Transcendental Number

A transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one.

A complex number can be tested to see if it is transcendental using the Wolfram Language 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 Gelfond-Schneider 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 th zero of , is the Thue-Morse constant, is the universal parabolic constant, is Chaitin's constant, is the gamma function, and is the Riemann zeta function.

transcendental number	reference
Chaitin's constant
Champernowne constant
	Hermite (1873)
,	Nesterenko (1999)
Gelfond's constant	Gelfond
Gelfond-Schneider constant	Hardy and Wright (1979, p. 162)
exponential factorial inverse sum	J. Sondow, pers. comm., Jan. 10, 2003
	Le Lionnais (1983, p. 46)
	Chudnovsky (1984, p. 308), Waldschmidt, Nesterenko (1999)
	Chudnovsky (1984, p. 308)
	Davis (1959)
	Hardy and Wright (1979, p. 162)
smallest root, 2.4048255...	Le Lionnais (1983, p. 46)
Komornik-Loreti constant	Allouche and Cosnard (2000)
Liouville's constant	Liouville (1850)
	Hardy and Wright (1979, p. 162)
	Hardy and Wright (1979, p. 162),
	Lindemann (1882)
	Borwein et al. (1989)
Plouffe's constant	Smith 2003, Margolius
	Hardy and Wright (1979, p. 162)
for rational and	Margolius
Thue-Morse constant 0.4124540336...	Dekking (1977), Allouche and Shallit
Thue constant
universal parabolic constant

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 , , , (the Euler-Mascheroni constant), , or (where is a modified Bessel function of the first kind) are transcendental.