3
$\begingroup$

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.

$\endgroup$
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.