Talk:Euler's identity/Archive 1

message: Substituted at 06:32, 7 May 2016 (UTC)
thread_title: Assessment comment

message: The first comment on perceptions, referring to 0, is currently not justified,
as the formula is written as ...=-1 and not ...+1=0.
(Only quite implicitely the 0 is present through the definition of "-".) — MFH: Talk 19:30, 10 May 2005 (UTC)
thread_title:  perceptions... 

message: Euler's identity is not "true by definition", because e(i pi) is not "defined" to be -1. 
it just happens that if you define ez as the sum of the infinite series 1 + z + z2/2! + ... , 
and if you define sin(x) as the sum of the infinite series x - x3/3! + ... , and similarly cos(x) as
1 - z2/2! + ... (this is only one possibility of defining sin and cos on the complex numbers; other equivalent
definitions are possible), then   
: ei z = cos(z) + i sin(z) 
is a consequence of that definition, and Euler's identity is again a corollary to that formula.
:Some analysis courses essentially introduce π as half the absolute of the period of the exponential function, which makes Euler's identity something that's easy to prove, if not immediately obvious.  You could say it is by that definition of π that e2πi = 1.  Prumpf 05:14, 4 August 2005 (UTC)
thread_title:  true by definition? 

message: Who said "The thought should console (as it bloody well ought) that e^(pi.i)+1=0?
Can't find it on Google.
--Slashme 14:27, 21 November 2005 (UTC)
thread_title: Quote

message: The current version of the article says \pi is a constant in a world which is Euclidean, or on small scales of non-Euclidean geometry otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences). 
While the "otherwise" part of the sentence is true, it does not talk about \pi at all.  You could as well define \pi to be the infinite string 3.14159..., and then claim that \pi is not constant if you use hexadecimal numbers.  The string 3.14... does not define \pi if you interpret it in hexadecimal, and the expression  "ratio of circumference..." does not define \pi if you talk about circles in a non-Euclidean plane. 
-- Aleph4 16:47, 13 May 2005 (UTC)
thread_title: Pi not constant?

message: I was reformatting the footnotes in this article when I noticed that one of the notes reads simply "Derbyshire".  Could someone who has a clue please make sense of this? --Slashme 11:45, 23 May 2006 (UTC)
replies: text: See the references!  But you are right, there should have been a page number; this is now included. —Daphne A 11:02, 1 June 2006 (UTC)
thread_title:  Derbyshire 

message: I just noticed the replacement of PNGs by SVGs. I was rather surprised that my browser correctly displays these. To me it's a completely unknown format, and I'm not sure if all visitors can visualize this format without problems. Is it a good idea to switch from well established formats by "bleeding edge" latest developments? (Personally, I have nothing against promoting file formats supported by all browsers except for IE... ;-) — MFH:Talk 17:47, 29 May 2006 (UTC)
replies: text: The SVG format doesn't actually appear on your browser; it's translated to PNG by MediaWiki first. Try opening one of the thumbnails in a new window and you'll see what I mean. Melchoir 14:37, 1 June 2006 (UTC)
thread_title:  PNG > SVG 

message: Can we give an estimate of e to several significant digits? At page pi we have an estimate of that number to at least 5 digits... can we have the same here? 4.242.108.238 05:24, 19 December 2006 (UTC)
replies: text: 2.71828, from the article or "page" e. Jclerman 06:35, 19 December 2006 (UTC)
thread_title:  Estimate of e? 

message: When I look at the first, simpler diagram, I feel I can almost understand the equation, which is wonderful. But then I read "Starting at the multiplicative identity z = 1" I go "Hunh?" The equation doesn't have a z in it (I can see how the point on the diagram is 1 (0i)). And I've never seen the word "multiplicative" in my life before. I think I know what it means, but I can't imagine what a "multiplicative identity" might be in that sense. Can someone please clarify? --Hugh7 00:12, 23 January 2007 (UTC)
replies: text: How about this? Melchoir 01:33, 23 January 2007 (UTC)
replies: text: Reminder (from earlier comments):
Jclerman 02:05, 23 January 2007 (UTC)
replies: text: The image would perhaps be easier to understand if the axes were labeled. Jclerman 08:08, 15 September 2006 (UTC)
replies: text: No problem, but can you be a little more specific? Melchoir 17:04, 15 September 2006 (UTC)

text: Without labels on the horizontal and vertical axes, simple (lay)users might think they are x- and y-axis, while more complex (pun iuntended)users might guess they are real- and imaginary-axis. The caption mentions i and z but doesn't relate them to the image. Jclerman 17:24, 15 September 2006 (UTC)
replies: text: Okay, I can put in something to that effect in a day or so. Melchoir 03:39, 16 September 2006 (UTC)
thread_title: the diagram

message: The article says that 
:The number π, which is ubiquitous in trigonometry, Euclidean geometry, and mathematical analysis. 
I think this is not correct.   Euclidean geometry only talks about constructible numbers, doesn't it?  --Aleph4 11:25, 26 April 2007 (UTC)
replies: text: No. Fredrik Johansson 13:23, 26 April 2007 (UTC)

text: It does if you take "Euclidean geometry" to mean "the geometry of Euclidean space", in contrast with "non-Euclidean geometry". On the other hand, \pi is useful in non-Euclidean geometry as well, so "the geometry in Elements" may be a more accurate interpretation... (and in that case, I must apologize for responding too quickly) Fredrik Johansson 23:53, 28 April 2007 (UTC)

text: Euler's identity is a basic result for complex numbers. There's a section on applications of complex numbers in the complex number article. --Zundark 21:11, 7 May 2007 (UTC)
thread_title:  ubiquitous in Euclidean geometry 

message: Can I ask- what is the point in i? Why did anyone bother to make up any of this? Euler's identity, however 'beautiful' it is, doesn't help anyone do anything. Please give a section on applications.
—Preceding unsigned comment added by Reagar (talk • contribs) 
:Euler's identity is a basic result for complex numbers. There's a section on applications of complex numbers in the complex number article. --Zundark 21:11, 7 May 2007 (UTC)
thread_title: Practical Applications

message: The image caption says "Starting at e0 = 1, travelling at the velocity i relative to one's position for the length of time π, and adding 1, one arrives at 0." Is this correct. How can the velocity be imaginary? Vints 06:53, 14 June 2007 (UTC)
replies: text: When you live in the complex plane, your position, velocity, acceleration, etc. are all complex numbers. Melchoir 19:48, 14 June 2007 (UTC)
replies: text: So if r = eiφ, r' = i*eiφ = i*(cos(φ)+i*sin(φ)), and if you change coordinate system this velocity is i relative to your own position? I mistook "one's" to mean 1's. Vints 06:19, 15 June 2007 (UTC)
replies: text: Yeah, I don't know what would be a better wording -- "your" doesn't really work in an encyclopedia. I don't know what you mean by changing coordinate system... Melchoir 18:00, 15 June 2007 (UTC)
replies: text: The velocity r' = i*eiφ is only equal to i when φ=0. For example when φ=π/2 the velocity is -1. So what does "relative to one's position" mean? Vints 08:30, 17 June 2007 (UTC)
replies: text: It just means "multiplied by", if you want it in algebraic terms rather than geometric. Melchoir 18:11, 18 June 2007 (UTC)
thread_title:  Image caption - velocity 

message: I was taught Euler's identity as the simpler e^{i \pi} = -1. Is e^{i \pi} + 1 = 0 really more common? And does anyone else think the equation loses some beauty when it so obviously simplifiable? Foobaz·o< 05:55, 12 July 2007 (UTC)
replies: text: I agree that the less verbose form is preferable, and probably more common. The article used e^{i \pi} = -1 until an anon user changed it on 26 January 2006 without even an edit summary to explain the reasoning. We should just change it back, but there some adjustments that need to be made to the rest of the text to make it consistent. --Zundark 09:05, 12 July 2007 (UTC)
replies: text: I strongly prefer the current form.  That is where much of the beauty comes from: this is discussed in the text at fair length.  Additionally most (all?) of the references use that form.   TheSeven 20:39, 16 July 2007 (UTC)
replies: text: You're right, most google hits for the identity use the same form as the article. It appears my perception of beauty is in the minority. Foobaz·o< 14:53, 17 July 2007 (UTC)
thread_title:  Simplified identity 

message: At the end of The Feynman Lectures on Physics, vol. 1, chapter 22, he says: "We summarize with this, the most remarkable formula in mathematics: eiθ = cos θ + i sin θ.  This is our jewel."  I think the Feynman quote belongs on the Euler's formula page instead.  Feynman also doesn't say in this book that it is remarkable for those reasons.  Not that they aren't true; they just weren't said by Feynman.
: What exactly does Feynman mean by: "We summarize with this, the most remarkable formula in mathematics... This is our jewel."? This has been bugging me for a while. - 06:16, 22 February 2006 (UTC)
replies: text:  I think, this formula links vast areas of mathematics - analysis, calculus, geometry, trigonometry, complex numbers, pi, e, 0, 1 - so deeply understanding it can help you on every occasion. As I study Informatik (something like computer science + information technology), I saw this formula very often in the last years, as for example the whole fourier analysis is based on this, which in turn is heavily used in the fields of engineering, computer graphics, digital audio and many others. --80.136.80.190 14:57, 23 July 2007 (UTC)
thread_title: Feynman

message: I'd like to see a sentence or two describing the mathematical meaning (interpretation) of the equation, i.e., something to answer the question "that's nice, but what does it mean?". Something simple would suffice, e.g., The formula represents a rotation in the complex plane of the unit vector through an angle of π radians (180°). — Loadmaster 18:23, 16 August 2007 (UTC)
thread_title:  Meaning 

message: The article says: 
After proving the identity in a lecture, Benjamin Peirce, a noted nineteenth century mathematician and Harvard professor, said, "It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." 
I have a complex variables text in front of me and it has exact same quote, but it says it was B.O. Peirce's math teacher who said it and not after proving the Euler's identity, but after finding sqrt(exp(pi)) to be the principal value for i^(1/i) i.e. i-th root of i. It's on p.130 of Complex Variables With Applications by A. David Wunsch, 3rd edition ISBN: 9780201756098 ------Added Tue, Apr 15 2008  —Preceding unsigned comment added by 99.225.240.84 (talk) 00:59, 16 April 2008 (UTC)
thread_title: Peirce's quote

message: Is this section really worth having?   TheSeven (talk) 09:10, 21 June 2008 (UTC)
replies: text: Yes, of course. The expansion is used to derive the identity. --AB (talk) 22:28, 29 June 2008 (UTC)
replies: text: Not really; the expansion is used to derive Euler's formula, and Euler's formula is then used to derive the identity (as shown in the article).  My preference is to delete the section.   TheSeven (talk) 14:25, 13 July 2008 (UTC)
replies: text: Yes, all right. --AB (talk) 15:41, 13 July 2008 (UTC)
replies: text: Okay!  No one else commented; so I have deleted the section.   TheSeven (talk) 22:00, 24 July 2008 (UTC)
thread_title:  Section on Taylor Series Expansion 

message: Sbyrnes321's animated graphic at the top of the page shows (1 + i*pi / n ) ^ n converging to -1 as n grows large.  However, the graphic moves too quickly to read the actual real and imaginary values at each iteration.  The attached pdf: <> shows the numeric value of the Complex Result for several increasing values of n.  As n approaches infinity, you can see that the Real Part approaches -1 while the Imaginary Part approches 0, illustrating that e^i*pi = -1. DAnderson (talk) 06:01, 22 September 2008 (UTC)
thread_title:  Illustration of Euler's Identity 

message: theres a worrying result you get with some simple manipulation of this identity:
e^(i.pi) = -1
e^ 2(i.pi) = 1
ln e^2 (i.pi) = ln 1
ln 1 = 0
2.i.pi= 0
 i does not equal 0
  pi does not equal 0
therefore 2=0??
i cannot argue with the way the identity is reached, but surely we should be explaining why these results can come after. —The preceding unsigned comment was added by Edallen399 (talk • contribs) 18:07, December 21, 2005 (UTC).
:\ln(e^x) = x \,\!  for all real x.\,
But
:\ln(e^{2i\pi}) \ne 2i\pi.
Paul August ☎ 01:35, 22 December 2005 (UTC)
replies: text: Yes, this is the trouble you can get into with the definite article. The very definition of a logarithm of x to base b is an exponent that brings b to x; but there's more than one! Melchoir 02:17, 11 June 2006 (UTC)

text: e^{2i\pi}
= cos(2\pi) + isin(2\pi)
= 1 + i0
= 1
Which makes sense because you just rotated -1 by 180 degrees, and is the same as e^0  —Preceding unsigned comment added by 99.253.93.99 (talk) 04:49, 11 June 2009 (UTC)
thread_title:  2=0?? 

message: On the page of Ian Henderson's "proof for the Layman", someone "complains" that this page is not even cited in the discussion. Let's fix this, by remarking that with the power series "formula" for e^x, sin x, cos x, Euler's formula and thus Euler's identity are trivial.
All the geometric preliminaries do not contribute to this, they are only used for sin(pi)=0,
but using for this the geometric definition of sin, while equality with the power series is not shown.
The other external link to a proof (.../jos/...) (which in fact concerns rather Euler's formula and thus should be moved there) deserves about the same critics (with algebraic instead of geometric preliminaries), IMHO, the crucial step being the comparision of the derivatives in zero on the last page (implicitely (via Taylor series) admitting that both sides are analytic). — MFH: Talk 21:25, 10 May 2005 (UTC)
replies: text: I don't think those PhysicsWeb Links work, at least they didn't for me... Guerberj (talk) 21:03, 7 August 2009 (UTC)
thread_title:  External links 

message: The article title refers to e^{i \pi} + 1 = 0 \,\! as an identity.  Is that correct?  According to the Wikipedia article on identity (mathematics), an identity is an "equality that remains true regardless of the values of any variables that appear within it".  And according to the Shorter Oxford English Dictionary [2007], an identity is an "equation which holds for all values of its variables".  Those two definitions are effectively the same.  And since e^{i \pi} + 1 = 0 \,\! has no variables, it would seem to be stretching things to call it an identity.
Nahin's book is entitled "Dr. Euler's Fabulous Formula".  I.e. Nahin refers to 
e^{i \pi} + 1 = 0 \,\! as a formula.  Here is a quote from p.3.
…  e^{i \pi} + 1 = 0 \,\!  is actually not an equation.  An equation (in a single variable) is a mathematical expression of the form f(x)=0, for example, x^2 + x - 2 = 0, which is true only for certain values of the variable, that is, for the solutions of the equation.  For the just cited quadratic equation, for example, f(x) equals zero for the two values of x = -2 and x = 1, only.  There is no x, however, to solve for in e^{i \pi} + 1 = 0 \,\!. So, it isn't an equation.  It isn't an identity either, like Euler's identity e^{i \theta} = cos(\theta) + i cos(\theta) \,\!, where \theta is any angle, not just \pi radians. That's what an identity (in a single variable) is, of course, a statement that is identitically true for any value of the variable. There isn't any variable at all, anywhere, in e^{i \pi} + 1 = 0 \,\!: just five constants. … So, e^{i \pi} + 1 = 0 \,\! isn't an equation and it isn't an identity.  Well, then, what is it?  It is a formula or a theorem.
I think that Nahin is correct in saying that e^{i \pi} + 1 = 0 \,\! is not an identity, but incorrect in saying that it is not an equation.  Certainly, e^{i \pi} + 1 = 0 \,\! fits the Wikipedia definition of equation (mathematics).  It also fits the defintion in the Shorter OED: "a mathematical formula affirming the equivalence of two symbolic or numerical expressions (indicated by the sign =)".
Could Wikipedia call it a formula, as Nahin does?  There already is an Euler's formula; in fact, there are many (List_of_topics_named_after_Leonhard_Euler#Euler.E2.80.94formulas).  Additionally, I think of a formula as connoting a description of something in terms of other things; for example,  V =\frac{4}{3} \pi r^3 describes the volume of a sphere in terms of the radius: this is consistent with the discussion in the article on formula.
So, I suggest that the present article be renamed "Euler's equation".  Indeed, I had always called it an equation, before seeing it called an identity in WP.  I know of no other source that refers to it as an identity.  And some of the references refer to it as an equation, including Keith Devlin.
TheSeven (talk) 12:26, 14 August 2009 (UTC)
replies: text: It's an identity, an equation, and a formula. Nahin's claim about what is an equation is definitely bogus. As far as I'm aware, "Euler's formula", "Euler's identity" or "Euler's equation"  have long been used to refer to either of what Wikipedia calls Euler's formula and Euler's identity; exclusively calling the former "Euler's formula" and the latter "Euler's identity" is a convention that originates on Wikipedia. Fredrik Johansson 15:09, 14 August 2009 (UTC)
thread_title:  Is this an identity, equation, formula, or something else? 

message: I tried to see what this formula would do and got something like this:
eπi+1=0
eπi+e0=0
eπi=-e0
πi=0
π angle 90°=0
0=0
But I wonder if eπi at the top should have been written as -eπi
because e0=0 but not e0=-1  ("or" for that matter eπi=e0
but if we could say -eπi=-1
then this statement would be legal -eπi=-e0
in other words we could start with saying this: -1 + 1 = 0
I am probably doing something terrible wrong here, but I think this correct (but maybe not correct if you look for more dimensions than RE(real)).
Anyway I find the start of the formula a little funky, could exi ever go negative therefor I suspect that for imaginary it should have
as follow: -exi
or even perhaps (1 angle 180°)exi
extra stuff:
e0=1
i=1 angle 90°=0 (in RE)
Could be interesting to hear what you people would say about this :D  —Preceding unsigned comment added by 158.39.125.196 (talk) 19:58, 16 November 2008 (UTC)
replies: text: Lots of good questions here! I don't think I can do justice to them all... I advise you to ask Wikipedia:Reference desk/Mathematics. You'll probably get a good response there. Melchoir (talk) 23:07, 17 November 2008 (UTC)

text: Your transition from eπi=-e0to πi=0seems unwarranted to me. With complex numbers, exp(z) is not a bijection, and exp(z1) = exp(z2) does not imply z1=z2. --Thomas Arelatensis (talk) 15:18, 10 June 2009 (UTC)
thread_title:  -1 + 1 = 0 ? 

message: e^{i \pi} + 1 = 0, \,\!
Rearranging gives :
e^{i \pi}  = -1, \,\!
Square both Sides :
e^{2i \pi}  = 1, \,\!
Take a natural log of both sides :
2i \pi = 0, \,\!
i \pi = 0, \,\!
Substitude into the original equation :
e^{0}  = -1, \,\!
How can this be?
—Preceding unsigned comment added by Pi (talk • contribs) 20 November 2007
:The problem is that log is multivalued for complex numbers. If you use the principal value, then \log(e^{2i\pi})=0. --Zundark (talk) 10:12, 20 November 2007 (UTC)
replies: text: This may just make me sound stupid, but doesn't squaring something generally affect the domain of the result? 174.18.154.104 (talk) 23:52, 16 February 2010 (UTC)
thread_title:  Does it make 1 = -1 ? 

message: Which identity is considered more mathematically beautiful?
:e^{\frac{(i+j+k)}{\sqrt 3}\pi} + 1 = 0, \,\!
:({e^i e^j e^k})^{\pi{\sqrt[2]{3}}^{-1}} + 1 = 0, \,\! (this one doesn't even work, does it?)
:e^{\pi(i+j+k){\sqrt[2]{3}}^{-1}} + 1 = 0, \,\!
:e^{\sqrt[2]{3}^{-1}(i+j+k)\pi} + 1 = 0, \,\!
Leethacker1347 (talk) 17:09, 9 December 2009 (UTC)
replies: text: e^{i \tau} = 1\,\!
Is this easier to extend to quaternions?  — Preceding unsigned comment added by 71.167.69.162 (talk) 16:29, 26 May 2011 (UTC)

text: ({e^i e^j e^k})^{\tau{\sqrt[2]{3}}^{-1}} = 1 \,\! Lanthanum-138 (talk) 03:54, 9 June 2011 (UTC)
thread_title: Quaternions

message: Why is phi used in e^iphi = cos(phi) + isin(phi)?  When I first read this, I though it meant the golden ratio (1.618...).  Why not just use x, instead of a Greek letter?  Wouldn't that just be easier, and simpler to understand?  Or is there some significance to phi? My 2 Cents' Worth (talk) 15:57, 19 May 2010 (UTC)
replies: text: In some contexts it's the usual convention to write angles using Greek letters, and phi is among the popular choices. It has no special significance.—Emil J. 16:52, 19 May 2010 (UTC)

text: No. Actually I'm rather surprised our article on polar coordinate systems uses theta and not phi - the article on cylindrical coordinates uses phi. Maybe the preference for either is regional. Huon (talk) 20:27, 19 May 2010 (UTC)
replies: text: It is still true for \phi = \frac{1+\sqrt{5}}{2} \,\!, isn't it? It's just a preference, nothing to worry about. Lanthanum-138 (talk) 03:56, 9 June 2011 (UTC)
thread_title:  Why Phi? 

message: Better? Worse? Are there improvements that could be made to the image? To the caption? This is my first Wiki image, so I may have messed up some technical aspects :-) --Steve (talk) 17:14, 5 May 2008 (UTC)
replies: text: It's a promising idea; I think there's a potential for beauty that isn't quite met. Have you considered not animating it? Melchoir (talk) 18:31, 5 May 2008 (UTC)

text: Cool image. Foobaz·o< 21:52, 5 May 2008 (UTC)
replies: text: I like it - it does explain/show something and one can actually learn from it. Stotr (talk) 15:48, 6 May 2008 (UTC)
replies: text: OK, I've tentatively replaced it. Melchoir, I can't think of how to make it more beautiful and/or clear, but then again, my background is science, not graphic design. If you have more specific advice, I'd be happy to try it that way. Or, I can post the Mathematica code (it's just a couple lines) and you can tinker yourself, if you'd prefer. :-P --Steve (talk) 16:03, 6 May 2008 (UTC)
replies: text: Sure, I'll give it a shot some time. I'm thinking the animation can be avoided by showing many iterations at once, and a still image could be an svg, which looks smoother. The axes could be less busy, the "N =" could be better incorporated spatially, and color could be exploited a little more. Melchoir (talk) 16:34, 6 May 2008 (UTC)
replies: text: Wow, those are all really good ideas. I don't know how to do many of them in Mathematica though. (You're welcome to use a different program, or maybe you're better at Mathematica then me.) In case it helps, here's the Mathematica program I used: . I'll leave it posted there for the foreseeable future. You (or anyone else) are welcome to try it later, or never. Good luck! :-) --Steve (talk) 17:20, 6 May 2008 (UTC)
replies: text: Thanks! I haven't ever played with Mathematica, no. The SVGs I've created were all cooked up in Adobe Illustrator, although any decent vector editor would do. For the diagrams with any nontrivial geometry, such as yours here, I usually write a C program to output Postscript, which is then imported into Illustrator for finishing touches. This workflow is probably heresy to people who actually understand Postscript, and it's not terribly efficient, but it works! Melchoir (talk) 10:19, 10 May 2008 (UTC)

text:  I don't know the animation, but trying it for a full 2 pi turn is kinda beautiful. Drf5n (talk) 21:59, 8 August 2011 (UTC)
thread_title:  Proposed replacement diagram 

message: I may only be speaking for myself, but I would find that  \lim_{x \to \infty} e^{-x} = 0  is more counter-intuitive than e^{\pi} = 23.14069... \,\! when considering Euler's Identity.
* Erm. Why? As x grows large, you will have the reciprocal of a very large number... why is it a stretch to see this as approaching zero? DocSigma 05:02, 6 Feb 2005 (UTC)
replies: text: I agree.  -- Aleph4 16:47, 13 May 2005 (UTC)

text: I was struck by the non sequitur about "counter-intuitive" myself. There is no viewpoint so naive as to make this property remarkable. I support removing the comment. 66.214.64.122 23:23, 29 May 2005 (UTC)

text: I agree current wording is worse, but even the original wording is probably original research. All this "naive" non sense is from people who have a much more advanced understanding of the mathematics than those the comment referred to, and of course to them it is not counter intuitive. However, consistently when teaching students that do not have a firm grasp of imaginary numbers, they express surprise at the result. Since I don't have a published source to draw that from, I do admit I added it when I was not as stringent about the no original research bit. So go ahead and remove it. - Taxman Talk June 30, 2005 19:07 (UTC)
I finally deleted this paragraph. It slipped my mind till now. Paul August ☎ 06:22, July 15, 2005 (UTC)
He was talking about the strangeness regarding how  \lim_{x \to \infty} e^{-x} = 0  converges to zero, but Euler's equation suggests a cyclical result. This becomes obvious when you see Euler's equation graphed on the axes of Real numbers, Complex numbers, and x.  — Preceding unsigned comment added by 72.222.201.45 (talk) 16:20, 9 March 2012 (UTC)
thread_title:  Counter-intuitive? 

message: The current order of topics is as follows: Intro, 1 Mathematical beauty, 2 Explanation, 3 Generalizations, 4 Attribution, other... I suggest a new order of these topics to make it similar to other mathematics articles, i.e: Intro, 1 Explanation, 2 Generalizations, 3 Attribution, 4 Mathematical beauty, other... This is still a mathematics article and as such, it must be prioritized for comprehension of the meaning and not the popular culture behind it. Hamsterlopithecus (talk) 15:06, 25 September 2012 (UTC)
thread_title:  Order of the topics in the article 

message: I feel "History" would be a more helpful and appropriate section title. 86.173.146.3 (talk) 15:56, 19 February 2014 (UTC)
thread_title:  "Attribution" vs. "History"  

message: The article currently claims that the identity is from the Introductio. I was curious about how Euler wrote it, so I found a translation in the library, and I don't see the identity; Euler seems more interested with the general formula. What's the deal? Melchoir 20:11, 6 February 2006 (UTC)
replies: text: I've added a new section that briefly discusses this; see what you think.  TheSeven 22:37, 5 March 2007 (UTC)
replies: text: Thanks! That's a great column. It's a shame that we don't have more information, but what you wrote is a lot better than the alternative. Melchoir 23:03, 5 March 2007 (UTC)
thread_title:  History 

message: :(-1)^{x} = e^{(i \pi)x}  — Preceding unsigned comment added by Becker-Sievert (talk • contribs) 07:13, 14 July 2014 (UTC)
thread_title:  x is a number

message: Of note - a crop circle appearing 22 May 2010 appears to contain an ascii coded reference to Euler's Identity - see  http://img64.imageshack.us/img64/3570/img3736o.jpg
Starting radially from center along line marked with arrow,  each anticlockwise "branch" is a 0 and each clockwise "branch" is a 1  
Read in order they become 
01100101
01011110
00101000
01101000
01101001
00101001
01110000
01101001
00101001
00110001
00111101
00110000
and converted in ASCII e^(hi)pi)1=0  
Note:  this makes no suppositions or assumptions about the origins of the circles themselves and is not intended to be a discussion about who's making them - only the factual statement that Euler's Identity appears to have been encoded in one of them.  —Preceding unsigned comment added by 68.98.122.107 (talk) 02:52, 25 May 2010 (UTC)
replies: text: Not really if you've had millions of years more evolution allowing you to warp space for travel over astronomical distances, monitor and/or modify a promising species for mere thousands of years, and plug into the earliest communications systems ;-)  69.144.93.138 (talk) 06:39, 22 October 2014 (UTC)
thread_title:  Made an appearance in a crop circle?   

message: "... where the values of the trigonometric functions sine and cosine are given in radians."
This is poorly worded. The values of trigonometric functions are ratios and have no units. I believe what was meant is that the value of x is given in radians. 2601:588:4200:1C59:A840:AA16:3B89:C50B (talk) 20:28, 20 August 2015 (UTC)
replies: text: how about "inputs" instead? --JBL (talk) 20:34, 20 August 2015 (UTC)
thread_title:  A Problem With the Wording 

message: The article contains the following sentence: "Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics". The sentence was removed, with this explanation: "Remove editorialization". Editorialization is defined as the act of expressing an opinion as if it were an objective statement.
The sentence comprises two main parts. The first part is this: "the equation is given in the form of an expression set equal to zero". The first part is clearly an objective statement. The second part is this: "which is common practice in several areas of mathematics". The second part is not as clearly an objective statement; it is, though, something that anyone familiar with mathematical analysis would surely agree with. As an illustration, in Maple, an equation of the form "f(x)=0" can be entered as just "f(x)", i.e. the "=0" is implicit.
I believe that the sentence adds to the appreciation of the sense of wonder that Euler's identity gives. Moreover, some version of the sentence has been in the article since February 2006. Hence, I have restored the sentence.
Lastly, the edit that removed the sentence was marked as minor. The edit is not minor, and should not have been marked as such.
TheSeven (talk) 18:08, 1 October 2016 (UTC)
replies: text: While I can't speak for the editor who removed the sentences, I have a feeling that the truly objectionable one was the second sentence that had been recently added. This sentence, the first one of the pair, was kinda ok on its own, but juxtaposed with the second sentence it did seem to be out of place and its removal made sense as a "package deal".--Bill Cherowitzo  (talk) 18:30, 1 October 2016 (UTC)
replies: text:  , thanks for starting a discussion.  If you don't mind, I'm going to skip responding to the stuff about the edit summary and m and focus instead on the substantive question.  On this, Bill Cherowitzo correctly summarizes my thoughts.  To expand: the sentence in question sort of looks harmless, but it's of the same type as the (egregiously bad) sentence that was just added: it is an unsourced and probably unsourceable (note I do not say "false" or "wrong") opinion, which manages to both be of unclear relevance (what does that property have to do with beauty?) and partially redundant (it is just restating the presence of 0 with some additional unneeded specificity).  If there were some connection to be made between this not-obviously-interesting convention and mathematical beauty, that would be one thing; but there is no such connection made in the article, nor do I believe that any such connection relying on reliable sources could actually be made.  (Incidentally, the older version of this sentence, from 2006, is of similar quality to the other sentence I removed; having been around for a long time is not a sign of value.)  I believe that the article is better with the sentence removed. --JBL (talk) 19:06, 1 October 2016 (UTC)

text: Mathematical beauty, like any other kind of beauty, lies primarily in the eye of the beholder. This will make it hard to source the sentiment expressed in this sentence. The very thing that adds to the beauty of this expression for you, subtracts from it for me. For me there must be some element of "exceptionality" present to qualify for beautifulness. Something that is beautiful should stand out from the commonplace and setting an equation to zero is just too common a manipulation to impart any beauty (for me at least). --Bill Cherowitzo  (talk) 18:10, 4 October 2016 (UTC)
thread_title:  An equation set equal to zero 

message: Obviously "an example of mathematical beauty" fails to adequately summarize the body.  Various editors (mostly a banned user, but occasionally others) have removed the trailing superlative phrase.  I hereby open discussion on alternative wordings that capture the significance (as reported in reliable sources) in a way that will avoid the whining.  Suggestions?  --JBL (talk) 21:04, 1 February 2019 (UTC)
replies: text: JBL, as I noted in my edit summary: "perhaps a supreme example" is a judgment, or at best a speculation, that should not be expressed in the encyclopedic voice, because it can neither be proven nor disproved (hence is unverifiable). See WP:V and WP:NPOV. You stated that it had been extensively discussed on the Talk page, but I have not found that discussion. Perhaps you could point it out, as the one discussion I did see about this didn't appear to have achieved any consensus. (And yes, you were correct that I inadvertently included an edit from mid-January in my count when I stated earlier that you were at 3RR  but you are now.) General Ization Talk   21:14, 1 February 2019 (UTC)

text: As for suggestion of an alternative wording, removing the superlative, (as in "It is considered to be an example of mathematical beauty, as it shows a profound connection between the most fundamental numbers in mathematics") seems perfectly reasonable.  Any superlative phrase will represent, apparently, your opinion, unless you can cite a source that says that it is a "supreme" or otherwise superlative example of something, i.e., that no better example is possible.  You haven't done so to date. General Ization Talk   21:18, 1 February 2019 (UTC)
replies: text: I agree with JBL, more needs to be said in the lead to convey the content of the article. However, I am unhappy with the phrase "supreme example" as it sounds rather sophomoric to me. I had put in a replacement using the term "exemplar" but alas this got wiped out by a large revert. I will propose it again, "It is considered to be an exemplar of mathematical beauty as it shows a profound connection ..." This usage is well justified by the citations given in the article, which includes a poll of mathematicians that voted it "The most beautiful equation in mathematics". --Bill Cherowitzo  (talk) 06:18, 2 February 2019 (UTC)
replies: text: +1 for "exemplar".  It expresses the idea of "perhaps supreme", which is supported by the citations, concisely and without sounding like the article is overly enamored with the equation. Danstronger (talk) 21:26, 2 February 2019 (UTC)
replies: text:  This sounds good to me, too.  (I hope I was not the one who wiped it out!)  --JBL (talk) 21:48, 2 February 2019 (UTC)
thread_title:  "perhaps a supreme example"