# Were transcendental numbers considered rare, pre-Cantor?

Because the real numbers are uncountable and the real algebraic numbers are countable, there are uncountably many transcendental numbers. So there are far more transcendentals than rationals. With the language of set theory these facts are easy to see, but historically it was very difficult to construct transcendental numbers (or prove that certain numbers were transcendental). It is my impression that, historically, nowhere-differentiable functions were considered pathological and rare among the continuous functions. Was this similarly true for transcendental numbers among the rationals: were they considered pathological and rare?

• Who was particularly interested in transcendental numbers pre-Cantor? One might find hints in their correspondance, if any of it's published. – Jack M Nov 10 '14 at 19:02
• @JackM I know Liouville was; I believe he was the first to give an example of a transcendental number. If he believed all transcendentals "looked like" his numbers I would be unsurprised to hear he thought they were rare indeed. – user106 Nov 10 '14 at 19:08
• The relevant paper by Liouville doesn't express any opinion on "how common" transcendentals are, but it's worth noting that Liouville states his theorems as general methods for constructing transcendentals. He was therefore aware that there were at least infinitely many transcendentals. To me this suggests that the answer to the question is probably "no", but it's hard to say. – Jack M Nov 10 '14 at 19:29
• Given that for a fixed transcendental $t$ it was certainly well-known that $at$ for whatever algebraic (that is non-transcendental) $a$ is also transcendental, I have a hard time imagining a notion of comparison where transcendentals would be rare relative to non-transcendentals. Also $\pi$ and $e$ rather would not have been considered pathological, which should have been suspected transcendental quite a while before proved to be so. – quid Nov 10 '14 at 21:39
• @MikeMiller: I have heard more than a few times that nowhere differentiable functions were considered bizarre, rare, pathological They can be considered bizarre and pathological simply because there are large bodies of applications where they don't occur or are of no interest. And before the modern notion of a function had finally evolved, people thought of these things as expressions that you could write down using symbols. Depending on what symbols you use, it may indeed be true that such functions are in some sense "rare." – Ben Crowell Apr 18 '17 at 21:32