Talk:Transcendental number

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This article has comments.

"too well approximated"[edit]

Is it supposed to read "well approximated," or is there a specific meaning to "too well approximated"? Thanks for any response. —Preceding unsigned comment added by 173.180.139.172 (talk) 00:03, 30 October 2010 (UTC)

I expect the author of that sentence did mean “too well”, in the sense of “too well to be algebraic”. Maybe the wording is confusing, but I would have thought it is adequately explained by what comes behind the comma. Hanche (talk) 12:30, 30 October 2010 (UTC)

the proof that e is transcendental is too rushed at the end[edit]

the proof proceeds in a simple-to-follow fashion up until the end when the P2 partition of the algebraic equation for e is supposedly shown to satisfy: |P2 / k!| < 1

at this point an auxiliary function is defined, separated into two factors, but then the rest of the explanation degenerates rapidly.

it isn't clear what is meant by "using upper bounds" or how this is related to the following limit regarding G^k / k!

it would be nice if someone would unpack this part of the proof somewhat so that it can be digested as easily as P1 / k! being in Z-{0} —Preceding unsigned comment added by 76.182.194.195 (talk) 09:30, 15 March 2010 (UTC)

older comments[edit]

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)

This raises an interesting point, because it follows from what you've just said that transcendental numbers must be countable. If they have to be defined by some sort of method, then they have to have a "generating rule" and these rules can be embedded in space-time geometry - most obviously, by when and where they were first defined or thought of. Thus, the rules and hence the numbers can be counted.

- Meltingpot

Meltingpot 09:57, 12 September 2007 (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)


Inline <math> style[edit]

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 \mathbf{\phi} or \mathbf{\varphi}, 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)

missing digit in example[edit]

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

Liouville's constant. Scott Tillinghast, Houston TX (talk) 04:17, 23 September 2010 (UTC)

Hilbert's seventh problem[edit]

What does "The general case of Hilbert's seventh problem, namely to determine whether ab 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 ab=√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)

The error is in the statement of the problem, it is required that b is not only irrational but also algebraic, the article was fixed when I looked again. You can consider this a proof that log3/log2 is in fact transcendental. —Preceding unsigned comment added by 81.151.98.151 (talk) 01:38, 24 July 2008 (UTC)

Values of the gamma function[edit]

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)

{{quantity}}[edit]

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: Straight Line Steady.svg

structure:

space:  

quantity: {{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 {{BranchesofChemistry}} and

and

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)

π+e or πe transcendental[edit]

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
\{\pi,e\}=\frac{\pi+e\pm\sqrt{(\pi+e)^2-4\pi e}}{2}

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

proof of uncountability[edit]

"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)

Existence of non-transcendental irrationals[edit]

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)

Proof, and references[edit]

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)

amongst[edit]

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)

Axel Boldt edit[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)

Herkommer number[edit]

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

sin(a) transcendental?[edit]

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)

"Almost all" reals transcendental[edit]

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)

You cannot say that "most real numbers are irrational and transcendental", because an infinitude of both exists. Although it seems like you can say that there are twice as many integers as there are natural numbers, because each natural number has a negative, this is not correct, as there is an infinitude of both. You cannot compare infinity with itself - you cannot divide infinity by infinity (∞/∞) - in other words, you cannot treat it as a number; you can only talk about a limit. So for real numbers - an infinitude of rationals and irrationals, algebraics and transcendentals exists, and the smallest line segment contains infinitely many of each. Majopius (talk) 21:31, 26 March 2010 (UTC)

This is not correct. There are different sizes of infinities measured by cardinal numbers. In particular, there is no doubt that there are more transcendental numbers than algebraic numbers. Hanche (talk) 16:24, 27 March 2010 (UTC)
A way to number/count the set of integer points on a plane - a spirallike line.
Although at first it seems that there are different sizes of infinities, we can actually show that they can sometimes be made equivalent to eachother. Consider, for example, the set of integer points on a plane. It is reasonable to say that the number of these points is 4∞2, because the plane is a 2D figure with dimensions of 2∞ * 2∞; the number of integer points on a line, like I already said, can be considered to be 2∞ - the line is a two-sided infinity, the ray is one-sided. Yet, we can equate the set of integer points on a plane to the natural sequence - set of integer points on a ray, as shown in the picture. So we get a strange result at first glance: that 4∞2 = ∞, which, however, becomes apparent due to the nature of infinity. Majopius (talk) 23:23, 16 April 2010 (UTC)
Except your example is irrelevant. The set of algebraic numbers is countable and the set of transcendental numbers is uncountable. These sets cannot be put into bijection with each other. Sławomir Biały (talk) 14:51, 8 December 2010 (UTC)

Contradiction between Algebraic numbers and Transcendental number[edit]

The article Transcendental number states that

In mathematics, a transcendental number is a real or complex number which is not algebraic, i.e., not a solution of a non-zero polynomial equation with integer coefficients.

whereas in the article Algebraic numbers the following is stated:

In mathematics, an algebraic number is a complex number that is an algebraic element over the rational numbers. In other words, an algebraic number is a root of a non-zero polynomial with rational (or equivalently, integer) coefficients.

Is it only integer coefficients or rational coefficients? Hakeem.gadi 09:29, 10 April 2007 (UTC)

There is no contradiction, since the roots of a polynomial with rational coefficients are also the roots of a polynomial with integer coefficients. Consider the rational polynomial an/bn * x^n + ... a1/b1 * x + a0/b0. It has the same roots as the integer polynomial An * x^n + ... + A1 * x + A0, where Ai = ai * product(bi,i,0,n). In fact, if the coefficients are known to be algebraic themselves, then the roots will also be algebraic. We could of course add the phrase "(or equivalently, rational)" to the transcendental page heading, but it's best to keep it simple. --Macrakis 12:00, 10 April 2007 (UTC)

New Scientist[edit]

The latest issue of New Scientist seems to be claiming that many mathematicians now think that the Gelfond-Schneider theorem is on the verge of being extended. As a result a whole raft of new theorems are on the verge of being proved about transcendentals. Someone please check it out.--Michael C. Price talk 05:55, 25 July 2007 (UTC)

Proof that the set of transcendental numbers is uncountably infinite[edit]

It may just be me, but i think describing this proof as "simple" is a bit off as it requires at least a basic understanding of honours level mathematics.

212.139.100.238 15:00, 7 October 2007 (UTC)

Gelfond's constant[edit]

Part of the article says Gelfond's constant eπ is transcendental, but a later part says all sums and products and powers of e and π are transendental EXCEPT Gelfond's constant. Which is it?

Crasshopper 16:55, 29 October 2007 (UTC)

The later part in fact says that sums, products and powers of e and π are not known to be transcendental, except for Gelfond's constant. Algebraist 15:46, 22 November 2007 (UTC)

eπ/2 is not equal to ii. eπi/2 is. —Preceding unsigned comment added by 84.248.150.161 (talk) 18:40, 14 March 2008 (UTC)

eπi/2 is equal to i. So ii is e-π/2 as the article says. Chenxlee (talk) 02:39, 15 March 2008 (UTC)

Pi as a root of a quadratic equation[edit]

Is pi a root of any quadratic equation at all? If so, give me an example of a quadratic equation that has pi as a root.--98.199.76.184 (talk) 22:19, 13 April 2008 (UTC)

Sure, it is a root of (x-pi)*(x+pi) = x^2-pi^2. But of course that is not a polynomial with rational coefficients. --Macrakis (talk) 00:31, 14 April 2008 (UTC)

A number that is infinitely small, like PI, is considered a transcendental number. So should a number that is infinitely big, be considered for categorization into transcendental numbers.[edit]

The point of having a "Numbers for which it is unknown whether they are transcendental or not" pretty much explains its self. Putting my suggestion there is valid, and is the only correct spot on the page to suggest a link, to discuss the topic.

THE FACT THAT ITS NONSENCE in YOUR OPINION doesn't mean that it dosen't deserve a chance to be discussed to an indepth degree.

Even if you have a PHD in "WHATEVER", doesn't change the fact that a bit of imagination, along with a proper level of common sense would go along way in helping old people that are unwilling to accept new ideas and math. Most ideas aren't accepted by the old generation, even if they are PROVEN to be correct. They are accepted because the OLD PEOPLE DIE, and the younger generation that has already accepted it now have the control. Having some PHD say that something is nonsence, is nothing new. WHY DON'T YOU MATHEMATICALLY PROVE THAT ITS NONSENCE, MR PHD. But coming up with a NEW math, new ideas, and sharing them takes more guts and effort than most with a PHD will ever have. Too afraid of getting heckled.

The number PI that is infinitely big is a transcendental number

Like it or lump it, discuss it.

WikiMath: How to realize, prove, disprove, and or show the math behind a new type of number. A Transcendental number.

I wonder how many years it will take to get this accepted in circles of proper number organization. Such is the case when you have MR. PHD offended that his perfect math picture is upset by new ideas and schools of thought. NONSENCE!  :)

Much like MR.PHD does, almost 75% of people who edit on wikipedia don't even bother to show PROOF of why they feel a certain way. They just take the lazy route and insult, and delete. Wow, I want to learn from someone like that! What good is a PHD if your not even willing to use it and teach others, or correct others AND SHOW THE MATH, pun intended. Geez. Gravitroid (talk) 03:04, 6 July 2008 (UTC)

Wikipedia is not the place for original research. Your "infinite numbers" are not part of the standard mathematics of algebraic and transcendental numbers, so discussion of them doesn't belong here. Wikipedia is also not an educational institution, so I feel no obligation to teach you mathematics here. There are many good sources for that. --Macrakis (talk) 09:33, 6 July 2008 (UTC)
The discussion of them is not here. It WAS in the link you deleted.

Under the title "Numbers for which it is unknown whether they are transcendental or not" I will repost the link now. If you choose to discuss this topic, follow the link. WikiMath: How to realize, prove, disprove, and or show the math behind a new type of number. A Transcendental number.

Gravitroid (talk) 09:43, 6 July 2008 (UTC)

BUT I do have one simple question, MR.PHD,

In your professional opinion, if this kind of number doesn't belong in standard mathematics of algebraic and transcendental numbers,,,,

WHERE DOES IT BELONG????????????


Where would you file a number that is infinitely big as it is infinetly small? Can you even classify this number? Where IS the place for correct discussion of this number?

Your telling me that wikipedia has no known location for discussion, or description, or article of a number that can be described in simple terms?

On wikipedia aren't all types and formats of numbers pretty much allowed?

So tell me, among all the wealth of locations in wikipedia for this....

Where would you put a number that is infinitely big as it is infinetly small?

Gravitroid (talk) 10:29, 6 July 2008 (UTC)

You ask:
Your telling me that wikipedia has no known location for discussion, or description, or article of a number that can be described in simple terms?
That's correct. Wikipedia is not a place for general discussion. Please review our policies on original research and notability. Even if we assume that the numbers you're talking about are well-defined (which is not clear), just because a class of numbers can be described "in simple terms" doesn't make it notable. --Macrakis (talk) 10:57, 6 July 2008 (UTC)

OK fine.

But this is not a class of numbers,,,

Im talking about a single number... ,,, a number that is infinitely big as it is infinetly small.

Just one number. Not a class, not a set, not an array, .....

You seem to have things in control around here, you consider yourself a part of wikipedia, like you work there, or your apart of some elite that is wikipedia? I am obviously not allowed to be apart of that.

Ok, whatever, I can accept that.

But can you accept the fact that you, can't answer this:

WHERE DOES IT BELONG????????????

Where would you put a number that is infinitely big as it is infinetly small? Can you even classify this number?

A theoretical number? An imaginary number? An inifinite number? An ________ number?

I don't even care to put it in this category anymore, I DON'T CARE. I HAVE NOTHING TO GAIN.

I just want to find answers to my questions.

Perhaps if I had a PHD i could answer my own questions.

IF I ever figure out where this number belongs, I'll let you know.

It seems your more interested in getting rid of the number, than contemplating its existence.

I give up. Gravitroid (talk) 11:54, 6 July 2008 (UTC)

The standard system of complex numbers does not include infinitely small numbers, or infinitesimals. In any event, pi is not infinitely small. Infinitesimals are found in the extended number system developed by Abraham Robinson. Scott Tillinghast, Houston TX (talk) 03:25, 25 September 2010 (UTC)

Euclidean geometry[edit]

Recent addition: “Another way of looking at transcendental numbers is that they are numbers that do not arise from euclidean geometry or ordinary algebraic expressions.” What is this supposed to mean? The number π certainly “arises” from Euklidean geometry, since circles do. Or is this a reference to non-constructible numbers? But many non-constructible numbers are algebraic. In my opinion, the sentence is too vague to be of any use.

Hanche (talk) 16:39, 13 July 2008 (UTC)

If nobody comes up with a good explanation and rationale for the reference to Euclidean geometry, I am going to remove it. Hanche (talk) 11:58, 21 July 2008 (UTC)

Should the definition include irrationality?[edit]

The intro of this article states that "a transcendental number is an irrational number that is not algebraic." However, the algebraic number article trivially proves that every rational number is algebraic—equivalently, that every transcendental number must be irrational anyway. That is, defining a transcendental number as "an irrational number that is not algebraic" is equivalent to defining it as "a number that is not algebraic." It seems to me that it could simply say, "a transcendental number is a number that is not algebraic," and later, "every transcendental number is irrational because…." Thoughts? --bdesham  13:05, 18 July 2008 (UTC)

I agree that it should not be part of the definition. But the irrationality of transcendental numbers should probably be mentioned somewhere in the introduction. Hanche (talk) 09:23, 20 July 2008 (UTC)

edit glitch?[edit]

The opening line of the article says that transcendental numbers are complex numbers that are not algebraic. Unless you consider reals to be a subset of complex numbers, this statement is misleading. Since later in the article it talks about "real or complex numbers" I think it would be clearer, especially for novice readers, to state that transcendental numbers are "real or complex numbers" that are not algebraic.

Al-mandarb (talk) 03:40, 16 August 2008 (UTC)

I don't see it as a glitch: All real numbers are indeed complex numbers. By writing "real or complex" one only propagates the misunderstanding that the complex numbers do not include the reals. (I think that anyone who looks at the examples will quickly notise that they are all real numbers.) Hanche (talk) 18:11, 18 August 2008 (UTC)

Who first defined transcendental numbers?[edit]

The History section starts out as "Euler was probably the first person to define transcendental numbers in the modern sense." Just today, I read in the book Mathematics in Ancient Greece (Dover Books on Mathematics) (Paperback) by Tobias Dantzig that in fact it was Leibniz (spelled as Leibnitz in the book -- which is also an accepted spelling) that introduced the concept of transcendental numbers (as well as transcendental functions); the comment is, as I recall, on p. 142, but I don't have the copy of the book available here, and in any case, one can easily find the page by looking in the index. He called the algebraic numbers "analytic numbers", according to the book. I assume this information is correct, but maybe someone can confirm it from other sources. Mateat (talk) 01:57, 8 November 2008 (UTC)

Question[edit]

Is zero a transcendental number? carmicheal99 12:47 PST 27 December 2008 —Preceding unsigned comment added by 71.83.230.207 (talk) 20:47, 27 December 2008 (UTC)

No. x=0 is an algebraic equation. --macrakis (talk) 21:30, 27 December 2008 (UTC)

Confusion about history[edit]

I'm sure my confusion here stems from the difference between algebraic and transcendental functions vs. numbers, but the History section states:

"Euler was probably the first person to define transcendental numbers in the modern sense. The name 'transcendentals' comes from Leibniz in his 1682 paper where he proved sin x is not an algebraic function of x."

And then:

Joseph Liouville first proved the existence of transcendental numbers in 1844..."

Wait a minute. If Leibniz proved that sine was not algebraic, then once Euler defined the term, wouldn't that have immediately implied the existence of transcendental numbers (namely, certain values of the sine function)? I guess what I really need to know is, what's the relationship between transcendental functions and transcendental numbers? I don't see that explained in either article. - dcljr (talk) 03:05, 2 August 2009 (UTC)

If you plug an algebraic number into an algebraic function, you get out an algebraic number. There's no general rule for transcendental functions, though. The only algebraic number x such that ex is algebraic is x = 0. But there are examples dating back to Weierstrass of transcendental functions that pop out a rational result whenever a rational number is input. And more recently it was shown that you can make transcendental functions that always return an algebraic number if you put in an algebraic number. And then there are functions where we don't know what happens if you put in an algebraic number, like exp(exp(x)). To answer your question, then, there really isn't much of a relationship between them, they're just both examples of transcendental elements of a certain field extension. Chenxlee (talk) 14:08, 2 August 2009 (UTC)
OK, thanks. Perhaps something about this should be mentioned in the article? - dcljr (talk) 22:57, 3 August 2009 (UTC)

Mahler-Sprindzhuk theorem[edit]

I find it difficult to believe this theorem as I understand Baker has stated it. I could more easily believe that almost all real numbers have measure of transcendence = 1.

For any irrational real x, its measure of irrationality ω(x,1) is at least 1. If the type of x is the supremum of the set {ω(x,n)} it would follow that ω(x,1) = 1 for almost all real numbers x.

A regular continued fraction can be constructed to represent a real number with an arbitrarily large measure of irrationality. I suspect that such a number can be constructed to have a measure of irrationality with any arbitrary value > 1. This sounds to me like a set of non-zero measure.

All numbers x with ω(x,1) > 1 are transcendental by Roth's theorem. Scott Tillinghast, Houston TX (talk) 23:59, 25 September 2010 (UTC)

Powers of one are not transcendental[edit]

The article claims: Numbers known to be transcendental: * ab where a ≠ 0, a is algebraic and b is irrational algebraic

if a = 1 (algebraic and non-zero) , then ab is 1 for all b including irrational algebraic b. This is not transcendental. Or am I missing something? Keirf (talk) 10:29, 18 November 2010 (UTC)

Originally that line said a ≠ 0,1. It seems to have misplaced the 1 at some point since then. I put it back. Chenxlee (talk) 11:15, 19 November 2010 (UTC)

Approximation of Liouville Numbers[edit]

I have a problem with the following sentence in the history section:

"Liouville showed that this number is what we now call a Liouville number; this essentially means that it [I assume a louivelle number] can be more closely approximated by rational numbers than can any algebraic number."

Many algebraic numbers are rational so can be exactly approximated (if you'll excuse the oxymoron) by a rational number leaving little room for louiville numbers to be more closely approximated by irrational numbers. I fact tagged this a while back as it seemed contradictory but it was removed by Hanche. I asked him about it on his Talk but haven't received a response.

Looking at the Louiville number article, I wonder if it should read:

"Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be more closely approximated by rational numbers than can any irrational algebraic number."

In any case pure maths isn't really my thing so I've not edited it as yet but need to either see the article or my misunderstanding corrected..... IanOfNorwich (talk) 21:03, 16 January 2011 (UTC)

If the number in question is rational, then it is a matter of approximating it by other rational numbers whose denominators are not too large. If a rational number has denominator q, then no other rational number with denominator less or equal can be closer than 1/q2. Liouville numbers do not follow this rule. A Liouvile number has in its neighborhood rational numbers that are remarkably close considering their denominators.

Let integer r≤q. Then s/r-p/q = (sq-rp)/qr. Because sq-rp is an integer, its absolute value ≥1 (unless 0). Hence |s/r-p/q| ≥ 1/qr ≥ 1/q2. Scott Tillinghast, Houston TX (talk) 05:52, 17 January 2011 (UTC)

Oops, I forgot to say r>0. I make it a habit to use positive denominators. Scott Tillinghast, Houston TX (talk) 06:02, 17 January 2011 (UTC)

Let's try

"Liouville showed that this number is what we now call a Liouville number; this essentially means that it can be more closely approximated by rational numbers with small denominators than can any irrational algebraic number."

Scott Tillinghast, Houston TX (talk) 08:03, 17 January 2011 (UTC)

OK, Thanks Scott, I think I follow what you're saying. At least I see the proof that no rational number with denominator <= q can be closer than 1/q2 to a rational number with denominator q. Though I have to confess I can't see how this helps without establishing the denominator of the liousville number!? I guess "small denominators" hints at how the proof works but seems a bit vague here - how small exactly?
What is this sentence trying to do - define Liousville numbers or outline the proof that they are transcendental? It doesn't seem to do either accurately. How about:
"This is an example of what are now called Liouville numbers." IanOfNorwich (talk) 20:48, 17 January 2011 (UTC)

When I say small denominator I mean small considering how close the rational number is to the number (irrational or rational) being approximated. Example: the Liouville constant has an approximation with denominator 10n!, but the error is < (10/9)10-(n+1)!, like the reciprocal of the 10th power of the denominator.

Only rational numbers have denominators, although algebraic numbers have a height, which is a related concept. The height of a rational number is the absolute value of its numerator or denominator, whichever is greater.

My sentence and its predecessor both attempt to be brief about something that hard to describe in a few words. Scott Tillinghast, Houston TX (talk) 06:12, 18 January 2011 (UTC)

Correction: I said 10th power of denominator; I meant (n+1) power. Scott Tillinghast, Houston TX (talk) 06:17, 18 January 2011 (UTC)

Wait a second[edit]

Currently, the article states that Leibniz proved that sin(x) is not constructible algebraically, which is to say (in today's terminology) that it is transcendental. But then we say that Liouville proved that transcendentals exist in 1844. These can't both be true. If Leibniz actually proved that sin(x) can't be found algebraically, than surely that's a constructive proof of the existence of transcendental numbers. Ethan Mitchell (talk) 00:15, 4 August 2011 (UTC)

A function being transcendental doesn't imply that it has to take transcendental values at any algebraic argument, it's possible to construct a perfectly nice transcendental function f such that f(x) is algebraic whenever x is algebraic. So the fact sine is transcendental doesn't a priori mean transcendental numbers exist. Chenxlee (talk) 13:36, 5 August 2011 (UTC)
Of course. I've reversed the order the order of the sentences, which I think adds clarity on this point. Also, it makes them chronological. Ethan Mitchell (talk) 20:43, 6 August 2011 (UTC)

Logarithms other than the natural[edit]

What is known about logarithms to other bases than e based on e.g. the Gelfond–Schneider theorem? Isheden (talk) 15:30, 8 January 2012 (UTC)

Baker's theorem, one of the most general results in transcendental number theory concerning logarithms, and which includes the Gelfond-Schneider theorem as a corollary, also applies to logarithms to other bases, as long as you use the same base in every logarithm. Chenxlee (talk) 20:15, 8 January 2012 (UTC)

The logarithm article reads "the Gelfond–Scheider theorem states that given two algebraic numbers a and b, logb(a) is either a transcendental number or a rational number p / q (in which case aq = bp, so a and b were closely related to begin with)." Is this what one can say about logb(a) or does Baker's theorem bring more insight? Perhaps it would be interesting to add to this article? Isheden (talk) 22:41, 8 January 2012 (UTC)

Comment regarding the lead[edit]

I am having a hard time understanding the wording of the following line, "The converse is not true: not all irrational numbers are transcendental, e.g. the square root of 2, and of other non-perfect squares, are irrational but algebraic numbers (2.5) and therefore not transcendental."

The "2.5" in there and that whole sentence is somewhat grammarically confusing and doesn't come across as informative or instructive. Anyway to write it up so it can be understood?

Thanks

173.238.43.211 (talk) 07:31, 28 April 2012 (UTC)

I have revised the lead in an attempt to make it more readable. Isheden (talk) 12:44, 28 April 2012 (UTC)

YAY ty. I just now see that. Helps very much ty.

173.238.43.211 (talk) 10:10, 29 April 2012 (UTC)

Mahler's classification[edit]

This section as it stands seems somewhat long and technical. It also has significant overlap with Transcendence theory. I propose that it be retitled "Classification of transcendental nujmbers", cut down, and detail transferred to the other article. Deltahedron (talk) 08:37, 22 July 2012 (UTC)

irrational^irrational?[edit]

Are all or part of the set of irrationals to the power of a irrational considered transcendentals? — Preceding unsigned comment added by Reddwarf2956 (talkcontribs) 13:38, 14 December 2012 (UTC)

An irrational raised to an irrational power can be an integer: for example, elog 2 = 2 where both e and log 2 are irrational (and indeed, transcendental). It can also be irrational but algebraic, for example e(1/2)log 2 = √2; or transcendental, for example, √2√2. Indeed, every positive real number can be expressed as in the form of an irrational raised to an irrational power. Deltahedron (talk) 22:25, 14 December 2012 (UTC)

Alternative definition[edit]

Can transcendental numbers be (loosely) defined as "Any irrational number that is not the root of a rational number. For example, pi (π) is transcendental, but \sqrt{2} is not, because it is the (square) root of a rational number (2)" ?

If so, perhaps such a definition should be added to the opening section - Hatster301 (talk) 11:50, 29 July 2013 (UTC)

No, I'm afraid that definition is incorrect. There are many algebraic numbers that are not the root of a rational number, e.g. \sqrt{1+\sqrt{2}}, a root of x^4-2x^2-1. In fact, there are many algebraic numbers that cannot even be expressed as a composition of nested arithmetic operations and roots. The simplest case is in the roots of some 5th-degree polynomials. --Macrakis (talk) 13:25, 29 July 2013 (UTC)