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Various irrationality and transcendence results have already been posted, but it is interesting to see that the mere existence of transcendental numbers was not proven until the nineteenth century. Of course the existence of irrational numbers was already known in ancient Greece, but it took until 1844 before we first knew with certainty that there exist transcendental numbers.

The notion of algebraic and transcendental numbers was not yet available in ancient Greece. It seems that the first mention of the term transcendental was made by Leibniz in the 17th century, although he was more interested in transcendental functions rather than transcendental numbers. As Bourbaki puts it (Elements of the History of Mathematics, page 74):

“The definition that Leibniz gives of "transcendental quantities" [...] seems to apply more to functions than to numbers (in modern language, what he does reduces to defining transcendental elements over the field obtained by adjoining to the field of rational numbers the given numbers from the problem); it is however likely that he had a fairly clear notion of transcendental numbers (even though these latter do not appear to have been defined in a precise way before the end of the XVIIIth century); [...]”

The first proof of the existence of transcendental numbers was given by Liouville in 1844, who constructed a class of numbers which he then proved to be transcendental (now known as the Liouville numbers). The Louiville numbers are a cornerstone in the field of Diophantine approximation, which has since grown into a rich and active field of study in contemporary mathematics.

Nowadays of course we get the existence of transcendental numbers as an easy corollary of a famous theorem of Cantor's, which states that $\mathbb{R}$ is uncountable. Since the set of algebraic numbers is only countable, it follows that there must exist (uncountably many) transcendental numbers. Knowing this simple proof, I was very much surprised to learn that the existence of transcendental numbers had only been settled for 30 years when Cantor first proved $\mathbb{R}$ to be uncountable.

(Interestingly, Cantor's first proof of the uncountability of the real numbers predates his famous diagonal argument by some 17 years. See also Robert Gray, Georg Cantor and Transcendental Numbers, The American Mathematical Monthly, Vol. 101, No. 9 (Nov., 1994), pp. 819-832, at the time of writing also available in its entirety at the website of the Mathematical Association of America.)