Talk:Transcendental number theory

Untitled

Not worth an edit war; but it is inexact to call x in P(x) a kind of formal variable. Charles Matthews 22:39, 16 Nov 2004 (UTC)

My problem is that P(x)=0 on its own, without x quantified, doesn't really mean anything. Your point is that it stands for the map x -> P(x), and in some contexts we might use this confusion. For example we say the function sin x. However the tone of this article is pretty formal and I think it is necessary to say P(x)=0 for all x, or perhaps better to use the notation later in the article P=0. Or could say P(x) is identically zero (rather more old fashioned I think). Billlion 07:32, 17 Nov 2004 (UTC)

Well, no, strictly, P is not a mapping but a formal expression. And the assertion is that it is the constant 0 (as formal expression, also); which is the notation for the polynomial with all its coefficients zero. Charles Matthews 08:25, 17 Nov 2004 (UTC)

Now we are in to some interesting pedantry! In the line above P(e)=0 clearly refers to the evaluation of P at e, so we are identifying the formal expression with a function evaluated at a real number. I am now more convinced P=0 is best, as P is clearly an object in the module of integer coeff polynomials. Billlion 09:51, 17 Nov 2004 (UTC)

Not to rude, but I think that the description in the first sentence on the Transcendental number article is best. The one I edited in here is very similar. Furthermore, this article needs some cleanup, in my opinion. Look below:

The quantitative approach asks one to find lower bounds

P(e) > F(A,d)

depending on a bound A of the coefficients of P and its degree, that apply to all P ≠ 0.

I, as a reader of the article, have some questions about this:

What is the function F?
The lower bounds of what exactly?
What is the function F?
I think A is a number that is greater than the magnitude (absolute value) of any coeffient of the polynomial function P, right?
What is the function F?
d is the polynomial degree of P, right?
What is the function F?

Do you see the point I am trying to make clear?

Also, where are the references? I can't check your work or believe anything unless there is some reading that I am inclined to read. (preferably there be an online one so that people do not have to go to the library, but at least 1 book so that it is more verifiable)
EulerGamma 21:29, 11 September 2006 (UTC)[reply]

Overlap with Transcendental number article

I'd love to help get this article into reasonable shape, but it seems inevitable to me that it is going to overlap a lot with the transcendental number article. I'm definitely not suggesting the two be merged but it'd be useful to get some suggestions on what to put into which article Chenxlee (talk) 20:02, 13 February 2008 (UTC)[reply]

Perhaps the entire section on Mahler's classification could be moved to this article. Scott Tillinghast, Houston TX (talk) 20:48, 16 December 2020 (UTC)[reply]

For historical reasons I think a section on Liouville numbers belongs in both articles. Pi and e I think fit best into the transcendental number article. Scott Tillinghast, Houston TX (talk) 02:06, 17 December 2020 (UTC)[reply]

There is already a separate article about the Gelfond-Schneider theorem. Scott Tillinghast, Houston TX (talk) 04:26, 23 December 2020 (UTC)[reply]

The very last edits Scott Tillinghast made on Wikipedia were to this article. He died [1] several months later. I have no doubt he would have continued to work on the article had he been able to, but due to the circumstances, a major change has been left in an incomplete state. Mainly I'm referring to the move of the material on Mahler's classification from Transcendental number to here. Wikipedia has, for example, numerous redirects of various terms related to Mahler's classification that lead to the transcendental number article. I've also run across a number of links relating to Mahler's classification outside of Wikipedia that point to the transcendental number article. I'm only just learning about this topic, and at first thought that Wikipedia had no content at all relating to Mahler's classification. I only became aware of this page after running across one of the off-site links and then tracing through the edit history at Transcendental number to discover when and where the material on Mahler had been moved. (I now see that this article is one of the "See also" links at Transcendental number, but otherwise there is no link to this page, and no hint of the Mahler classification material that was once there.)

As I say, I'm new to the topic, and am not sure I'm the most suitable person to decide what should go where. I do think that Mahler's classification should at least be mentioned at Transcendental number, with a pointer to this as the main article. Or if some people think it would be better to restore the material there, that could be done as well. If the material stays here, redirects should be tracked down and fixed. Will Orrick (talk) 21:49, 5 November 2022 (UTC)[reply]

There also are bibliographic references at Transcendental number that should be migrated over to here. Will Orrick (talk) 22:26, 5 November 2022 (UTC)[reply]

constructing all the transcendental numbers?

i hate to sound like a retard but does

Other techniques: Cantor and Zilber Liouville proved in the 1850s that transcendental numbers exist and even gave examples, but his ideas can be used to give only a countable number of transcendental numbers. Indeed, when he published his results the notion of countability and uncountability had not yet been developed. In the 1870s Georg Cantor started to develop set theory and in 1874 published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable.[15] Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result.[16] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number[17][18], the proofs in both the aforementioned papers give methods to construct transcendental numbers.[19]

mean methods to construct (as in the constructivists sense) all transcendentals? —Preceding unsigned comment added by Cheat notes (talk • contribs) 04:41, 15 January 2009 (UTC)[reply]

If you use the method gleaned from Cantor's diagonal argument then you can get every transcendental number, and that's proved in Gray's paper if you can find a copy. I'm fairly sure that the same holds true if you use his 1874 proof, too. Chenxlee (talk) 14:37, 21 January 2009 (UTC)[reply]

Added new reference to Gray's paper that gives everyone access to entire article. Also, added reference to Cantor's 1874 article. In Gray's paper, it is proved that every transcendental can be generated by a suitable enumeration of the algebraic real numbers. If you start with a computable transcendental (see computable number), then the proof generates a computable enumeration of the algebraic reals. However, if you start with a highly non-computable transcendental, then the proof generates a highly non-computable enumeration of the algebraic reals. Basically, the enumeration of the algebraic reals will be as complex (or constructive) as the transcendental is (and vice versa). Also, the remark above about using the Cantor's 1874 proof in a similar way is true. I can answer any questions about the article since I wrote it. I'm pleased to see that the article is being discussed. --RJGray (talk) 02:52, 11 August 2009 (UTC)[reply]

A branch of number theory?

Is number theory really defined broadly enough to include transcendence theory as a branch of it, as the lead sentence here says? I don't get that impression from the article Number theory. 208.50.124.65 (talk) 22:18, 18 June 2014 (UTC)[reply]

Certainly it is. See for example Baker's book Transcendental Number Theory. Deltahedron (talk) 19:53, 19 June 2014 (UTC)[reply]

Alan Baker's book

The most recent version of his 'Transcendental Number Theory' was the paperback edition of 1990. I suggest that all citations point to this edition. Scott Tillinghast, Houston TX (talk) 02:02, 17 December 2020 (UTC)[reply]