Talk:Euler's identity

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Mathematics grading:	B+ Class	High Importance	Field: Analysis
More prose needed in "Derivation" section; expand "Generalization" section. Then consider nominating for Good Article status. Tompw 19:22, 7 October 2006 (UTC)

[edit] Gauss' quote

Was he refering to the fact tht it's so beautiful When he said if it's not obvious then you'll never be a high-class amthematician?Wolfmankurd 08:06, 26 May 2006 (UTC)

[edit] Quote

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)

[edit] Feynman

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: e^iθ = 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)

[edit] Sacred geometry

I removed the following paragraph twice:

There has been substantial debate in the philosophy of mathematics on the "real meaning" or "deep meaning" or even sacred geometry reflected by the Identity's relationship of key constants and operations (multiplication, exponentiation, addition, equality). Some assert that it describes cognitive properties of an embodied mind - and advocate a cognitive science of mathematics. At other extremes, some assert it represents rational social consensus of mathematicians, or is simply a fundamental fact of the physical universe, and that algebra itself is a natural consequence of its structure. If so, the formula would be more than simply remarkable - it would be 'divine'.

There has not been any substantial debate about sacred geometry related to this identity in the philosophy of mathematics. If I have missed the relevant literature, please point me to books, articles, conference presentations etc.

have you read Tymoczko, 1998? "The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind." ([Thomas Tymoczko]?)

The way that traditional cultures refer to this "external mathematical reality" is with "sacred geometry" - whether or not mathematicians call it that.

Of course that is the central question of the philosophy of mathematics. I asked specifically about references relating Euler's identity to the concept of "sacred geometry", and I am still waiting. I dispute the claim that "sacred geometry" is a commonly used term; EB doesn't list it at all. AxelBoldt

do a google. you'll find a fair bit. The idea is somewhat contrary to Christian dogma, and occurs in Buddhism and Judaism and certain Hermetic beliefs - sometimes in Christian dogma it is associated with Satan, i.e. the pentagram, etc. One of the major sources of anti-semitism, actually, was the belief that Kabbalic rituals were "Satanic".

I just Googled for < "sacred geometry" Euler >. None of the resulting pages made any connection between the two, I'm afraid. Matthew Woodcraft

Furthermore, the paragraph presents the issue as "some assert..." — "at the other extreme....", as if those two were the only positions on the question, while in fact many other popular positions are left out.

not much room... philosophy of mathematics gave some room to this.

Well, then put a link to that page here and be done with it. AxelBoldt

ok, but the "remarkable" nature of the identity was here before I edited it, and another paragraph to establish that this "remarkable" nature may have some other origins is important.

[edit] Algebra, Popper etc.

Algebra cannot be a natural consequence of this equation, because the equation records a fact about the complex numbers, while in algebra many

there can be no such thing as "a fact about the complex numbers" since the complex numbers, and complex analysis, is a notational convenience to begin with. Your concept of reality is wrong. Fix it. ;-)

You seem to think that the questions of the philosophy of mathematics have been finally answered by your little pet theory; you're wrong. There will never be consensus on those questions. You also don't seem to understand that there can be facts about notational conveniences, and that notational conveniences are part of reality. AxelBoldt

no, there can't be facts about notation conveniences in the Popperian sense, as they are only falsifiable w.r.t. the rest of the notation - at best this is internal consistency. And no, notational conveniences are not part of "reality", they are part of colonialism or a certain paradigm of science at best. And no, again, there is no claim that the questions have been "answered by my little pet theory", as the theory that mathematics arises from the mind is very old, and the theory of mind arising in cognitive science is very deep... so it is *your* "little pet theory" that is under discussion, and its irrelevance in the face of cognitive science and philosophy of mathematics combined. As to your prediction that there will "never be consensus", that could be established merely by killing all over-educated people. To disprove this thesis, of course, you must kill them all yourself. Which brings us to the question of reasonable method...

In other words, you believe that colonialism is not part of reality. Can I quote you on that, 24? AxelBoldt, Sunday, March 31, 2002

other fields, rings and groups are studied which have nothing whatsoever to do with the complex numbers and with Euler's identity. The "divine"

that's foolish. How can fields, rings, and groups be totally independent of the operations of addition, multiplication, exponentation, and especially equality and equivalence? Euler's identity summarizes exactly these issues, and it is the way complex numbers "disappear" in the identity's resolution that makes it interesting. Also, fields rings and groups were more or less an invention of Galois - prior to that, Euler's identity summarized what was known. Suggestion, read cognitive science of mathematics and the references.

Euler's indentity summarizes issues about addition, multiplication, exponentiation and equality of complex numbers. Just because we use the word "addition" in every abelian group doesn't mean that those additions share all properties of complex addition. Euler's identity says precisely nothing about the multiplication in the monster group. It cannot even be interpreted in any way in that context, because there's no exponential map and no addition and no zero element in that context. AxelBoldt

why is *complex addition* the standard meaning? It isn't required for Euler's identity in particular, as the "e to the i pi" isn't a complex value according to Euler's formula but rather is "equal to minus one".

But i is a complex number, and the exponential function e^x is a function defined on the complex plane. Formulas don't just sit there, they are valid in a certain context. The context in which Euler's identity is valid is the complex number field.

The "monster group" is a post-Eulerism that wouldn't exist if not for Galois's theory, which is not necessarily a guide to mathematics pre-Euler. I think the naive terms "plus" or "times" meant less to Euler than Galois... who may well have overly generalized them.

So who cares about the subset of mathematics that was known at Euler's times? It has nothing to do with the discussion. You claim that Euler's identity underlies all of algebra, and the Monster group (and countless other examples) disprove that claim. AxelBoldt

connection is completely out of place and does also not relate to what was said earlier: if Euler's identity were just a social consensus, or a property of human cognition, then it would exactly not be divine. AxelBoldt, Sunday, March 31, 2002

and if it were *neither* of those, it *would* be 'divine' in the same sense as the Planck length, etc,. - something part of the fundamental structure of the universe, unchangeable, etc.

there is no need to use the loaded term "divine" for "unchangeable". Furthermore, again you are simplifying matters: Euler's identity would not have to be a fundamental structure of the universe; Platonists would argue that it necessarily holds in any possible universe. AxelBoldt

fair enough... although a god or "divine" concept can be bound by a universe, and in Plato's time, to an even smaller entity. Although you are definitely splitting hairs here, as the difference between "the universe" and "any possible universe" is a distinction that not all theories of note recognize... why should there be more than one universe? There is value in deliberately loading the term, as it makes a connection to theology, where such matters have been more thoroughly discussed...

"The formula is a consequence of (or, viewed alternatively by some theories in the philosophy of mathematics, assumed in) Euler's formula " -- really? -- Tarquin 10:50 Jan 5, 2003 (UTC)

No, not really, but in the wonderful mind of user:24, which you can also see at work on this very talk page. AxelBoldt 02:05 Jan 8, 2003 (UTC)

Am I not correct in saying that "Euler's Identity" is shortform for "Euler is Identity", whilst "Eulers Identity" would be the correct way of putting it? I remember a very good educational video on Channel 4 (UK) back when I lived over there that explained the eccentricities of the apostrophy - it had a very addictive little tune that I haven't been able to get out of my head in the 15 or so years since I saw it.. but I'm sidetracking: This must be wrong, right?

    --Smári 19:01, 1 Mar 2004 (UTC)

You are incorrect. See: Apostrophe (punctuation) - Bevo 20:23, 1 Mar 2004 (UTC)

Perfect. Then no need to worry. :) --Smári McCarthy 00:44, 2 Mar 2004 (UTC)

The last revision by 63.189.8.249 states: "It was however known long before to Chinese mathematicians." -- This is highly unlikely, as the concept of imaginary numbers was unknown to Chinese mathematicians at that time. (It may be a case of confusion about Chinese remainder theorem.) I have removed it for now, until a source is cited in favor of it. --Autrijus 18:06, 2004 Aug 15 (UTC)

I would have to concur with the removal until a valid source/evidence can be found supporting the statement. - Taxman 18:32, Aug 16, 2004 (UTC)

[edit] Counter-intuitive?

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)

I also consider this remark on the main page (e^\pi vs e^i\pi) quite ... useless, say. The simple insertion of "-" would also change the result, by twice the order of magnitude (speaking of ratios). (Funny... twice the order of magnitude, by putting the square of the exponent....)

I think it would be quite justified to suppress this annotation, — MFH: Talk 19:24, 10 May 2005 (UTC)

I agree. -- Aleph4 16:47, 13 May 2005 (UTC)

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)

I agree that the paragraph especially as currently worded is not that useful. I will delete it. If someone wants to re-insert it they should look at earlier versions which I think are worded better. Paul August ☎ June 30, 2005 18:52 (UTC)

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)

[edit] Pi not constant?

The current version of the article says $π$ 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 $π$ at all. You could as well define $π$ to be the infinite string 3.14159..., and then claim that $π$ is not constant if you use hexadecimal numbers. The string 3.14... does not define $π$ if you interpret it in hexadecimal, and the expression "ratio of circumference..." does not define $π$ if you talk about circles in a non-Euclidean plane.

-- Aleph4 16:47, 13 May 2005 (UTC)

Yes. The area or circumfrence of a circle on a minifold, for instance, is independent of pi, and depends if that manifold is a sphere, for isntance, on the magnitude of the curve of the shpere, or more particularly , to the ratio of of the size of the sphere to the size of the circle.He Who Is 22:40, 24 May 2006 (UTC)

[edit] true by definition?

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 e^z 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

e^{i 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 e^2πi = 1. Prumpf 05:14, 4 August 2005 (UTC)

[edit] perceptions...

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)

[edit] External links

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)

[edit] 2=0??

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)

For more information, you might want to review the following articles:

-- 24.218.218.28 01:48, 22 December 2005 (UTC)

I spotted this and it seems quite interesting. I've looked at all those aarticles, and nothing was explained. Is not the very definition of a logarithm of x to base b the exponent that brings b to x? It seems counter-intuitive that the very concept that gave rise to the idea of a logarithm would collapse anywhere, even over the complexes. He Who Is 02:08, 11 June 2006 (UTC)

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)

[edit] Does this article need an image?

I cooked one up in about 30 seconds. Is it original research, or do I need a source? Is it appropriate at all? Melchoir 02:51, 28 January 2006 (UTC)

I like the image! Maybe the (black) line should consist of two directed lines? That is, the semi-circle might have an arrow head (at -1, pointing down) and the straight-line segment might also have an arrow head (at 0, pointing right). Not sure--just an idea. —The preceding unsigned comment was added by 81.103.219.85 (talk • contribs) 16:12, January 29, 2006 (UTC).

I like the image too, but it is a little by cryptic for the uninitiated. I think the idea of adding arrows as suggested is good. You might also want to add some context, like coordinates, axes, the unit circle, the origin, etc. -- Metacomet 16:15, 29 January 2006 (UTC)

Ideally, I'd like to keep the diagram being interpretable by aliens from another galaxy. (Yup, I know that's not really constructive.) —The preceding unsigned comment was added by 81.103.219.85 (talk • contribs) 16:33, January 29, 2006 (UTC).

Another brilliant idea from the community. -- Metacomet 16:38, 29 January 2006 (UTC)

My point was to keep the diagram language-independent. This would, for example, exclude numerals. Be sarcastic if you want, but I think that art is preferable that way. And greatly more so here, when dealing with an equation that seems to hold some universal transcendent truth. Think about it. —The preceding unsigned comment was added by 81.103.219.85 (talk • contribs) 19:00, January 29, 2006 (UTC).

Actually, one of the reasons I think this article needs an image is to demonstrate that Euler's identity isn't so "transcendent" after all; it's just telling you that the length of the unit semicircle is pi, which really shouldn't come as a surprise. Anyway, I'll try putting in a bit more context. Melchoir 22:33, 29 January 2006 (UTC)

Woo. Melchoir 22:45, 29 January 2006 (UTC)

The revised image is a big improvement. Well done. -- Metacomet 15:53, 31 January 2006 (UTC)

I think it is still too cryptic. I would suggest adding a line segment going from the origin to the circle, making an acute angle with the horizontal axis, labled with a $\phi \,$ , and its point of intersection on the circle labled $e^{i \phi}.\,$ Paul August ☎ 16:34, 31 January 2006 (UTC)

I think those are good suggestions. It might also make sense to add either a caption to the image, or some descriptive prose to the article that explains the meaning of the diagram. -- Metacomet 17:38, 31 January 2006 (UTC)

I much like the image as it is. Adding an angle like that is extra confusion. Keep it simple. Also, the more complicated image that you propose is already in the article for Euler's formula.

I do think that the identity shows something universal, as explained in the section entitled "Perceptions of the identity". Including Greek letters tends to go against that. —The preceding unsigned comment was added by 81.103.219.85 (talk • contribs) 19:41, January 29, 2006 (UTC).

WP is an encyclopedia, not a book on transcendental philosophy. -- Metacomet 19:52, 31 January 2006 (UTC)

Getting back on topic, I didn't want to label the general point, e^i phi, because it's already been done at Euler's formula. It might be more appropriate to simply include that image in the Derivation section. Melchoir 20:53, 31 January 2006 (UTC)

Come to think of it, that's what I've done. Melchoir 20:55, 31 January 2006 (UTC)

Ok that's fine now. Paul August ☎ 21:28, 31 January 2006 (UTC)

[edit] September

As a math novice, the diagram (the first one) needs some explanation. This article needs to discuss what Euler's identity is about in a large context. Does it have applications? Was it just happened upon? What does all that stuff with the circle mean? -- Reid 03:49, 15 September 2006 (UTC)

I've added a caption. The larger context of the identity is, unfortunately, that it looks pretty. I doubt that Euler ever wrote it in the form that gets so much press. But it's related to the historically important identity

log( - 1) = π i,

which is less photogenic but quite significant because before Euler came up with it, no one knew what the logarithms of negative numbers were. Or, to be more precise, different mathematicians had different answers. I don't have a source in front of me, but I know that one exists... somewhere! Melchoir 04:10, 15 September 2006 (UTC)

The image would perhaps be easier to understand if the axes were labeled. Jclerman 08:08, 15 September 2006 (UTC)

No problem, but can you be a little more specific? Melchoir 17:04, 15 September 2006 (UTC)

Without labels on the horizontal and vertical axes, simple (lay)users might think they are x- and y-axis, while more complex 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)

Okay, I can put in something to that effect in a day or so. Melchoir 03:39, 16 September 2006 (UTC)

Good summary folks - I think that the relevant parts of this discussion should be worked into the text of the article.Reid 01:54, 19 October 2006 (UTC)

[edit] Derivation

The "derivation" given here involves just one logical step, substitution. To expand on the substitution is not only unnecessary, it's a red herring; the real work is being done by Euler's formula. Melchoir 09:13, 30 January 2006 (UTC)

This is true for people who are comfortable with high school mathematics. The previous derivation, though, could be partially understood even by people who can do little more than arithmetic and syntactic substitution. So I prefer the previous derivation: let everyone feel that they have understood a little bit.

In the current version, I've kept all the information; if you don't know the values of cos and sin at pi, they're right there. Melchoir 16:57, 30 January 2006 (UTC)

I'm pretty sure that the earlier version is easier for people without a mathematics background. See too "The Science of Scientific Writing" by Gopen & Swan (it's on the web).

I still don't understand what was wrong with this version. Why do we have to display every formula? The choice of which equations to display and which to leave in-line is an important writing tool; it helps guide the reader. From this we get this. In the current version, displaying everything breaks the continuity of the argument. And why isn't x = π being displayed? It's an equation, isn't it? Melchoir 21:02, 31 January 2006 (UTC)

Metacomet, will you explain why you said "equations should not be in-line with prose" and also why you think x = π is an exception? I most prefer this version of the derivation.

[edit] Perceptions of the identity, recently

We can't claim "Many people find..." without a source.
Crowing the word "fundamental" four times is too much. Even once makes me uneasy.
Even worse, "arguably the most fundamental"?
"the study of logarithms"? Who actually studies logarithms?
"Furthermore, an equation with zero on one side is the most fundamental relation in mathematics"? Please, let's not mistake conventions for truths.

I'm reverting back. Melchoir 09:00, 5 February 2006 (UTC)

1. Come on--does "many people find the moon inspiring" also need a source? Go to any pure math department and ask people. In any case, the quote from Benjamin Peirce gives cited support.

2. I think think that the statements are justified. If you disagree, will you explain why?

3. (Ditto.)

4. Agreed. This was copied from an earlier text.

5. This was also copied from an earlier text. Surely, though, this is more than convention! Perhaps you might argue that equality is the most fundamental relation. Okay, but if you have a zero, then you can almost always cast the equality as ___ = 0, which seems more fundamental.

Daphne A 14:14, 5 February 2006 (UTC)

Thanks for responding! I'll be glad to explain further...

1. Yes, the moon bit also needs a source, and Pierce is not "many people". I'm personally aware that plenty of people find the identity beautiful, but there ought to be some support for that. If you can find a source, it would really improve the article (hint hint). For now, it's best to avoid unsupported hedging.

2,3,4. It used to say "three most fundamental functions in arithmetic". Incrementation is more "fundamental" than any of those, and historically, among the fundamental functions of arithmetic people have listed numeration, squaring, cubing, extracting roots, and halving as examples-- but IIRC, not exponentiation. It used to say "arguably the most fundamental mathematical constants". Why should that list include 1 and i but not −1? Where's infinity? How about the square root of 2? Going to history again, these were important concepts long before e was even discovered. It used to say "fundamental in the study of logarithms"; I think you agree with me here.

5. Finally, it used to say "Furthermore, an equation with zero on one side is the most fundamental relation in mathematics". I don't think there's a most fundamental relation, but if there is, it's probably equality, morphism, implication, or set membership. Equations with zero on one side don't even make sense in mathematical contexts without a zero, and that's a huge chunk of mathematics. You can't do it "almost always"; in fact, it's forbidden whenever you don't have a subtraction operation. For examples, multiplicative groups, the natural numbers, the cardinal numbers, the ordinal numbers...

Okay? Melchoir 19:52, 5 February 2006 (UTC)

I've made some revisions, but I think that there should be more. The description of e, for example, is almost vacuous. Taking your points in turn....

1. I don't agree with you, but I've not changed this in the revision. Peirce would not seem to be speaking for himself alone. And you say that you know people who effectively agree with him (as I do). A quick search turned up a claim by Contance Reid, in From Zero To Infinity, that the identity is "the most famous formula in all mathemetics". (I'm not sure about that; e.g. the formula for the area of a triangle is more widely known.) In any case, I think that the present wording is stronger than (i.e. semantically implies) the "Many people ..." wording; if so, then surely the stronger wording would need a citation! Writing "Many people ..." seems more accurate and, to me, reads better. The only poll that I've seen was done by Physics World, but this only got 120 responses:

http://physicsweb.org/articles/world/17/10/2/1

(Euler's identity seems to have tied for first, but the other first, Maxwell's equations, aren't pure math).

2,3,4. Okay. Your point about increment is intriguing, since the identity does include that. Should this be mentioned?

5. Is there a branch of mathematical analysis that does not have zero?

Daphne A 12:21, 6 February 2006 (UTC)

I like your latest revision a lot better; it retains a certain sense of wonder while keeping its head out of the clouds. Moving to specifics...

1. Please add your sources to the article! If you're worried about formatting and presentation, I can help with that. As for the poll, a sample of 120 is better than a sample of 2; you should put it in. You may doubt Reid, and so do I, but his quote is still relevant; that should go in too.

2,3,4. No, it shouldn't. The statement "Euler's identity does not include X" is not a challenge.

5. You could argue that the Greeks did a kind of analysis without admitting zero. I'm not sure about that, but it sounds right. More importantly, is there any hope of ever producing a reliable source that says "an equation with zero on one side is the most fundamental relation in Y"? I think not. It's your opinion, and that's fine. I don't feel a strong need to convince you otherwise, but it doesn't belong in an encyclopedia article. That sentence is the only bit that still does have its head in the clouds.

I'll edit that last point to make it more realistic. Melchoir 19:46, 6 February 2006 (UTC)

I've removed the explanations of 0 and 1. Everyone knows what these are, and it is incorrect to suggest (as the prior text seemed to do) that 1 is nothing more than a multiplicative unit.

I've also put back in a sentence about equations with zero on one side. I think that there should be something about this, but the wording could likely be improved.

As Melchoir says, the trick is to retain a sense of wonder without getting carried away. I don't find this easy! I'm still not sure that including the Reid quote is a great idea (as per above), but have included it here.
Daphne A 09:37, 9 March 2006 (UTC)

[edit] History

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)

I've added a new section that briefly discusses this; see what you think. TheSeven 22:37, 5 March 2007 (UTC)

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)

[edit] We need a proof

I think a good improvement to the page would be to include a calculus based proof for e^(pi*i) + 1 = 0. Unfortunately, I'm not yet skilled enough to do this myself. - Christopher 22:49, 9 February 2006 (UTC)

I am not sure you can prove this identity with calculus. The identity is in fact a direct consequence of Euler's formula, which states:

$e^{ix} = \cos(x) + i \sin(x) \,$

where

$i = \sqrt{-1}$ is the imaginary unit.

So it is really a matter of whether you can prove Euler's formula. It is not too difficult to prove using the Taylor series expansions for the exponential, sine, and cosine functions, and the fact that

$i^2 = -1 \,$

This proof is actually included in the article for Euler's formula – see Proof.

Once you prove Euler's formula, then Euler's identity follows quite readily. In fact, the derivation is already included in this article.

-- Metacomet 03:26, 10 February 2006 (UTC)

First, I don't like to see $i = \sqrt{-1}$ - the expression $\sqrt{-1}$ can maybe stand for any solution to x²=-1 — yet, where to search this solution before already "having" i ? — but anyway, IF this equation has a solution, it is not unique, thus it cannot be used to define what i is.

Second, to the calculus proof: it depends of what you mean by calculus (series should be part of), and what you mean by "e^x". I you define the latter by the power series (which is imho the best definition) then writing explicitely the first 5 terms clearly shows you how grouping together real and imaginary terms make up the power series of (-)cos(π) and sin(π), thus (this shows a way to write a proof for) the result. — MFH:Talk 16:47, 27 March 2006 (UTC)

Actually, $x = \sqrt{-1}$ technically has no solutions, real or complex, since the radical, $\sqrt{}$ , points to the principal, or positive squared root of the radicand. $i$ does not fit this description, since a positive number is defined as any number x|x>0. i is complex, and thus incomprable to real numbers, making it neither positive nor negative. Therefore, $i = \sqrt{-1}$ should be written $i 2 = - 1$ , as it was before. 71.65.9.68 01:53, 18 May 2006 (UTC)He Who Is 01:54, 18 May 2006 (UTC)

Of course a real definition isn't going to work outside the reals. See Square root#Square roots of negative and complex numbers for the square root of negative one. Melchoir 04:58, 18 May 2006 (UTC)

[edit] Derbyshire

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)

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)

[edit] PNG > SVG

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)

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)

[edit] Estimate of e?

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)

2.71828, from the article or "page" e. Jclerman 06:35, 19 December 2006 (UTC)

[edit] the diagram

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)

How about this? Melchoir 01:33, 23 January 2007 (UTC)

Reminder (from earlier comments):

Jclerman 02:05, 23 January 2007 (UTC)

The image would perhaps be easier to understand if the axes were labeled. Jclerman 08:08, 15 September 2006 (UTC)

No problem, but can you be a little more specific? Melchoir 17:04, 15 September 2006 (UTC)

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)

Okay, I can put in something to that effect in a day or so. Melchoir 03:39, 16 September 2006 (UTC)

Whoops! Yeah, I forgot about that entirely. Sorry! Uh… maybe it's no longer necessary? Melchoir 02:16, 23 January 2007 (UTC)

Many thanks. And I take it the more complex ;) users' guess is correct, so that the second diagram is a more general case of the first? -- Hugh7 23:39, 26 January 2007 (UTC)

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