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

2 Answers

As I understand, people did not think much about comparing the size of infinite sets before Cantor, so for most of them the question you are asking had no meaning, and they did not think about it. Perhaps few people felt intuitively that a countable set is "smaller" than a continuum, but Cantor was the first to state and prove this.

Edit. But there were actually exceptions to this. Nicola Oresme in 1360 (!) really argued that there are much more irrational numbers than rational ones, and even that a "random number", whatever it could mean, is irrational. For more details on this see the answers to this question https://mathoverflow.net/questions/269893 and references there. Of course his arguments could not be rigorous at that time. And he was discussing irrational numbers. That transcendental numbers exist at all was only proved in 19 century, by Cantor and Liouville, with different proofs.

• Indeed, back then there was essentially no consideration given to any kind of structure (or size) of sets of numbers such as this. Probably the only thought anyone might have had at this time was that the transcendentals are dense in the reals (phrased as "every interval contains a transcendental number"), but since this is trivial (add rational numbers to a known transcendental number) and didn't (appear to) lead to anything of interest, it might never have been written down before the 1870s. – Dave L Renfro Nov 12 '14 at 17:57

is a real or complex number that is not algebraic—that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e. Though only a few classes of transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable. All real transcendental numbers are irrational, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental; e.g., the square root of 2 is irrational but not a transcendental number, since it is a solution of the polynomial equation x*x − 2 = 0.

Thus, with insight, now that we know Cantor's proof of the uncountability of the real numbers, we can easily conclude that we have a lot more transcendental numbers than non-transcendental ...

But this is the history made from the point of view of modern time; as explained by Alexandre in his answer, at the time of Liouville's proof of the existence of transcendental numbers (1844) there was no clear understanding of the possibility (if any) of comparing the "size" of infinite collections [you can see this post].

• " Cantor's proof does not give us any "procedure" to manufacture transcendental numbers ". This is not true. The nested intervals argument can be easily turn into an "algorithm" that outputs as a result a transcendental number. – Andrés E. Caicedo Nov 11 '14 at 17:46