Talk:Transcendental number

From Wikipedia, the free encyclopedia

Contents 1 older comments 2 Inline <math> style 3 missing digit in example 4 Hilbert's seventh problem 5 Values of the gamma function 6 {{quantity}} 7 π+e or πe transcendental 8 proof of uncountability 9 Existence of non-transcendental irrationals 10 Proof, and references 11 amongst 12 Axel Boldt edit 13 Herkommer number 14 sin(a) transcendental? 15 "Almost all" reals transcendental

[edit] older comments

misplaced text on misspelt page:

(This has now been incorporated into the article on the correctly spelled page.

This is annoying me: This and other "Liouville numbers" are artificial examples, rather than numbers that directs our attention to it in a natural way. The first important number --- an eminently "natural" example ---. Are "artificial" transcendental numbers less important than so called "natural" ones? I think not. Just think on Lioville's discovery. I guess the first ones who thought about transcendentality of numbers were Johann Heinrich Lambert and Adrien-Marie Legendre in late 18th century. In the beginning of 19th century all mathematicians have agreed that their hypotheses are correct. Liouville gave a final step, regardless of non-naturality of his numbers. It is the idea that counts. Best regards. --XJam 23:18 Dec 23, 2002 (UTC)

I'm not even sure the "artificial" / "natural" distinction makes sense. The Liouville number seems constructed -- but so is e, if you define it as a summation. Even its definition as the base of natural log is artificial -- we've defined it as "the number that is the base of natural log". Why is that more natural than "the number whose nth digit is 1 if n is ... bla bla" ? -- Tarquin 23:24 Dec 23, 2002 (UTC)

Yes, why? :-) Perhaps π is the most "natural" as the other numbers... Perhaps. --XJamRastafire 23:36 Dec 23, 2002 (UTC)

I have a feeling that besides pi and e, the original author would be hard pressed to give an example of a "natural" number which is transcendental - since there doesn't seem to be an obvious sense of what "natural" is supposed to imply here - constructible? computable? commonly used? "useful"? or what? If I read the article a few more times, I'm sure it will irk me enough to "boldy edit" the distinction out of existence. Chas zzz brown 00:46 Dec 24, 2002 (UTC)

Since a trans.number is an infinite decimal, we can never write it down precisely as a decimal expansion -- thus we cannot define it as a decimal (unlike, say, 10. I can just point to "10" and say -- baboom! that's 10). So one has to define a trans.number with some sort of method or expression. The construction of Liouville's number is no worse a method than pi, which is usually defined as a tan^-1 expansion IIRC. be bold! -- Tarquin 02:02 Dec 24, 2002 (UTC)

I believe that Liouville indeed proved for the first time that transcendental numbers existed, so I'm going to revert the last change. I'm also going to change the sentence about the artificialness a bit. AxelBoldt 03:38 Jan 4, 2003 (UTC)

Changing the link irrational to irrational. First edit ever... -- Cyp (on some parts of the internet), 22:20 25 Jan 2003 (Danish timezone, probably GMT+0100)

---

[edit] Inline <math> style

I vote for Axel's manner of avoiding inline <math> but I am also afraid that shouting out won't help much since there are already pretty much articles which have exactly that. I just have one tiny observation about Greek letter phi. <math> mode goes like this or , but in wiki line style goes just like φ. What to do in this and such cases? --XJamRastafire

I agree with you and axel. ... unless... we set up the TeX parser to *always* produce HTML inline. In the meantime we can change it back. I suppose we have to match the phis. -- Tarquin 10:41 Feb 6, 2003 (UTC)

By the way, HTML allows not only φ (&phi;) but also Φ (&Phi;).

i made {{phisymbol}} to select fonts that rendered lower case phi with the bar right through for situations like this. Plugwash 01:40, 10 July 2005 (UTC)

[edit] missing digit in example

It looks to me like the last example (does it have a name?) is missing one digit. It should be 0.11010001000000010000000000000001000...

[edit] Hilbert's seventh problem

What does "The general case of Hilbert's seventh problem, namely to determine whether a^b is transcendental whenever a ≠ 0,1 is algebraic and b is irrational, remains unresolved" mean?
It has been solved by Gelfond when b is irrational algebraic - the answer is "yes". And it looks obvious to me that if a=√2=1.41421... and b=log(3)/log(2)=1.58496... then a^b=√3=1.73205... is irrational algebraic, b cannot be rational (otherwise there is an integer power of 2 which is also an integer power of 3), so the answer to the general case stated here is "not always". Perhaps someone can point out my error in understanding.--Henrygb 23:48, 8 Aug 2004 (UTC)

[edit] Values of the gamma function

I would guess that Γ(1/6) is known to be transcendental; this should follow from connections with complex multiplication, or, even more simply, from what is known about Γ(1/3) (which should be known to be algebraically independent of π, so that things like relations with Γ(1/2) can be used via the duplication formula). Charles Matthews 16:08, 31 May 2005 (UTC)

[edit] {{quantity}}

I think this template be rather called "numbers", as all it contains is numbers.

Also, I am a bit weary of templates. Would creating a Category:Quantity be a better idea? Or wait, there is already Category:Numbers. Anyway, my point is, I wonder, what is the purpose of the "quantity" template? Looking forward to an answer. Oleg Alexandrov 01:04, 11 July 2005 (UTC)

I really don't have a preference about templates versus categories. I added transcendental to the template just because I thought it should be in there with the other classes of numbers. But about the weird decision of calling this template "quantity", I do agree that it's a pretty weird name, but it sort of fits with the way the mathematics article is organized: it divides mathematics into quantity, structure, space and change. So you'll see a corresponding {{structure}} in abstract algebra and {{subst:Change}} in calculus. -Lethe | Talk 03:05, July 11, 2005 (UTC)

So, here they are:

change:

structure: Template:Structure

space:

quantity: Template:Quantity

I would advocate deleting these templates. What do you think? Oleg Alexandrov 03:40, 11 July 2005 (UTC)

I agree that the templates probably aren't going to do much; is the idea that someone who wants to learn about functional analysis will also want to learn about structural proof theory because they have some vague relation having to do with structure? Seems unlikely to me. However, a little poking around turns up that there are a lot of these templates. Like

and

and

and

and

v • d • e  Fields within the natural sciences
Astronomy | Biology | Chemistry | Earth science | Physics

What will you do? launch a full-on attack on these useless templates? or just that math related ones? -Lethe | Talk 05:06, July 11, 2005 (UTC)

I would want to attack the four math templates listed above only. And again, it is not yet clear whether the math ones should be deleted, even if I would think so. Thus, if I post this as discussion at Wikipedia talk:WikiProject Mathematics, would you agree to provide the arguments you listed above as supporting their deletion? I ask this to make sure this cause has at least some hope. :) Oleg Alexandrov 13:40, 11 July 2005 (UTC)

While I agree that these templates are a bit weird and of dubious utility, I'm not sure I completely support their removal. I might, but I probably need more convincing. The fact that lots of other technical subjects seem to have similar templates means that removal from the math pages would damage the consistency of wikipedia across technical subjects, and I do think we should value some uniformity of format at this project. So what am I saying, either we have to delete all the templates or none of them? No, that's probably too severe.

How about this Oleg, can you imagine a mathematics template which we could agree may be useful? Maybe a much coarser template, and only a single one instead of four of them. And maybe not organized so bizarrely (structure, quantity, change, and space???? wtf!). I'd feel better if we still had one organizing template, to ensure consistency with the other technical subjects. -Lethe | Talk 17:31, July 11, 2005 (UTC)

Hehe. I think we are having useful discussion here. :) I will copy this to Wikipedia talk:WikiProject Mathematics tonight. Oleg Alexandrov 16:39, 12 July 2005 (UTC)

[edit] π+e or πe transcendental

A recent addition to the article says: "at least one of π+e and πe must be transcendental, since both π and e are". Could we have a reference or a proof, please? Maybe it's something obvious, but I don't see it.... Macrakis 09:43 4 Aug 2005

if both π·e and π+e are algebraic, then π and e are zeros of the polynomial x² − (π + e)x + π·e. zeros of a polynomial over algebraic numbers are themselves algebraic. contradiction. -Lethe | Talk 23:50, August 10, 2005 (UTC)

more concretely, using the quadratic formula, we have

-Lethe | Talk 21:26, August 11, 2005 (UTC)

[edit] proof of uncountability

"The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the set of algebraic numbers is countable. But the reals are uncountable; so the set of all transcendental numbers must also be uncountable." i don't see how the first sentence is relevant, nor how it is related to the second. the second could stand alone, but i think should be explained a little more (maybe a link to uncountable or to cantor's diagonal argument). Xrchz

Yes, there should be a reference to the uncountability of the reals. But then to establish the uncountability of the transcendentals, you still need to show that the reals minus the algebraics are uncountable. This is easy because the algebraics are countable, which is what the first sentence establishes.--Macrakis 11:49, 19 September 2005 (UTC)

[edit] Existence of non-transcendental irrationals

The following was recently added to the first paragraph of the article:

However, not all irrational numbers are transcendental; √2 is irrational but is a solution of the polynomial x2 - 2 = 0.

It is of course true, but it seems superfluous to me. What do others think? --Macrakis 19:39, 29 October 2005 (UTC)

Harmless? Charles Matthews 22:09, 29 October 2005 (UTC)

Mostly harmless? Rick Norwood 00:39, 30 October 2005 (UTC)

When I first saw that diff, I thought like Macrakis. But then I chose to interpret that text as a very simple example of what a transcendental number is about. It somehow drives a good point, here is a number which is not transcendental, here is the algebraic equation it solves, and it is irrational, which clarifies the sentence right before. So, I feel a bit uncomfortable with that sentence, it might need moving, but overall it makes a good point. Oleg Alexandrov (talk) 03:31, 30 October 2005 (UTC)

[edit] Proof, and references

The given proof that e is transcendental uses the notation of integrals with no integrands... this notation looks very strange to me, is the proof generally presented like this? It seems that the integrals could easily have a f(z)dz under them, just to make the notation correct. I didn't change it because I don't know if there is a reason for it. When I checked the external links to see if the proofs given there used similar notation, I discovered that those pages were in german! Is that really appropriate for the english article? --Monguin61 09:08, 14 December 2005 (UTC)

I'd prefer I to the long-s here, but the notation is not actually 'wrong'. Charles Matthews 09:26, 14 December 2005 (UTC)

[edit] amongst

I've noticed several edits recently that replace "among" with "amongst". The latter looks slightly archaic to me. The two dictionaries I checked both say the two words are synonyms, but list "among" first. Is there a distinction in meaning that I am unaware of, or is this just a matter of personal preference? Rick Norwood 15:34, 14 December 2005 (UTC)

New Fowler's Modern English Usage says only that amongst is somewhat less common in American English. Charles Matthews 16:12, 14 December 2005 (UTC)

[edit] Axel Boldt edit

Alel Boldt has just made a major edit without discussion. I see good things and bad things in the edit. Among the bad things are the removal of the idea that some people, when they use the word "transcendental", mean "real transcendental" and the implication that the set of irrational numbers is not a subjet of the real numbers. If we go with this, we will need to rewrite the articles number and mathematics to be consistent. Thoughts? Rick Norwood 20:35, 27 March 2006 (UTC)

Yes, irrational numbers are usually taken to be real, so the statement that all transcendentals are irrational is not good and has been fixed. To define transcendentals are reals however means that several statements here and elsewhere become false, for instance the Gelfond-Schneider theorem or the fact that algebraic functions of one variable, applied to transcendental numbers, always yield transcendental numbers. AxelBoldt 00:31, 28 March 2006 (UTC)

[edit] Herkommer number

Can somebody please verify naming and transcendentality of the Herkommer number (preferably by providing references)? Mon4 12:31, 10 July 2006 (UTC)

[edit] sin(a) transcendental?

The article says that sin(a) is transcendental for any rational a. sin(π/6) = 1/2, so it is not transcendental. How could we fix this?

π/6 is certainly not rational, its not algebreic either for that matter. Plugwash 00:01, 26 November 2006 (UTC)

I missed the rational in that sentence as well. Actually my edit was correct but useless: sin(x) is of course algebraic for any rational root of the sine function, since the only rational root of the sine function is zero itself. Moreover, it's algebraic for any root of the sine function (whether it be rational or not) but that's not what the sentence said. Yet, (quote from Leibniz from the article):

sin(x) is not an algebraic function of x

Anyway, the word rational is subtle but crucial here. --CompuChip 10:40, 27 November 2006 (UTC)

[edit] "Almost all" reals transcendental

Should the article say something about (if I recall correctly) "almost all" real numbers being transcendental, since all finite algebraic formulae can be put in one-to-one correspondence with the integers using a Godel-number coding, and are therefore countably infinite, while the reals are uncountably infinite in any finite interval? Is there a precise way of putting this, for example, the algebraic numbers having measure zero in any finite interval? Or am I mistaken about this? -- The Anome 13:49, 18 March 2007 (UTC)

Well, it already says "The set of transcendental numbers is uncountably infinite." and proves it by showing that the algebraics are countable, which a fortiori makes them countable in any interval, so yes, "almost all" reals are transcendental, but given the multiple technical meanings of almost all (q.v.), I'm not sure that adds anything. --Macrakis 18:56, 18 March 2007 (UTC)