number to be produced by a geometric construction using the ancient Greek rules, it must be either rational or a very special kind of algebraic number known as a Euclidean number. Because the number is transcendental, the construction cannot be done according to the Greek rules.
Liouville showed how to construct special cases (such as Liouville's constant) using Liouville's approximation theorem. In particular, he showed that any number that has a rapidly converging sequence of rational approximations must be transcendental. For many years, it was only known how to determine if special classes of numbers were transcendental. The determination of the status of more general numbers was considered an important 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 special cases of numbers of the form are transcendental. Baker produced a further revolution by proving the transcendence of sums of numbers of the form for algebraic numbers and
The number e was proven to be transcendental by Hermite in 1873, and pi () by Lindemann in 1882. Gelfond's constant is transcendental by Gelfond's theorem since
The GelfondSchneider constant is also transcendental (Hardy and Wright 1979, p. 162). Known transcendentals are summarized in the following table, where is the sine function, is a Bessel function of the first kind, is the nth zero of
transcendental number  reference 
e  Hermite (1873) 
Lindemann (1882)  
Gelfond  

Nesterenko (1999) 
Hardy and Wright (1979, p. 162)  
Hardy and Wright (1979, p. 162)  
exponential factorial inverse sum S  J. Sondow, pers. comm., Jan. 10, 2003 
Hardy and Wright (1979, p. 162)  
Hardy and Wright (1979, p. 162)  
Hardy and Wright (1979, p. 162),  
Le Lionnais (1983, p. 46)  
Borwein et al. (1989)  
Dekking (1977), Allouche and Shallit  
Chaitin's constant  
Champernowne constant  
Thue constant  
Liouville's constant L  Liouville (1850) 
Le Lionnais (1983, p. 46)  
Chudnovsky (1984, p. 308), Waldschmidt, Nesterenko (1999)  
Chudnovsky (1984, p. 308)  
Davis (1959)  
Apéry's constant has been proved to be irrational, but it is not known if it is transcendental. 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
There are still many fundamental and outstanding problems in transcendental number theory, including the constant problem and Schanuel's conjecture.
Algebraic Number, Algebraically Independent, Algebraics, Constant Problem, Four Exponentials Conjecture, Exponential Factorial, Gelfond's Theorem, Irrational Number, Irrationality Measure, LindemannWeierstrass Theorem, Roth's Theorem, Schanuel's Conjecture, Six Exponentials Theorem
Allouche, J. P. and Shallit, J. In preparation.
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Baker, A. "Approximations to the Logarithm of Certain Rational Numbers." Acta Arith. 10, 315323, 1964.
Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers I." Mathematika 13, 204216, 1966.
Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers II." Mathematika 14, 102107, 1966.
Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers III." Mathematika 14, 220228, 1966.
Baker, A. "Linear Forms in the Logarithms of Algebraic Numbers IV." Mathematika 15, 204216, 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, 201219, 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. 103107, 1996.
Davis, P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function." Amer. Math. Monthly 66, 849869, 1959.
Dekking, F. M. "Transcendence du nombre de ThueMorse." C. R. Acad. Sci. Paris 285, 157160, 1977.
Gray, R. "Georg Cantor and Transcendental Numbers." Amer. Math. Monthly 101, 819832, 1994.
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Lindemann, F. "Über die Zahl
Liouville, J. "Sur des classes trèsétendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques." J. Math. pures appl. 15, 133142, 1850.
Nagell, T. Introduction to Number Theory. New York: Wiley, p. 35, 1951.
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Nesterenko, Yu. V. A Course on Algebraic Independence: Lectures at IHP 1999. http://www.math.jussieu.fr/~nesteren/.
Pickover, C. A. "The Fifteen Most Famous Transcendental Numbers." J. Recr. Math. 25, 12, 1993.
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Shidlovskii, A. B. Transcendental Numbers. New York: de Gruyter, 1989.
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