# Talk:Euler's formula

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## e to the pi i

Somewhere in the introduction the familiar (and amazing!) expression e**pi*i = -1 needs to be shown. Casey (talk) 22:51, 25 September 2013 (UTC)

## A proof based on De Moivres formula

Eulers formula, for real x, may be obtained from De Moivres formula, for integer n,

${\displaystyle (\cos _{\theta }+i\sin _{\theta })^{n}=\cos _{n\theta }+i\sin _{n\theta }}$

Let ${\displaystyle \theta ={\frac {x}{n}}}$, and take the limit as n tends to infinity;

${\displaystyle \lim _{n\to \infty }((\cos _{\frac {x}{n}}+i\sin _{\frac {x}{n}})^{n}=\cos _{\frac {nx}{n}}+i\sin _{\frac {nx}{n}})}$

Using the power series expansions,

${\displaystyle \cos _{\frac {x}{n}}=1+{\frac {0}{n}}+{\frac {?}{n^{2}}}+...}$
${\displaystyle \sin _{\frac {x}{n}}=0+{\frac {x}{n}}+{\frac {?}{n^{2}}}+...}$

gives,

${\displaystyle \cos _{\frac {x}{n}}+i\sin _{\frac {x}{n}}=1+{\frac {ix}{n}}+{\frac {?}{n^{2}}}+...}$

In the limit of the binomial expansion of ${\displaystyle (\cos _{\frac {x}{n}}+i\sin _{\frac {x}{n}})^{n}}$, it can be shown that the sum of all the terms arising from ${\displaystyle {\frac {?}{n^{2}}}}$ and higher power terms will go to zero as n goes to infinity. So,

${\displaystyle \lim _{n\to \infty }((\cos _{\frac {x}{n}}+i\sin _{\frac {x}{n}})^{n})=\lim _{n\to \infty }((1+{\frac {ix}{n}})^{n})=\cos _{x}+i\sin _{x}}$

Two pathes to the result are possible from this point;

Complex numbers are a field. Direct expansion

Consider the function,

${\displaystyle f(y)=\lim _{n\to \infty }(1+{\frac {y}{n}})^{n}}$

where y is a real number. Applying the change of limit variable ${\displaystyle my=n}$ gives,

${\displaystyle f(y)=\lim _{m\to \infty }(1+{\frac {1}{m}})^{my}=e^{y}}$

${\displaystyle e^{y}}$ may be represented as a convergent power series,

${\displaystyle e^{y}=\sum _{k=0}^{\infty }{\frac {{y}^{k}}{k!}}}$

The complex numbers form a field, so a power series may be constructed from complex numbers. So ${\displaystyle e^{z}}$ may be defined for complex z. f may also be expressed as a power series, which can be generalized to complex numbers. Therefore the coefficients must be equal and the expressions equal for all complex z. So,

${\displaystyle e^{z}=f(z)=\lim _{n\to \infty }(1+{\frac {z}{n}})^{n}}$

Expanding the limit using the binomial expansion,

${\displaystyle (1+{\frac {z}{n}})^{n}=\sum _{k=0}^{n}{\frac {n!}{k!(n-k)!}}({\frac {z}{n}})^{k}}$

Which may be re-arranged as,

${\displaystyle (1+{\frac {z}{n}})^{n}=\sum _{k=0}^{n}{\frac {z^{k}}{k!}}{\frac {n!}{n^{k}(n-k)!}}}$
${\displaystyle =\sum _{k=0}^{n}{\frac {z^{k}}{k!}}+\sum _{k=0}^{n}{\frac {z^{k}}{k!}}({\frac {n!}{n^{k}(n-k)!}}-1)}$

The limit of the second term can be shown to tend to zero as n goes to infinity,

${\displaystyle \lim _{n\to \infty }(\sum _{k=0}^{n}{\frac {{z}^{k}}{k!}}({\frac {n!}{n^{k}(n-k)!}}-1))=0}$

so,

${\displaystyle \lim _{n\to \infty }(1+{\frac {z}{n}})^{n}=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}=e^{z}}$

so where ${\displaystyle z=ix}$

${\displaystyle \lim _{n\to \infty }(1+{\frac {ix}{n}})^{n}=e^{ix}=\cos _{x}+i\sin _{x}}$

Thepigdog (talk) 12:04, 2 March 2014 (UTC)

What you're proposing is almost the same as the limit definition based proof in the article. In that proof there is a parenthetical "(The proof of this is based on the rules of trigonometry and complex-number algebra.[6])" which is actually referring to the arguments that (1) Multiplying by (cos theta + i sin theta) corresponds to a rotation by angle theta in the complex plane; (2) In this context we can correctly approximate (1 + i*a) with (cos a + i sin a) for small a. You might say, why not actually make those arguments, like you're proposing here? Why skip them via the parenthetical? It's because we want brevity and clarity, and also we want to make it accessible to people who are unfamiliar with limits and calculus, and also because the animation helps fill in those gaps. (In my opinion anyway.) --Steve (talk) 05:08, 5 March 2014 (UTC)
Yes I am not disagreeing with you. I like the limit definition based proof section. I just think that this is the most intuitive fundamental proof. It actually describes what is happening. The other proofs (calculus and taylor series) pull a rabbit out of the hat and say "look its equal". But I am not at all happy with the above yet. Actually I am still fiddling with trying to make it more rigorous and more understandable. Maybe it will come to nothing. Hope this is OK.
Thepigdog (talk) 05:30, 5 March 2014 (UTC)
The proof is kind of the way I would like. To me the limit definition section is not really a proof. But its a very nice demonstration. I suggest it should go earlier, and be replaced by the above proof. Maybe. What do you think?
Thepigdog (talk) 11:43, 5 March 2014 (UTC)
Why is the limit definition section "not really a proof"? Because it leaves out steps? (Yours leaves out steps too...)
I don't see how your proof is "intuitive", I think most non-expert readers will see it as a series of random formal manipulations that somehow magically arrives at the right answer. (Especially readers who don't have much experience with or understanding of limits.) But maybe you're going to add more prose? --Steve (talk) 13:27, 5 March 2014 (UTC)
For example, cos x + i sin x = (cos x/n + i sin x/n)^n. In the limit of large n, cos x/n approaches 1 and i sin x/n goes to zero, so cos x/n + i sin x/n approaches 1. Therefore cos x + i sin x = 1^n = 1. Oops!
My proof is wrong, but what's wrong with it? It gives the wrong result, but that's just the symptom, not the cause. I'm sure you can answer the question, but could a typical reader read through your proof and understand why it is correct to replace sin d with d for small d, but it's incorrect to replace sin d with 0 for small d? After all, both limits are correct.
There's nothing wrong with leaving out essential details if you say that you're leaving out essential details, but if you present a certain step as a simple logical step ("replace sin d by d for small d") when it's actually fraught with danger and only works in certain circumstances, it's a problem.
Here's an example of "intuitive". The De Moivres formula (cos x + i sin x)^n = cos nx + i sin nx is not intrinsically "intuitive" -- when someone looks at the formula they won't "see" anything but a mess of symbols, unless they know what to look for. Whereas an expert might see it as quite intuitive: "The left hand side is rotating the complex plane by x radians, n times, and the right hand side is rotating the complex plane by nx radians all at once. Obviously those transforrmations are the same." In other words the De Moivre formula is a distraction from a more important and also more intuitive fact: "Multiplying by cos x + i sin x corresponds to rotating a point in the complex plane by x radians." Well it's not intuitive at first, nothing is, but it is a better building block for constructing the mental model in which Euler's formula is intuitive, better than writing down De Moivres formula as a starting point.
Anyway, it's good that you have built mental models in which statements like "sin d approaches d for small d" are intuitive, but don't forget that there will be many readers for whom facts like these statements are not intuitive... --Steve (talk) 16:03, 5 March 2014 (UTC)
Firstly I must apologise for my approach. I started this as a bit of a doodle to see if I could construct something that was the way I could understand it. I never expected to change the actual text.
I will start by saying that I really like what the author is trying to do in the limit definition section. I think in an encyclopedia its just as important to be understandable as accurate.
Yes its not wrong to leave out stuff that is too technical.
However for me I wanted a series of logical steps that took me from de moivras to e raised to the power of i x, along the way explaining what raising to the power of a complex number is. I am thinking more about how my self as back when I was a 14 year old would have understood it.
That 14 year old would have liked your section but found it frustrating also.
Yes there are still gaps in my version. Bits where the actual mechanics are not completely explained. Maybe that can be improved.
For me personally I see the limit definition as a very valuable explanation before the proof.
You also make the point that we dont really need to start with de moivras formula, which is true. in fact in an earlier version I didnt. But as it evolved I found that seemed to be the best starting point, unless I wanted to explain the whole complex number / trigonometry / field relationship.
Yes the replacing sin d with d step bothers me. I wish I had a better way of explaining that.
Well I am not going to change anything in the article unless I have agreement. I don't feel qualified to do that.
Thepigdog (talk) 21:42, 5 March 2014 (UTC)
I am really not happy with the "d = sin d" step. Its not really a logical step, and it doesnt really explain why the substitution is valid. I've never been really happy with it. I will see what can be done with it.
Thepigdog (talk) 00:38, 6 March 2014 (UTC)
I have sorted out that step (I think).
My version of the proof assumes a knowledge of limits, while yours explains the limit idea as part of the proof.
Your "imaginary part" graph is also very intuitive. To me they are too alternate approaches to essentially the same proof, but aimed at different audiences. I know you don't want any change ;). Lets consider the matter.
Thepigdog (talk) 05:03, 6 March 2014 (UTC)
Reading the section "Using the limit definition" I felt there was something missing. I suggest the above as a sub section with a title "In detail". But I don't feel that strongly about it. The argument for it is that it provides a step by step proof. On the negative side it adds to the length of the article.
Thepigdog (talk) 05:57, 6 March 2014 (UTC)
For one thing, "e^z is by definition equal to the limit of (1+z/n)^n as n --> infinity" is a legitimate definition of e^z, described as a valid definition in the article and in many references. It is just as valid and common a definition as the definition you prefer, "e^z is by definition equal to 1 + z + z^2/2 + z^3/6 + ...". So you can delete the whole half of the proof where you go from one to the other.
Well, maybe it's worth having a proof somewhere on wikipedia that the limit of (1+z/n)^n is equal to 1 + z + z^2/2 + .... In this article, it would go in the definitions section, not the proof section, as it is a proof that the two definitions offered here are actually equivalent. Or it could be in the article exponential function (and this article could just have a link), or somewhere else, I don't know.
Once you split that part off, you will see that your proof is getting more and more similar to the "limit definition" section. Actually the main difference is that that section goes forward from (1+ix/n)^n to (cos x + i sin x), while your proof goes backwards from (cos x + i sin x) to (1+ix/n)^n. Also, you are giving more detail, especially the step "(The proof of this is based on the rules of trigonometry and complex-number algebra.[6])".
Actually there used to be more detail in the proof -- see [1]. It was shortened on request to improve the readability of the article as a whole, and because interested readers can look at the cited reference to get details...
Oh, and sorry if I'm biased :-D --Steve (talk) 15:14, 6 March 2014 (UTC)
For the general puplic e^z for complex z is the natural extension you get if you extend e^y to complex numbers. Both powers series and the limit fit this requirement, and there equal. Yes this could be covered under exp but its not. I suspect if I tried to add it the exp article the mathematical mafia would stop me ;)
It seems to me that bit is of interest particularly for eulers formula. Logically you are right it should go in the exp article but practically I would want it in this article. I would be happy if it was in a separate section down the bottom of this article.
The rest of the proof is simple and in some ways easier to understand than the existing limit proof. Not that I don't like the existing limit proof, its just a different style for a different audience.
Yes that old version uses the absolute value/arg approach. Thats nice, but for me not as direct a method. Actually I prefer it to the current version. Now I see where your current version comes from. But its a different approach from the above. Its a very nice version of the absolute value, arg approach, better than other versions I have seen.
I think most people dont like to follow cited references. You never know what you might find there. It might be too detailed and assume a level of knowledge you don't have.
Cant we have both versions? And the old version too. Different levels of detail for different people.
Personally I want,
• An introductory section with your cut down proof similar to the current Using the limit definition
• In the proof section;
• The old absolute value arg proof .
• The above de moivre based proof (as above without the table).
• At the bottom of the document before references
• Proof the limit verion and taylor series are equal and are the most natural extension of exp for real to complex numbers (the table in the above).
Aaaaaaaggghhh, I love having proofs in articles, just not long detailed ones. :-(
I can see now the mathematical mafia will not allow a 4th proof, even though its the simplest :-(
Thepigdog (talk) 22:18, 6 March 2014 (UTC)
Dear mathematical mafia.
Its not the length of the dance that counts, or the number of movements. Its how beautiful the dance is and how it enlightens the world.
Firstly we don't have more proofs because more proofs makes it more true. We have more proofs because each proof tells a different aspect of why it is true. You could have 5 or more proofs and each one would give a different insight as to why this beautiful result is true,
• From De moivre formula - Very simple and highlights the role or rotations. The most elemental and constructive proof.
• Arg / absolute value - The distance from the center is one. The angle changes under rotation. Easy to visualise.
• The taylor series proof - The practical efficient proof. The exponential function of complex number must be understood as a power series. So jump ahead and use this to give an elegant powerful proof that tells you nothing about why it is true, but magically gives you the answer.
• Calculus proofs - The derivatives of exponential and trigonometric functions are known. Use them to prove a result you already know is true.
• Arg - log proof - Logarithms and angles in polar co-ordinates behave the same way. So no need to take limits. Characterize and compare there behavior to give the result.
Thats 5 fundamentally different proofs of eulers formula and each one gives insight.
Secondly proofs or derivations may be pitched at different levels of understanding. A demonstration pitches the idea of the proof for the audience not yet ready to understand the full story.
A derivation builds an explanation of core concepts (like limits) into the proof and explains them along the way.
A proof may have gaps in an encyclopedia, because not all steps are interesting. But ultimately you don't make a subject more understandable by missing out key steps. You give the illusion of understanding without true understanding. True accessibility comes from organizing the information so the readers sees, at each stage, what they need to move to the next step.
Grumble grumble grumble.
Thepigdog (talk) 13:48, 7 March 2014 (UTC)
Doggie, my whole problem with this is that the only method I know of to show that De Moivre's formula is true is by way of Euler's. Then to use that to prove De Moivre's appears ostensibly circular. We had two very simple proofs that made no assumptions other than i 2=-1, one based on a simple 1st-order derivative, the other based on a simple 2nd-order diff eq. Both of those proofs, which were nothing but solid and instructive, were removed by the mathematical mafia. So don't feel alone, but I sure don't get your "proof" when I don't understand how to prove De Moivre without first knowing about Euler. 70.109.183.141 (talk) 14:46, 8 March 2014 (UTC)
70.109.183.141: The article does have a proof based on the 1st order derivative of e^ix -- [2]. What is the "removed by the mathematical mafia" proof based on the 1st order derivative that you're talking about? I'm sorry I don't remember.
I'm not the boss of this page, and I'm not necessarily opposed to more proofs, or different proofs. I was just voicing my initial opinions and asking questions and having a discussion. :-D
A basic problem is that this article covers more than just the proof; so if the proof section is too long, the page becomes unbalanced and hard-to-read. Perhaps there should be a spin-off article Proof of Euler's formula. --Steve (talk) 00:41, 10 March 2014 (UTC)

As I think you may have discovered by looking at the talk page archives, there has been ALOT of argument in the past about what proofs to include here (some of which I took part in). Everyone seems to have a strong opinion about which proof is best, and some people have strong opinions about which shouldn't be included. My favorite has always been a variant of the calculus proof. There was an old version of the page with a better incarnation of the calculus proof (IMHO) which got destroyed by the MM (mathematical mafia). Personally I think the current limit proof is not a proof at all as it's written. It basically says, look, if you put 1000 in for n that's really close to what it should be therefore, proof done (and here's an animation in case you're not convinced). That doesn't mean I don't think it should be in the article; it is a sketch of a proof. At any rate, I would support you changing that one to be more rigorous. Holmansf (talk) 11:43, 13 March 2014 (UTC)

The limit section is not a proof, it's a summary of a proof. Isn't that the case for all of them?
I used 1000 instead of "n" because over time, three different people had made edits to the proof that made it quite clear that they couldn't follow it. Maybe you forget what it was like when you were first algebra many years ago, but any schoolteacher will tell you that most people have an easier time with numbers than with variables, especially several variables at once. You are welcome to switch it back, but you should know that it will not actually make the arguments any more or less rigorous.
I mean, saying "let's use 1000 as an example, but these statements become more and more accurate if you replace 1000 with larger and larger numbers" is exactly the same as saying "let's use n, and take the limit as n goes to infinity". Maybe the second one sounds more professional, and certainly the second one is easier for a professional mathematician to read, but I do think that the first one is easier for a budding mathematician to understand. Just my opinion, I don't really know!
You should go through the page history to find the calculus proof you prefer. I'm open-minded (more open-minded than you might think, given my previous posts). --Steve (talk) 15:53, 13 March 2014 (UTC)
No, I think the limit one is substantially farther from being a full proof than the other two which are essentially complete. Both the others contain all essential steps in some form (well the calculus one is missing a justification that the derivative of ${\displaystyle e^{ix}}$ is ${\displaystyle ie^{ix}}$; if you look at the comments above I argued quite hard to include a few lines explaining this but was thwarted by the MM). The limit proof is just a description of what is meant by the word "limit", and then a parenthetical hand wave: (The proof of this is based on the rules of trigonometry and complex-number algebra.) The entire meat of the proof is dismissed in that statement. Holmansf (talk) 10:06, 14 March 2014 (UTC)
This is the old version of the calculus proof which I prefer. If no one shows up to complain about it in the next few days perhaps I will put it back in.

"Several other proofs are based on the following identity obtained by differentiating the power series definition of eix. Indeed, since this series converges absolutely for all complex numbers we can differentiate it term by term to obtain

{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}e^{ix}&={\frac {\mathrm {d} }{\mathrm {d} x}}\sum _{n=0}^{\infty }{\frac {(ix)^{n}}{n!}}\\&=i\sum _{n=1}^{\infty }i^{n-1}{\frac {x^{n-1}}{(n-1)!}}\\&=ie^{ix}.\end{aligned}}}

Now we define the function

{\displaystyle {\begin{aligned}f(x)=(\cos x-i\sin x)\cdot e^{ix}.\end{aligned}}}

The derivative of ƒ(x) according to the product rule (note that the product rule can be proved to hold for complex valued functions of a real variable using precisely the same proof as in the real case) is:

{\displaystyle {\begin{aligned}{\frac {d}{dx}}f(x)&{}=(\cos x-i\sin x)\cdot {\frac {d}{dx}}e^{ix}+{\frac {d}{dx}}(\cos x-i\sin x)\cdot e^{ix}\\&{}=(\cos x-i\sin x)(ie^{ix})+(-\sin x-i\cos x)\cdot e^{ix}\\&{}=(i\cos x+\sin x-\sin x-i\cos x)\cdot e^{ix}\\&{}=0.\end{aligned}}}

Therefore, ƒ(x) must be a constant function in x. Because ƒ(0)=1 in fact ƒ(x) = 1 for all x , and so multiplying by cos x + i sin x, we get

${\displaystyle e^{ix}\ =\cos x+i\sin x.}$"

Holmansf (talk) 10:31, 14 March 2014 (UTC)

## Pronunciation Guide

Any chance someone can add and source a pronunciation for "Euler's"? That seems to be a matter of dispute. Mt xing (talk) 02:18, 22 April 2015 (UTC)

This should hardly be in contention. The pronunciation of "Euler" is already adequately given in the linked article Leonard Euler. —Quondum 03:17, 22 April 2015 (UTC)

## (I think that) the main page should mention that i^(2×x÷π)=cos(x)+i×sin(x)=e^(i×x).

If you need proof, press the link. Equation Simplifier 2601:2C1:C003:EF7A:157E:69D0:9FC2:8328 (talk) 02:13, 2 April 2016 (UTC)

## Possible Vandalism

"Deniz is the best basketball player in the world and friends with Mr. Euler it is documented that deniz had a big contrubution to euler's formula"

This doesn't show up when I try to edit the page so it's difficult to remove this.

Does anyone have any idea what might have happened here? 92.220.103.141 (talk) 20:24, 31 August 2016 (UTC)

Already done Probably it was removed just before you tried to edit. Thanks for reporting it. Gap9551 (talk) 20:32, 31 August 2016 (UTC)

The sentence still shows up in the very first paragraph, and disappears when entering edit mode. Something still seems to be very wrong here.. — Preceding unsigned comment added by 141.226.218.40 (talk) 07:59, 1 September 2016 (UTC)

The text is not seen in my browser when logged in or when not logged in. Possibly there is a caching issue on your browser, or conceivably a proxy server. Try WP:BYPASS. Johnuniq (talk) 08:57, 1 September 2016 (UTC)

## Proposed merge with Cis (mathematics)

The cis function is rare and redundant to Euler's formula. A former AfD, Wikipedia:Articles for deletion/Cis (mathematics), was closed as no consensus. GeoffreyT2000 (talk, contribs) 05:34, 12 December 2016 (UTC)

• Oppose "Rare" is not "non-notable".
It is likely that any readers searching for cis() will find cis(), not Euler. There they are already given an adequate explanation of it in both trigonometry and history, with an appropriate pointer to this article. Readers finding Euler will find Euler, and that's what they need, im likely blissful ignorance of cis(). How does a merged article improve upon this? This is not paper. We are not short of pages. Andy Dingley (talk) 02:14, 13 December 2016 (UTC)
• Oppose. Obviously, Euler's formula and the cis() function are related but they have different histories to be told and different use cases, and we therefore would not do either of them a favour by discussing them in a single article. They kind of approach a problem from different angles. To someone, who hasn't learnt about exponential functions, the redundancy does not exist. To some, cis() is a sometimes very convenient abbreviation or shorthand notation, for others it is a vehicle in math education.
Regarding the previous AfD (which was still about an article in a much weaker state and with lacking sources), there were prior discussions in the past decade and they suggested to discuss the cis() info in a separate article, because readers felt that the info on cis() did not belong into the Euler article and it was inadequately covered there.
As has meanwhile been established by plenty of reliable sources, cis() has 150 years of history and is a notable topic by itself. Yes, it is not in main-stream use, but this doesn't make it non-notable. After all, we're an encyclopedia and it is our duty to document things from a neutral point of view and not suppress information.
--Matthiaspaul (talk) 12:43, 20 December 2016 (UTC)
• Oppose Many Wikipedia topics are rare. That is not a reason to make them go away because this is an encyclopedia and so should cover the full circle of knowledge. Andrew D. (talk) 14:24, 20 December 2016 (UTC)
• Oppose It's not even that rare. It comes in handy in computer documentation where superscript typography is not so easy. Here is a relatively high traffic example: http://en.cppreference.com/w/cpp/numeric/complex/exp I think the article's extremely negative intro needs to be toned down.--agr (talk) 15:19, 20 December 2016 (UTC)
• Oppose I just came here looking for cis and not for eulers formula. Someone might want to remove that sign above the article. 95.91.212.79 (talk) 01:15, 6 January 2017 (UTC)
• Oppose. One certainly could discuss cis within an article on Euler's formula, but cis (mathematics) has so much material on the usage of that specific notation that I think that it works better as a separate article. You wouldn't want to either get rid of that or try to stuff it all in here (which would make this article unbalanced). ―Toby Bartels (talk) 15:00, 10 January 2017 (UTC)
• I oppose. Cis(x) is an equation, and its uses are not limited to calculating eix. Also, Euler's formula can be written as exi=cos(x)+i×sin(x) without even acknowledging that there is a shorter way to write cos(x)+i×sin(x). I do not feel that the subjects of these two articles are closely enough related to merit merging.
98.195.88.33 (talk) 01:12, 28 January 2017 (UTC)

Just for the records and to help avoid unnecessary further discussion, User:GeoffreyT2000 closed the discussion as "Don't merge" on 2017-03-24T02:08:43‎. [3] Somewhere we have some nice wrapping template for closed discussions, but I can't seem to find it right now...

--Matthiaspaul (talk) 21:18, 27 March 2017 (UTC)