# Talk:e (mathematical constant)/Archive 1

Archive 1 | Archive 2

## Early questions

show a aravege person from dere to here and do it right as u can if u cant do it right then dont do it at all because u will not gte anywhere in your life

Betcha ain't no average person, 'cause "average" people can spell better than you. --Army1987 15:03, 19 October 2005 (UTC)

Why is the page called "E (mathematical constant)"? I've seen this number named "e" (lowercase) almost everywhere, and the only place, other than Wikipedia, where I've seen "E" (uppercase) is Mathematica. So, I think this page should be renamed to "e (mathematical constant)". --Fibonacci 00:47, 27 Apr 2004 (UTC)

Fiddly technical reasons, I'm afraid. Added this to the page. --Aponar Kestrel 16:27, 2004 Jul 31 (UTC)

In The Book on Numbers by John Conway and Richard Guy, the number e is persistently called Napier's number. I know that John Napier more or less discovered logarithms, but is this really the correct name? -- JanHidders

I don't think that's too common; Weisstein lists it as "Napier's Constant", but the main entry is under "e". Encyclopedia Britannica doesn't list "Napier's Number" or "Napier's Constant" at all. Most people call it "the base of the natural logarithm", I believe.

e is still called Euler's number in many texts too introductory to worry about confusion with &gamma (Euler's constant).

Could somebody explain what `e' is useful for? It was always difficult for me to explain it to mathematical newbies.

And could somebody explain ei*π = -1, and why it is so? -- Taw

See The most remarkable formula in the world (where it is poorly explained to the layman, sorry!) -- drj

It's used mainly because it arises "naturally" in calculus, and is related to useful functions (eg., trigonometric and hyperbolic functions). A connection with pi is inevitable, as pi is related (via polar coordinates) to -1 and the trigonometric functions.

Zundark - as far as I know, you should be entitled to claim that you invented the word "miscorrection" :) Great stuff! - MMGB

I'd like to take credit for it, but Google finds about 140 pages with this word. --Zundark, 2001 Nov 26
Really?? To me it would appear to be an Oxymoron but oh well :) - MMGB
No. It'd mean "correcting" something you believe to be wrong but it's actually right. In linguistics that's called hypercorrection.

D'oh! I really need to quit editing pages when I'm so tired I can hardly type straight. At least I got the sum notation definition right...--BlackGriffen

Could this be moved to e (base of natural logarithm), to make it more consistent with other disambiguated titles, and to allow for the pipe trick? -- Oliver P. 00:13 Feb 22, 2003 (UTC)

How about e (number)? -- Tarquin 09:08 Feb 22, 2003 (UTC)

I think that e (number) would be easily confsed with E numbers The Anome

e (mathematical constant)? -- Oliver P. 15:18 Feb 22, 2003 (UTC)
Sounds okay. -- Tarquin
Okay, I've moved it there. -- Oliver P. 00:12 Feb 24, 2003 (UTC)

I think, e should always be spelled e not E...

"The number e is relevant because one can show that the exponential function exp(x) can be written as e^x"

I'm really confused what this is supposed to mean. Exp(x) is just another notation for e^x. The only thing I can guess is that this is supposed to mean that exp(x) is defined as unique function y' = y and y(0) = 1, and e^x means limit of a^x as a approaches e, where e is given as the limit in the article. But this is a rather convoluted observation to make. Revolver
It's not just notation. For a^x where x is not rational, you have to define it in terms of exp(x) and natural log. So you need some construction of exp(x) to even define e^x for irrational x. -- Walt Pohl 18:17, 1 Mar 2004 (UTC)
No, you DON'T have to. You can define a^x to be limt->x at, and for rational t=m/n you can easily define am/n to be n√am. Since there are infinitely many rational t's in any neighborhood of any real x...--Army1987 21:42, 26 Jun 2005 (UTC)
I don't follow. If you have exp(x) defined, you have e^x, and vice versa. THEY'RE THE SAME FUNCTION. It's just a different notation. To say, "you need some construction of exp(x) to even define e^x for irrational x." is kind of a tautology, like saying, "you need some construction of sine function to define sin(x) for irrational x." Well, sure, that's true for anything. While it's true a^x can be defined in terms of e^x for a &neq; e, it's just as true that e^x is already defined for irrational x as soon as e^x is defined, you don't need "exp(x)", well you do, since it's the same function. I really miss the point. The claim only makes sense if "exp(x)" and "e^x" are actually defined in 2 different ways. I gave an example of this above. You can define exp(x) all sorts of ways (2 in this article, as solution of diff eq, yada yada), to say two notations are equal without saying how they're defined isn't saying anything. Revolver 01:19, 12 Jun 2004 (UTC)
I see, you're saying a^x = exp(log(a^x)) = exp(x*(log(a)), so e^x = exp(log(e^x)) = exp(x*(log(e)) = exp(x), that's certainly true, it's just not saying anything terribly interesting, IMO, beyond "log is the inverse function of exp" or something like that. So, I still don't get it. Revolver 01:23, 12 Jun 2004 (UTC)

I enjoyed both forms of the expansion (one was removed today). Of course they can be derived from each other, but I still found the patterns that each exhibit interesting. Here are the two forms:

${\displaystyle e=\sum _{n=0}^{\infty }{1 \over n!}={1 \over 0!}+{1 \over 1!}+{1 \over 2!}+{1 \over 3!}+{1 \over 4!}+\cdots }$
${\displaystyle e=\sum _{n=1}^{\infty }{n \over n!}={1 \over 1!}+{2 \over 2!}+{3 \over 3!}+\cdots }$

Can we put them both back into the article? Bevo 16:39, 21 Feb 2004 (UTC)

My vote is to leave out the second one. It is trivially equivalent to the first one, and I had been planning to remove it myself. -- Dominus 16:51, 21 Feb 2004 (UTC)
The reason I removed the second series is that it is completely identical to the first, i.e. if you write out the terms, they are exactly the same, so they are really the same series, not different, just expressed differently. You can see this by cancelling n to get 1/(n − 1)! and then do a shift of index, (drop on the index, up inside the sum). If the series were truly different, I would consider keeping it. Revolver 06:22, 27 Feb 2004 (UTC)

## "Keller's Expression"

The article says:

In 1975, the Swiss Felix A. Keller discovered the following formula that converges in e ("Keller's Expression" Steven Finch, mathsoft):
${\displaystyle e=\lim _{n\to \infty }\quad {\rm {}}{\frac {n^{n}}{(n-1)^{(n-1)}}}-{\frac {(n-1)^{(n-1)}}{(n-2)^{(n-2)}}}\quad {\rm {for}}\quad \left|n\right|>2.}$
This formula was published for the first time 1998 on Steven Finch's website www.mathsoft.com/asolve/constant/e/e.html. He refers to it as “Keller's Expression”.

This is a delightful formula, but I have some problems with the way the description is written. First, the referenced document is not available (404). Second, it seems unlikely that the formula was first discovered in 1975. Even if it was never published before (which I rather doubt) I think that's more likely to be because it is so simple to prove. The formula looks mysterious at first glance, but really it turns out that the left term approaches n·e and the right term approaches (n-1)·e; this can be proved in about two steps of simple algebra, directly from the definition of e. Google search doesn't turn up anything relevant for "Felix A. Keller" or for "Keller's Expression". So we have a formula here which could be discovered in ten minutes of idle tinkering by any bright undergraduate, but it's being credited to Mr. Keller as though it were a big discovery. That seems strange to me. Formulas usually only get names when they are important or at least surprising (Stirling's formula, Euler's identity) and this one is neither. -- Dominus 14:38, 10 Mar 2004 (UTC)

I got rid of this again in October 2004, and again today. -- Dominus 02:09, 20 Mar 2005 (UTC)

I've removed another insertion of "Keller's Expression" (together with proof) today. Paul August 16:32, 23 January 2006 (UTC)
I actually included a form of "Keller's expression" in the article, under the proviso that it is derived from simple algebraic manipulation of the basic limit def. I think it should be in there for its intrinsic formal beauty. Perhaps I should add a note about "keller's expression" beside the formula to prevent this type of thing?--Hypergeometric2F1[a,b,c,x] 06:53, 24 January 2006 (UTC)
Perhaps, but it would need a reference. Where did you get that limit from? Did they call it "Keller's expression"? Paul August 14:35, 24 January 2006 (UTC)
I got it from mathworld. I dont recall a reference, but it is the same thing as Keller's expression, just made into a more pleasing form.--Hypergeometric2F1[a,b,c,x] 04:36, 25 January 2006 (UTC)

## Proofs

I think we should add more proofs, eg, that the given continued fraction representation is correct.

Euler's proof of that is a doosy. I can think of a simpler, non-rigorous proof but I didnt think that we were suppost to overload wikipedia math articles with proofs. Perhaps the longtime math contributers can chip in on this.--Hypergeometric2F1[a,b,c,x] 04:38, 25 January 2006 (UTC)

I think there needs to be a proof that e^x is it's own derivative. I can do it as far as getting f(x) = b^x then f'(x) = f(x)*f'(0) but don't know how to prove that b = e makes f'(0) = 1 (so that f'(x) = f(x)) Kousu 05:26, 17 June 2006 (UTC)

I don't think that belongs in this article. But to prove that the derivative of ${\displaystyle e^{x}}$ at 0 equals 1 one one can use the fact
${\displaystyle (1+x)^{1/x} for ${\displaystyle x>0}$ to deduce by rasing both sides to x that
${\displaystyle 1+x
which shows that if the limit ${\displaystyle (e^{x}-1)/x}$ exists it is no smaller than one. The inequality ${\displaystyle (1+x)^{1/x+1}>e}$ should make it work the other way. Oleg Alexandrov (talk) 08:34, 17 June 2006 (UTC)
Agree with Oleg. The proof is obvious just by deriving each term of its power series expansion. Soltras 21:33, 17 June 2006 (UTC)

Just for the record, I removed the bit about the Pyramids and Greeks. It smells of nonsense, was originally added in bold text, and I can't find any other references to either part of it anyway. (And given that the Greeks were not known for their imprecision in mathematics, I can't imagine they'd mistakenly use 2.72 for e if they knew about it.) --Aponar Kestrel 06:38, 2004 Jul 31 (UTC)

## Expansion

I removed the <big> tags surrounding the approximation of e as this caused the number to be breaked at the resolution of 800x600. If there is a need to include longer expantion of e we should break it ahead of time I think. Two possibilities are:

 e ≈ 2. 71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66968
${\displaystyle {\begin{matrix}e\approx &2.71828\,18284\,59045\,23536\,02874\,71352\\&\ \,\,66249\,77572\,47093\,69995\,95749\,66968\\\end{matrix}}}$

--filu 13:08, 24 Apr 2005 (UTC)

Not terribly important, but with 30 digits shown, and digit 31 being '6' the number shown would be more accurate if rounded up (and is thus misleading), yet wouldn't show the same digits that longers representations do. (It's just that I assumed that this was the motivation for using 64 digits on the pi page, and was mine for using 64 here.) Frencheigh 00:27, 25 Apr 2005 (UTC)

Thanks for pointing this out. I am not sure I undertand though what you mean by "yet wouldn't show the same digits that longers representations do". I will for now just remove the last digit, to not have issues with rounding. Besides, the number looked kind of unesthetic with the 64 digits split onto two lines. Oleg Alexandrov
What I meant was if we'd rounded it, our 30th digit wouldn't have been the same as the 30th digit on, for example, the Wikisource page with 10000 digits. (So changing the number of digits, as you did, was precisely what I had in mind.) Frencheigh 01:49, 25 Apr 2005 (UTC)
Now I've fixed that.--Army1987 15:52, 19 July 2005 (UTC)

Just a technical note, currently the article states

(the first 10-digit prime in e is 7427466391, which surprisingly starts as soon as the 101st digit)

As well as being a subjective statement, it is incorrect to someone who knows about the frequency of 10-digit primes. The prime counting function π which has values listed on Prime number theorem, the number of 10 digit primes is π(1010) - π(109) = 455052511 - 50847534 = 404204977. Out of 9000000000 10-digit numbers, this gives an average density of 404204977/9000000000 which is about 4.5%. So we would expect one in 22 randomly-selected 10-digit strings to be prime. It is surprising the first 10-digit prime starts as late as 101 digits in. I am changng the statement for this reason. Andrew Kepert 03:58, 11 May 2005 (UTC)

You should simplily use π(1010)/1010, since 10-digit strings can begin with zero, too (see 31/2 below). However, that's really surprising. In other irrational numbers, there are 10-digit primes quite soon:
pi = 3.1415926535897932...
γ (Euler-Mascheroni constant) = 0.577215664901532...
21/2 = 1.4142135623730... (incredible but true!)
31/2 = 1,732050807568877293
If my calculations are right, only 0.9% of normal numbers doesn't have a 10-digit prime starting before than 101st digit. Therefore I'm going to change the sentence.--Army1987 11:00, 11 August 2005 (UTC)

## move to Euler's number

I agree with the move. But maybe this needs to be discussed in a wider setting, as this is a prominent article and very much linked to. Oleg Alexandrov 1 July 2005 02:26 (UTC)

Ed Poor has now moved this page to from E (mathematical constant) to Euler's number. Is everyone ok with that? I have no strong feelings either way, but the move has created a lot redirects which should be fixed (especially the double redirects). I don't know as yet if Ed intends to to do that. I'd be willing to help with the redirects, but i want to be assured that we have a consensus for the name change first. Paul August 19:43, August 2, 2005 (UTC)

• No, I'm not happy with the move. It is rarely called Euler's number< I think. Bubba73 20:07, August 2, 2005 (UTC)
• I am weakly opposed to the move. If it were not for the programming feature that makes e (mathematical constant) impossible, I would be strongly opposed to it. Euler's number is ambiguous, with γ, with Euler number and with the topological Euler number, and disputable (with Napier, at least). e is the name of the article subject. Septentrionalis 20:12, 2 August 2005 (UTC)
I just wanted to avoid the accusation that the article was "incorrectly named" - but looks like it's already been moved back and the template re-applied. If you guys like it that way, fine. Let it look stupid. I've rattled enough cages for one week. Uncle Ed 21:05, August 2, 2005 (UTC)

Everyone calls this number e. Calling it anything else is just confusing. Charles Matthews 21:12, 2 August 2005 (UTC)

And the pi page is called that, not Lyudolph's constant or suchlike. Charles Matthews 21:14, 2 August 2005 (UTC)

I agree. And the message should be gotten rid of by fixing the software. Fredrik | talk 21:18, 2 August 2005 (UTC)
No. Making the software case sensitive even for the first letters would create a great many confusion. For example, links in the first word of a sentence would all need to be piped, etc. And there are really very few pages (apart from this one) which would benefit from that.--Army1987 09:41, 15 August 2005 (UTC)
I'm not saying the software should be fixed that way (although I should, because the problem you mentioned is trivially circumvented by having either case automatically redirect to the other, and automatically disambiguate when necessary). Besides, that wouldn't be a matter of changing the software -- the feature is already supported but disabled. Anyway, it would be sufficient to add a meta-directive for the few 100s of articles with this problem, that changes the displayed title. Fredrik | talk 11:53, 26 August 2005 (UTC)
• I'd prefer e (mind the small caps and the italic), but since that's not possible I'd go for Euler's number. If I remember correctly, It was introduced on highschool under this name. --R.Koot 21:27, 2 August 2005 (UTC)
• I think e is by far the better known name. I accept that we can't have an article whose name starts with a lower case letter, but E is still better than Euler's number IMO. DES (talk) 21:39, 2 August 2005 (UTC)
• Oppose move. With Euler-this and Euler-that, it would never occur to me that the thing I know as e should be searched for as Eulers constant.linas 21:44, 2 August 2005 (UTC)
• I'd like "e (number)". It's the simplest and easiest available. Agree with Charles; it's hardly called "Euler's number" as far as I have heard. Dysprosia 22:27, 2 August 2005 (UTC)
• According to Wikipedia:Naming conventions (technical restrictions), there is a workaround for the initial lowercase letter, and it suggests people might be working on a real fix. I prefer e (mathematical constant) as the title, with or without capitalisation, and would suggest base of the natural logarithm as an alternative. Prumpf 22:48, 2 August 2005 (UTC)

The name of the thing is Euler's number whether you know this or not. Finding it will happen by going to e and being disambiguated, this is no argument for using a horrible title. e, or e or e is not its name, but simply the symbol used in formulas and should redirect to the name of the object, which is not the same as the symbol. e represent Euler's number in formulas, sometimes as it is also used for other things. I've never heard of Lyudolph's constant, but if that is the correct name, then that the corresponding pages should be treated similarly.--MarSch 18:10, 14 August 2005 (UTC)

WTF? Almost nobody knows Lyudolph's constant with this name, whereas virtually anybody knows it as pi. I would strongly oppose to such a move.--Army1987 21:37, 14 August 2005 (UTC)

It has been my experience that when mathematicians, physicists, and engineers refer to thsi value, they all call it "e", no one calls it "Euler's number" except in a historical context. You say The name of the thing is Euler's number as an established fact. Pray tell, which international standards body passed on this? Was there a decreee from the God of Newton, the God of Liebnitz, and the God of Cantor? can you site any source that says this is the name of this concept and no other name is valid, or anythign of the sort? DES (talk) 20:01, 14 August 2005 (UTC)

I agree with DESiegel. The name is just e. I've heard the locution "Euler's number" but it's rarely used. Yes, e is sometimes used to mean other things, but then so is π. Move to "e (number)" or "e (mathematical constant)", and make the others redirects. --Trovatore 21:15, 14 August 2005 (UTC)

I can't see any reason for this move. If you hate the technical limitation template for some mysterious reason I can't imagine, this page'd better be called "base of natural logarithms", but IMO "E (mathematical constant)" is OK and should remain.--Army1987 21:37, 14 August 2005 (UTC)

I am strongly opposed to this move. I agree with Charles Matthews's and DESiegel's comments, above. -- Dominus 15:06, 15 August 2005 (UTC)

Remarkable Euler was one of the most prolific mathematicians ever, and has quite enough things named after him. Perhaps it's a difference of cultures — whether of schools or languages or countries I cannot say — but I am unaccustomed to hearing my old friend e referred to as Euler's anything. I am very much accustomed to hearing the number γ referred to as "Euler's constant". Both are numbers and both are constants, so having distinct meanings would be awful. (That's never stopped mathematicians before, but still…) Given the redirect machinery, the decision makes little difference technically; if I'm wrong, please correct me. So I base my decision on other grounds: Calling the page "Euler's number" seems NPOV in light of this discussion, since e may or may not go by that name, depending upon who you ask; therefore the page must remain at "e (mathematical constant)". KSmrq 12:46, 2005 August 16 (UTC)

I assume KSmrq means to say above that Calling the page "Euler's number" seems" POV. — Paul August 17:11, August 16, 2005 (UTC)
I have heard this called Euler's number, but only very rarely. I'd stick with e (mathematical constant).---CH (talk) 22:17, 22 August 2005 (UTC)
Try this link [1] for some sources. On top of that this very article starts by saying "Euler's number (or Napier's constant) ...", it doesn't go "e is ...". It only goes to show. What about Euler's constant? Is that at γ (mathematical constant)? ... --MarSch 10:18, 26 August 2005 (UTC)
I'm also sure this thing is called Euler's number, so could it just be a Dutch thing? --R.Koot 11:19, 26 August 2005 (UTC)
It could be an education thingy. --MarSch 11:29, 26 August 2005 (UTC)
It started "Euler's number..." only because Ed Poor edited it that way after the article was moved. I've changed it back. For the record, I'd prefer this to be at e (number). Fredrik | talk 11:45, 26 August 2005 (UTC)

## Definition of e

${\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.}$

I don't get how that would equal e. I'm trying to think about it... if n tends to infinity, 1 / n tends to zero... which leaves just 1^n, but surely that'll just equal 1? I don't get it...

I prefer the definition of e via the integral of 1/x. It makes much more sense. Deskana 11:17, 27 August 2005 (UTC)

Well, taking the limit of a sequence is a much more fundamental operation than integration. Kprateek88 14:04, 7 September 2006 (UTC)
You get 1 if you "just insert" infinity, but that's not what a limit (mathematics) means. It means that you let n approach infinity, and consider what happens when it gets arbitrarily large. Try it for increasing n:
(1+1/1)1 = 2
(1+1/2)2 = 2.25
(1+1/3)3 = 2.37...
(1+1/4)4 = 2.44...
(1+1/5)5 = 2.49...
(1+1/6)6 = 2.52...
This sequence gives results arbitrarily close to e. Fredrik | talk 17:17, 27 August 2005 (UTC)
I see. That's really cleared it up. Thanks for that! Deskana 09:56, 28 August 2005 (UTC)

I have another question then. Why do those approach e? I understand the integral one, because when you do the integral you get ln(t), putting e and 1 in you get ln(e) - ln(1) which of course equals 1. Why do the others work? Deskana 10:05, 28 August 2005 (UTC)

See Characterizations of the exponential function#Equivalence of the characterizations--Army1987 21:02, 28 August 2005 (UTC)
Informally, you can see that this limit gives e via the defining property of the exponential function exp(x)=ex. That is, this function is its own derivative: exp'(x) = exp(x), so exp'(0)=exp(0)=1. The LHS of this is approx (exp(h)-exp(0))/h=(exp(h)-1)/h (for small h) so for large n=1/h, exp(h)=exp(1/n) is approximately 1+1/n, and so e is approximately (1+1/n)n. Of course, any mathematician (e.g. me) will tell you that this argument is bullshit. (Formal definition of "bullshit" here: the effort required to tighten this into something meaningful is greater than that required to do it properly in the first place.) 8-) HTH -- Andrew Kepert 09:21, 29 August 2005 (UTC)

The first definition of e says "The limit". The second one says "The sum of the infinite series". The second one is as much a limit of a sequence as the first one is. Kprateek88 14:01, 7 September 2006 (UTC)

## Important numbers

I disagree with the statement calling "e" (along with "pi" and "i") some of the "most important" numbers. I won't raise a POV argument (though one conceivably might)... just this:

Tell me which of these numbers is less important: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15...

I defy anyone to make a case that "e" is anywhere near as important as any integer from 0 up to... oh, I don't know... a million? (probably more than that) ...

I don't know how you'd rephrase that--I called them "special numbers", but that's probably not too accurate. If anyone knows the name for these types of numbers, that'd be awesome. Matt Yeager 06:31, 11 November 2005 (UTC)

It is among the most important because it appears in extremely many places. Perhaps a better word would be fundamental. But special makes no sense. - Fredrik | talk 10:46, 11 November 2005 (UTC)
After a bit of research, I found out what the technical term for "weird" numbers like pi and e is... non-algebraic numbers. So that's in there now. Matt Yeager 21:54, 11 November 2005 (UTC)
That is not correct. Non-algebraic numbers are called transcendental numbers. Further more this is just one property of π and e, they are also irrational numbers, non-integer numbers... The imaginary unit is algebraic. They may not be as important as 0 and 1 but much more important than any other integer, which is just a sum of 1's. --R.Koot 00:13, 12 November 2005 (UTC)
If e is transcendental, why not put that in the article like it is for &pi?
Well, I dunno about "algebraic/non-algebraic", I'm no expert mathematician... but what is the last part of your paragraph about? You really truly believe that i is more important than, say, 2? REALLY? Can I quote you on that?
REALLY?
I'm lost for words.
Actually, whether or not i is more important is irrelevant. Whether e is, now that's relevant. And I can't imagine many people agreeing with you on this. (And by the way, I don't think any of your other terms apply, because I doubt that e is as important as, say, the sq. root of 2, which is also an irrational, non-integer number.) Matt Yeager 08:00, 13 November 2005 (UTC)
You may quote him on that, and many mathematicians would agree. i is more important than 2 because 2 is just the sum of 1 with itself, whereas i is the fundamental unit required to define complex numbers. Complex numbers are one of the most important developments of mathematics.
e is among the most important constants in mathematics because:
• "it appears in myriad mathematical contexts involving limits and derivatives" (according to MathWorld, which considers it the most important constant second to π)
• it connects complex numbers with exponentiation and trigonometry, in Euler's formula
• it appears in solutions to many differential equations
• it appears in results from number theory
• it is used to define important special functions such as the Gamma function and error function
I'm sure someone can think of more reasons. Fredrik | talk 11:25, 13 November 2005 (UTC)

Hmm, alright, I see what you're saying. But then again, if all 2 is is (1+1), then all e is is (1/(1-1)! + 1/1! + 1/(1+1)! + 1/(1+1+1)! ... ), right? And wouldn't i be the sq. root of (1-1-1)? In that case, you could make a case that so long as you have all of the mathematical functions at your disposal, then with the aid of just one non-zero number, you could create all the others (except π (to the best of my knowledge), and maybe some other "weird" numbers). All the numbers we've mentioned (including e) can be picked up from, say, 7, by defining 1 as 7/7, defining 2 as 7/7 + 7/7, etc.

The point of all this is that e is just a number that can be derived from other numbers, just like 2, 3, 4, 5, etc. And if we're listing functions, I guarantee that I can come up with more common uses for the number 2 than anyone can for e.

What I'm trying to say is, there needs to be some sort of qualifying term applied to that first statement in the article. Right? Matt Yeager 22:23, 13 November 2005 (UTC)

Just take some undergraduate-level book on calculus and count the instances of all numbers used in it. If you sort that list descendently it would look something like this: 0, 1, e, π, i, 2, 3, 4, ... They are the numbers used most, or in other words the most important. In your 7/7 example you make the classic mistake of confusing notation and meaning. To expres π and e you need infinite sums of a function of 1, so they are not as trivial as integers. --R.Koot 23:34, 13 November 2005 (UTC)
Not being an expert mathematician, I can't defend my "classic mistake of confusing notation and meaning", so I must turn to your logic of the undergraduate calculus book...
Tsk tsk. The statement in question in the article does NOT say "Alongside the number π and the imaginary unit i, e is one of the most important numbers in calculus" (I surely wouldn't dispute that!); it says "Alongside the number π and the imaginary unit i, e is one of the most important numbers in mathematics."
Take some 1st grade book on arithmetic (which surely counts as mathematics!) and count the instances of all numbers used in it. I doubt you'll find any occurances of e, but you certainly will find a lot of occurances of 2. Are you going to suggest that calculus is more important (not more advanced, not more complex, but more important) than arithmetic?
If I'm not mistaken, you have made the classic mistake of confusing high-level mathematics and mathematics. Matt Yeager 00:01, 14 November 2005 (UTC)
Indeed, to make a statement about mathematics in general, you would need to count the occurences in all literature ever published. Because the amount of text on calculus, analysis, probability theory, applied mathematics, ... is far larger than texts on basic arithmetic a calculus book would be a representative sample. --R.Koot 00:12, 14 November 2005 (UTC)
Proof? (It's not just basic arithmetic, either--it's every part from the most basic math in kindergarten up to high school algebra and geometry, at least.) If none is forthcoming, I guess the statement gets deleted as an unsourced, POV piece of information. Matt Yeager 00:17, 14 November 2005 (UTC)
(Oh, and not to mention that none of those forms of mathematics you mentioned would even exist without arithmetic.) Matt Yeager 00:18, 14 November 2005 (UTC)
I do not see how a fact could be POV? Source: [2]. (Do you really believe there all the mathematicians in the world produce less material then the autors of kindergarten and high-school text books?) --R.Koot 00:28, 14 November 2005 (UTC)

Hmm... your logic doesn't work. You're comparing apples to oranges. Of course all the mathematicians in the world produce more than authors of K-12 math books. But the output of all the mathematicians in the world doesn't even scratch the surface of how great the output is of all the people in the world, most of which have no idea what e is but use 2 every day, so 2 wins if we compare general output. If we compare books (that are actually used), and the usage of those books, I believe 2 wins again, though that's debatable. Only when you compare the "material" output of mathematicians (whether anyone cares about it and reads it or not) to the number of books put out by K-12 authors (apples to oranges) does e appear to win.

The idea that your source supports your argument is questionable, too. All the site says is that e is the 2nd-most important constant (oh look, I just used 2). Is a constant the same as a number? Constant states that there really isn't a good definition of what a constant is, so it's hard to tell. The site you provided (when you click on the "constant" link) says that "In this work, the term 'constant' is generally reserved for real nonintegral numbers of interest" [3]. Dubious at best. Matt Yeager 03:57, 14 November 2005 (UTC)

I disagree with removing i from important numbers. I like it more the way it was before this change. I plan to move back. Oleg Alexandrov (talk) 06:49, 14 November 2005 (UTC)
I took it out again--the reason that "i" unfortunately has to be removed is because (as far as I know) "i" is not transcendental. As the version calling "e" one of the most important numbers in mathematics [4] is unsourced (or given a truly dubious source) at best and (more likely) untrue at worst, and the version calling "i" transcendental is false, as well (to the best of my knowledge), there unfortunately is no good alternative. Either "i" goes or the entire sentence does (which'd be quite a shame). Matt Yeager 23:26, 15 November 2005 (UTC)
Nobody said i is transendental, where did you get that from? Oleg Alexandrov (talk) 23:32, 15 November 2005 (UTC)
I think you missed what I was saying there. Because the article NEEDS to say that e is one of the most important transcendental numbers, OR to say nothing at all on e's importance (because the statement that it's one of the most important numbers is unsourced and untrue--see the above discussion), "i" can't be included--you can't say something along the lines of "Alongside pi, e is one of the most important transcendental numbers (oh yeah, and i is important too, though it's not transcendental)." Matt Yeager 23:41, 15 November 2005 (UTC)
Here is a reference for you: Ask Dr. Math FAQ: About Pi: "the five most important numbers in mathematics, 0, 1, e, pi, and i" (my emphasis). - Fredrik | tc 00:35, 16 November 2005 (UTC)
And a source that actually supports your argument finally emerges! However, your source gives no evidence whatsoever for his opinion. On the George W. Bush article, you wouldn't claim "Along with George Washington and Abraham Lincoln, George W. Bush is one of the best presidents ever" just because you found a quote by someone (even someone who totally knows his stuff on politics--say, Dick Cheney) that said as much. Why? Because the quality of a president is an OPINION. (You agree, true?) Please, please, explain why the importance of a number is any less of an opinion. (By the way, now that you've sourced your statement, you COULD just put in the statement, "is often considered to be one of the most important numbers," followed by the link, in the opening statement, and that'd be that.)
Oh yeah, and one last question (which hasn't been answered yet, at least not to my satisfaction). Unless you're dropping your claim that 2 is (relatively) unimportant because it's just 1+1, then why is e any more important, as it's just (1/(1-1)! + 1/1! + 1/(1+1)! + 1/(1+1+1)! etc.)?
If you can give good, complete answers to those two questions (the only two real objections left), I'll drop this whole thing. Alternately, do the whole "is considered to be" thing, and I'll likewise drop it (though I would like to see that last question answered).
(And a quick note on the poll below--that's an arbitrary way of prematurely ending discussion, thrown in without anything resembling "I suggest a poll--any objections?" posted here first, and I'm not going to recognize it. Sorry.) Matt Yeager 05:17, 16 November 2005 (UTC)
1. Saying that GWB is the best president is an opinion, it is a fact that e, π and i are among the most used numbers in mathematics.
2. For the integers you need a finite sum, for e you need an infinite sum of numbers calculated with divisions and factorials. --R.Koot 12:36, 16 November 2005 (UTC)
A poll is a good way of ending an edit war, because it shows what the community wants. Obviously the person who knows he is going to lose such a poll would object to it, so you can't. Not recognizing the outcome would most likely be considered vandalism. Sorry. --R.Koot 12:41, 16 November 2005 (UTC)

Of course to a certain extent the opinion of which numbers are more important is biased, but let me try a brief explanation. 1 is very important because it is the foundation of Peano arithmetic, which allows one to build the natural numbers, and from there the real numbers. The number i allows for the extension to the complex plane. π is a number which fascinated people from antiquity. The number e is at the base of natural logarithms which revolutionized calculations and lead to the invention of the slide rule.

All these numbers, 1, i, π and e produced a seismic shift in a sence in their time in mathematics. Think of the controversy/advances when it was realized that starting with basic axioms of counting you can build up the real numbers, that i is not just a magic imaginary number and that it allows solving any equations, that the circle cannot be squared, and that e does not solve any equation with polynomial coefficients.

Don't forget 0, at least as important as those others, and having a profound effect on mathematics. -lethe talk 20:10, 16 November 2005 (UTC)

By the way, please note that you have been stepping the bounds of a civilized debate lately. Repeatedly reverting this article does not help you make a point. Oleg Alexandrov (talk) 07:17, 16 November 2005 (UTC)

One at a time.

1. R.Koot: So what if it is a fact that e and company are among the most-used numbers in mathematics? http://dictionary.reference.com/search?q=important says nothing whatsoever about frequent use as a criterion for importance.
I don't see (in this context) a way to explain important in a different way than most used? --R.Koot 21:43, 16 November 2005 (UTC)
1. R.Koot: Hmmm... I guess I understand. Thanks!
2. R.Koot: See Wikipedia:Survey guidelines, particularily number 2.
Feel free to add other alternatives. --R.Koot 21:43, 16 November 2005 (UTC)
1. Oleg: I'm assuming the three paragraphs flush against the left margin are all from you. ... Alright, you've clearly stated your case--thank you very much, by the way. However, the one point I have left still remains, and I think that it's a good enough reason to keep the offending sentence out. The opinions of some Wikipedians and one external link do not merit enough to call one number more important than the other. "Considered to be one of the most important numbers", sure, no objections whatsoever. But just plain "one of the most important numbers"? There simply is not enough backing to that claim.

Oh and by the way... I'm sorry if I'm being a pain--I definitely see how my comment last night was a little over the line. Matt Yeager 21:04, 16 November 2005 (UTC)

Matt, your stance looks to me like NPOV pushed to the extreme. Maybe you can quitely drop it, no? Oleg Alexandrov (talk) 22:31, 16 November 2005 (UTC)
As there now is some evidence supporting the claim, I considered it, but really, there's no excuse for the statement being there. I'm going to do what I earlier suggested that one of you guys do and slightly edit the statement. If you all approve, then great, we're done. If not, well, that'll suck. Matt Yeager 02:44, 17 November 2005 (UTC)
I disagree with removing or editing "one of the most important numbers ...". Matt, drop it, really. Oleg Alexandrov (talk) 02:47, 17 November 2005 (UTC)
Since the "importance" of e seems to be a controversial opinion, I think Wikipedia:Neutral point of view policy requires that we *include* quotes showing people's opinions, even if there is no evidence whatsoever for that opinion. I'm going to do that now. (Should I group those quotes into an "Importance" section, or call that section "Quotes" ?
Unfortunately, I only have quotes from mathematicians saying e is the most important number (or at least one of the top 5). I look forward to seeing balancing quotes from people saying some other number is more important.
If we said something like "most frequently used" rather than "important", that would be a fact we could objectively check -- although I suspect lots of numbers are frequently used because of historical accidents, even though they are not fundamentally important -- numbers such as "360", "5 280", "25.4", etc.
--DavidCary 06:03, 17 November 2005 (UTC)
David, I can assure you that there is no controversy over the importance of e. Paul August 13:56, 17 November 2005 (UTC)
It's not a controversial opinion. It's just Matt Yeager against the world. -- Dominus 14:09, 17 November 2005 (UTC)

Works for me. I don't feel that any contradictory quotes are necessary, though. I'm going to try and fix up the formatting, though--the page looks a little off as is. Matt Yeager 06:17, 17 November 2005 (UTC)

I reverted the article to the version before the story of Matt & Co. started. Please, let us finish this discussion here and move on. There is nothing to argue about. Oleg Alexandrov (talk) 16:32, 17 November 2005 (UTC)

Here's a popularity test:

Not to be taken too seriously ;-) - Fredrik | tc 18:26, 17 November 2005 (UTC)

Hm, that's very different from the results for Googlefight: pi "most important number", vs. e "most important number"

Dear Oleg Alexandrov, it's not just this article on e. I don't like the phrase "... is one of the most important ...", in any article.
Apparently the people who wrote Wikipedia:Avoid peacock terms and Wikipedia:Be cautious with compliments and mass attribution feel the same way.
In my clumsy way, I attempted to follow their advice, which seems to be something like Avoid directly saying "X is really, really important, period" (without attribution) or "X is important because X did Y and Z". Instead, show the reader all the important things X did by saying "X did Y and Z", and quote-with-attribution any experts who say "X is really, really important".
Do you, like many others, disagree with that guideline in general? Or does it simply not apply in this particular case?
--DavidCary 06:50, 18 November 2005 (UTC)
You seem to be using a rule for rule's sake. That rule works rather well in controversal subjects like philosophy and the like. Let me state the question in a different way. Please find me a mathematician who would disagree that i, pi and e are the most important numbers in mathematics. Again, you are creating a problem where there is none. I think you will be very helpful in articles about politics, religion, etc. Here you are just taking people's time. Oleg Alexandrov (talk) 16:21, 18 November 2005 (UTC)

## Poll

1. The text should say

Alongside the number π and the imaginary unit i, e is one of the most important numbers in mathematics.

2. The text should say:

Alongside the number π, e is one of the most important transcendental numbers in mathematics.
• Support Kdammers 07:05, 25 January 2006 (UTC)

I don't think the definition of e^x should be in this article, that is why I took it out. It could say that e is related to e^X or something but the topic of e^X is so huge that it should warrant its own article (which it has) and should not bleed in to this one about the NUMBER e.--Hypergeometric2F1[a,b,c,x] 04:47, 22 December 2005 (UTC)

I disagree. The number e is not useful by itself, but rather, as the base of the exponential function. Cutting off e's most important application leaves this article without a good motivation. Oleg Alexandrov (talk) 05:08, 22 December 2005 (UTC)
This is why I said that something like "e is related to e^x see corresponding article" or something like that could be put in there. I think it is misguided to start of the "properties" section of the NUMBER E with a definition of the function e^x. But then again, no laymen one is probably going to look at this anyway so whatever.--Hypergeometric2F1[a,b,c,x] 05:18, 22 December 2005 (UTC)
I agree with Oleg. The properties of ${\displaystyle e^{x}}$ are properties of the number e. Since Hypergeometric seems to have acquiesced I'm going to restore that content. I think it is fine to start the properties section with it, but I'm open to consider other placement of this content. Paul August 06:07, 22 December 2005 (UTC)

Hypergeometric, please do not mark deletions of text as minor. -lethe talk 09:49, 22 December 2005 (UTC)

Hyper, you did a great job in adding new content, and I think we all appreciate your contributions, so thanks very much for that, and I'm sorry I didn't say so sooner. Wikipedia is a collaborative work, and that takes a bit of getting use to, and it is important to figure out how Wikipedia works, if we are all going to get along. For example, marking edits to article pages (no one cares about marking edits on talk pages) as minor or not, as appropriate, is a convenience and a courtesy to your fellow editors. It is a long-standing practice that other editors will expect and appreciate, and as you will see as you gain more experience, makes life a little easier for all of us. You will also have to expect people to edit what you write, (just like you edited what others wrote) and to make comments like the above. Sorry if you think some of us are "nitpicking" but unfortunately since I was born with the curse of a perfectionist personality, picking nits is one of the things I do, and since we are all volunteers, we have the luxury of dealing with either the trivial or the important as fits our inclinations. When you say you see a lot of us "sitting around doing nothing", well we are all free to do that as well, if we like. But If you look at any of the contribution lists — which you can do by going to a users page and clicking on "User contributions" — for any of the editors who commented above (or who contributed to the discussion on the Mathematics project talk page), I think you will find that they have all been doing rather a lot actually (notwithstanding the obligations of the real world ;-). Anyway thanks again for your contributions, I think they have made the article much better. Paul August 13:32, 22 December 2005 (UTC)

If you don't want to comply with my request, that is of course your prerogative. It is simply a courtesy to others who have a vested interest in the article you are editing. If you use it properly, it saves those others time and effort. If you abuse it, it costs us extra time and effort. I would prefer it if I could consider you a trusted editor, whom I don't feel obliged to monitor. I could spend more time editing, and less time policing and tidying. But please, there's no need to impugn my value as a contributor. I didn't mean to offend you with my request. I do not appreciate the implication that I "sit around and do nothing". Just because I'm not editing this article doesn't mean that A. I don't have a stake in what happens to this article, nor does it mean that B. I'm not doing anything at all. I have this week done a fair amount of work on for example axiom of replacement, boundary (topology), affine space, order topology. I also participate in group discussions, monitor my watchlist, and answer questions at WP:RD/Maths, and do general nitpicking and maintainence. But why am I defending my contributions against you? Bollocks. -lethe talk 15:27, 22 December 2005 (UTC)

Thanks for cleaning up my edits whoever did that. I retract my comments as I went a bit out of line. I'm getting my old EDM from my parents' house this Chistmas so I'll be referring to it for some contribs. By the way this is Hypergeometric Im just using my parents comp. -- Hypergeometric2F1[a,b,c,x] (signed by Oleg Alexandrov (talk))

Merry Christmas Hypergeometric, and a Happy New Euler number (whether raised to x or not). Oleg Alexandrov (talk) 18:56, 23 December 2005 (UTC)

## E Approximation

Is that E approximation really right?

Thanks for pointing that out. I fixed it.--Hypergeometric2F1[a,b,c,x] 05:48, 28 December 2005 (UTC)