I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do.

I've been thinking about irrational numbers recently, and a lot of my questions seem to come down to this: can we, in principle, show every transcendental number in some way?

Premise for the relevance of the question: Transcendentals must be shown rather than stated plainly, as can be done with (irrational) algebraic numbers. Perhaps I am wrong when I say this (please tell me if I am), but all definitions of π ultimately come down the ratio of a circle's circumference to it's diameter, and e must also be defined it terms of its use in a very specific, or rather, unique function. This is, more or less, what I mean by showing: that transcendental numbers must be given by their use in some definite structure ("structure" is here being used very broadly... again, I don't really have the proper mathematical vocabulary yet).

Now let me go back to my original question: can we always think of a context to fit any given transcendental number? Not necessarily a geometric construction, but some situation that provides a unique value that is a transcendental number.

Finally, links to any books, journal articles, etc., that may relate to or include this topic would be much appreciated.

I hope this made sense.

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    $\begingroup$ There are only countable infinitely many ways of showing a number (if I understood you correctly). This means that most of the reals cannot be shown. All the algebraic numbers obviously can, so it does look like transcendentals are a bit left out. Mind you, I'm fairly sure this has been asked and answered here before. As something called computable numbers or some such. $\endgroup$ – Jyrki Lahtonen Sep 27 '14 at 17:53
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    $\begingroup$ Just a comment: $\pi$ is usually defined in relation to a circle, but there are also ways to define it in relation to the prime numbers. $\endgroup$ – Robert Soupe Oct 26 '14 at 20:45

There's only a countable number of "definable real numbers", transcendental or otherwise, so if your notion of "showing" a number falls within the confines of definable, then the answer is no.

  • $\begingroup$ Thanks! This answers my question. I'm a little unclear about the difference between "definable" and "computable," (except that computables are a subset of definables, which is actually the main problem for me) but I'm sure I'll be able find that distinction somewhere. Also: is my description of transcendentals roughly accurate? I know it isn't in any formal sense, but is it true that the only way to "grasp" a transcendental is to find it in some kind of relation? Even a "computable" number doesn't seem to allow one to know that number, in the sense that one can π and √2. $\endgroup$ – user1801325 Sep 27 '14 at 19:42
  • $\begingroup$ I didn't see you give any description of transcendental numbers... You just seemed to be talking about "pinning down" a number with some definition, geometric or otherwise. $\endgroup$ – Dustan Levenstein Sep 28 '14 at 1:20
  • $\begingroup$ And computable means there is a computer program that can compute the number to arbitrary precision. Definable is a broader term which encompasses all numbers which can be uniquely determined by some conditions in the language of set theory. $\endgroup$ – Dustan Levenstein Sep 28 '14 at 1:22

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