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quchen
And here I was hoping it would be aboke Stokes' Theorem ;-(
mrbrowning
I was hoping that too, but I knew it would be about Euler's identity, since to people with only incidental exposure to the concepts that underly it it seems (justifiably) inscrutable and mysterious, thus its general popularity. It's funny that cultivating the mathematician's refusal to assign meaning to results can completely change which results you find fascinating.
betterunix
I would say that the Fundamental Theorem of Galois Theory is the most beautiful result of all mathematics, though Euler's identity is certainly a contender.
gohrt
What is Fundamental Theorem of Galois Theory in the form of an equation?
betterunix
The field extension lattice is isomorphic to the subgroup lattice; if you really wanted to, you could write this out symbolically (but I am not sure why you would want to, since it does not really convey the meaning of the theorem any better). I suppose you might say that such an isomorphism does not qualify as an equation, but that is a bit pedantic in my opinion since such isomorphisms have all the properties of an equivalence relation.
gohrt
Right, it's a beautiful theorem, not so much a beautiful equation.

Euler's identity is a beautiful equation, because it ties together several of the most fundamental objects of mathematics, with one occurrence of each, with no wasted boilerplate. The notation is part of the beauty. It looks darn good, on the surface in addition to the beyond the ideas behind the surface.

jckt
Good question. Though I suppose it's not too difficult to state in terms of the Abel-Ruffini theorem, but then again that watered down version would probably fail to mention the wide-reaching consequences of Galois...
blangblang
I'd offer that accessibility is a huge part of the beauty of Euler's identity. A few weeks into your average Calculus 2 class and it almost feels intuitive.
monjaro
Could you suggest any resource for understanding this theorem? I have a math degree, but I never came across any Galois theory.
decasteve
Not to mention the Inverse Galois Problem is one of the long unsolved problems in Mathematics. In the league of Fermat's Last Theorem.
cmvkk
Here's my favorite explanation of this formula:

http://betterexplained.com/articles/intuitive-understanding-...

pestaa
Loved the article, but there was this big jump between

    1 - x^2/2! + x^4/4! - ...
and

    cos x
(and similarly with sin x). Why exactly are these equal?

(Also, just a nitpick, shouldn't the addition be actually subtraction before both elippses to demonstrate the alternating sign?)

lmm
In advanced mathematics it's common to define cos and sin by these series (and pi is defined as the smallest strictly positive x with sin x = 0). (Of course that just reduces the question to "why do certain geometrical identities match this sin function")
dnautics
or you could use MacLaurin polynomial series (Taylor series at zero)
dbaupp
I'm probably missing something, but that is the Taylor series at 0.
dnautics
Yeah I mean to say you can derive the sin and cosine as abstract functions from their geometrical properties, get the form of their derivatives, and then derive their Taylor series, instead of defining them as infinite series, and then showing that the function looks like their geometric equivalents.
dnautics
I was also perturbed by the jump from the definition of e to the taylor expansion. I know how to get there the long way (define e first, derive properties of the exponential derivative, then construct the Taylor series), does anyone know a shortcut?
[dead] iTrollFreely
exactly. the dude has no idea what he's talking about, or didn't bother with even a cursory proofread. e^x as "x tends to infinity" is infinity.
lmkg
The Taylor series is actually the expansion of the limit in the line above. There's some trickery in proving that the limit converges, but you can derive one line from the other with some straightforward combinatorics.
betterunix
Here is another way (a bit informal):

e = lim_{n->infinity} (1 + 1/n)^n

Now, apply the binomial theorem:

1 + n * 1/n + n! / (2 (n-2)! n^2) + ... + n! / (m! (n - m!) n^m) + ...

Now, for each m, we have this sequence:

a_n = n! / (m! (n - m)! n^m)

Which converges on 1/m!, so we are left with this:

1 + 1 + 1/2! + 1/3! + 1/4! + ...

dnautics
it's a little ugly, because you have to have some strong conditions to use associativity on infinite series (and I forget what they are off the top of my head). Of course, this is true for splitting up the e^x into cos(x) and sin(x) as well.
[dead] iTrollFreely
it's evident you have no understanding of it. full of errors and logical fallacies.
j2kun
There's some even more important gaps regarding analytic continuations of functions to complex numbers (and the resulting power series expansions). You can prove it this way, but it's not at all rigorous by today's standards.
merraksh
Euler defined the function e^x in analysis as: e^x=lim(1+x/n)^n as x tends to infinity. So, we get:

It should be as n tends to infinity.</pedantic>

5teev
Small typo:

Euler defined the function e^x in analysis as:

   e^x = lim(1+x/n)^n
as x tends to infinity

Should be "as n tends to infinity".

anonymous
Personally, I prefer

e ^ i*tau = 1

But that's because I'm a tauist.

Roedou
The advantage of the traditional format is that it not only includes four fundamental constants (1, 0, e, i and π) but it also includes the four fundamental operators (addition, multiplication, exponential and equality.)

I guess that: e ^ i*tau + 0 = 1

would be a suitable hack to get that beauty back.

jessaustin
Haha, fundamental constant counting fail.

b^)

jhandl
Well, these are the four constants: 0, 1, i, e and tau. Yes, that really is four constants.
ChuckMcM
And after you read this you should read this: http://symbo1ics.com/blog/?p=1089 which was kind of fun as well.
carstimon
I prefer that point of view if you want to understand Euler's identity, and I find John Baez does it even better: http://math.ucr.edu/home/baez/trig.html The Baez article does leave it to the reader to convince himself that exp(i*theta) is good notation for a point on a unit circle.

I have problems with the attitude of the article you linked, though. Especially "Therefore, I’d like to complain to the thousands of people who find Euler’s identity stunning and beautiful." followed by a snide list of reasons why someone might find it beautiful. It's very common when doing math that something amazing is obvious an hour later. I believe that we are better served by reminding ourselves that (a) nobody knows everything, and (b) the basics facts are actually very beautiful.

ChuckMcM
I totally agree on the attitude on the blog, the author is clearly working through non-mathematical issues on their own (if you read some of the other entries you can see how those challenges affect their writing). That said, I tend to see it as a counterpoint to mathematics blogs that are bit too gushing the other way. And its amusing the path that is taken as well.
mohas
Can anyone name some of the actual uses of this equation in solving real world problems?
teeja
Sure. In any equation containing e^iπ, you can simplify by substituting -1.
thedufer
Well, this equation is really a consequence of the more general e^ix = cos(x) + isin(x). This, Euler's Formula, enormously simplifies sinusoidal equations. Most common trigonometric identities can be proven in only 3 or 4 steps if you spend 2 of them converting to/from the exponential form, but are far more complicated in the trigonometric form. Many problems in electricity, magnetism, and basic quantum physics would be drastically less wieldy (more unwieldy?) without it.

I don't know of any cases in which it makes things possible, but there are plenty of cases where it makes things practical.

tome
I think the actually remarkable equation is

    e^ix = cos x + i sin x
The cliched "e^(i pi) + 1 = 0" is a fairly mundane consequence of the fact that pi was chosen to make this equation hold.
viraj_shah
The latter is cliched because it incorporates an additional fundamental constant, pi. Who would have thought that the ratio of the circumference of a circle to the diameter when multiplied by the imaginary number and then exponentiated by another constant e would produce such a simple equation which also includes the multiplication identity and the addition identity? Yes, pi is chosen but it certainly encompasses the trigonometry and geometry (rotations, sinx, cosx, etc.).
tome
My point is that it is only fundamental because it leads to this equation!
senthil_rajasek
I wish I could take my up vote back. I read this article and the power series expansion of the exponential function was not clear. So I looked up the wikipedia article (http://en.wikipedia.org/wiki/Exponential_function) and http://en.wikipedia.org/wiki/Euler%27s_formula which were much more clearer.

Sadly, this article did nothing for me. I will remember to lookup wikipedia first...

[dead] kapooht
There is little explanation of the true mathematics behind Euler's identity; the article presents an idea and then proceeds by discussing the various segments, in little detail, of the identity. No respect is given to why Euler's e and its importance, its significance, or why it came about. No detail was given regarding why Euler decided to raise e to the x or why the imaginary number, i, appears in the equation.

In actuality, this article has not provided any useful information, and especially not helped anyone truly 'understand the most beautiful equation in Mathematics'. However, if someone is looking for, in my opinion, a real explanation, I feel as though Kalid Azad's explanation of Euler's identity (http://betterexplained.com/articles/intuitive-understanding-...) is fairly thorough and insightful.

comub
There is nothing particularly beautiful in this, it's just a trivially obvious identity (once you know the relevant theory, of course).
alok-g
>> Euler's brilliant mathematical mind replaced the real variable x with ix

Is there any proof that the equation remains true when x -> ix transformation is made? OK, I know there is formal proof for this; can someone explain please? :-)

JoeAltmaier
I've seen it taken to the i'th power:

e^(i*pi)i = 1^i

   or
e^-pi = 1^i

which seems very strange - e and pi are real numbers, so 1 to the i'th power must also be real?