Liouville showed that all Liouville numbers are transcendental. The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by Charles Hermite in 1873. In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable.
(not proven to be irrational). The Feigenbaum constants and , also not proven to be irrational. Mills' constant, also not proven to be irrational. The Copeland–Erdős constant, formed by concatenating the decimal representations of the prime numbers. Conjectures: Schanuel's conjecture, Four exponentials conjecture. ==Sketch of a proof that is transcendental== The first proof that the base of the natural logarithms, , is transcendental dates from 1873.
Liouville showed that all Liouville numbers are transcendental. The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers' existence was , by Charles Hermite in 1873. In 1874, Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable.
Cantor's work established the ubiquity of transcendental numbers. In 1882, Ferdinand von Lindemann published the first complete proof of the transcendence of .
Kurt Mahler showed in 1953 that is also not a Liouville number.
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.
A notable exception is (for any positive integer ) which has been proven transcendental. The Euler–Mascheroni constant : In 2010 M.
In 2012 it was shown that at least one of and the Euler-Gompertz constant is transcendental. Catalan's constant, not even proven to be irrational. Khinchin's constant, also not proven to be irrational. Apéry's constant (which Apéry proved is irrational). The Riemann zeta function at other odd integers, , , ...
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