
Understanding the most beautiful equation in Mathematics - dynamic99
http://functionspace.org/articles/6/Understanding-the-most-beautiful-equation-in-Mathematics
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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?)

~~~
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?

~~~
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.

~~~
pmiller2
The series has to be absolutely convergent. That is, you can rearrange the
terms of \sum_{i=0}^\infty a_n freely if and only if \sum_{i=0}^\infty |a_n|
converges. See <http://en.wikipedia.org/wiki/Riemann_series_theorem>

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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!

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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...

~~~
neotrap
If you had to look for the definition of exponential function, perhaps you
should start from here - <http://www.mathsisfun.com/basic-math-
definitions.html>

~~~
senthil_rajasek
that site you posted might actually be of use to someone more than the
original site.

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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.

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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.

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cmvkk
Here's my favorite explanation of this formula:

[http://betterexplained.com/articles/intuitive-
understanding-...](http://betterexplained.com/articles/intuitive-
understanding-of-eulers-formula/)

~~~
a-nikolaev
My favorite:

1) Using Taylor Series, show that exp(ix)=cos(x)+i*sin(x).

2) Then the result is trivial for x=pi

This image helps:

<https://en.wikipedia.org/wiki/File:Euler%27s_formula.svg>

~~~
stephencanon
Taylor series are ugly; do it directly from the definition of the functions as
solutions to specific differential equations =).

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pfedor
And just in case you weren't perfectly satisfied with the level of
mathematical rigor of the article, here is a complete, formal, machine-
verified and hyperlinked version of the proof:
<http://us.metamath.org/mpegif/eulerid.html>

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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.

~~~
aditgupta
This was more for basic understanding. There can be a follow-up article with a
more rigorous approach :) The same was done here -
[http://functionspace.org/articles/17/Solving----sum----
1----...](http://functionspace.org/articles/17/Solving----sum----1------infty
-------1-n-----2-------with-Sine-Function--Basel-Problem)

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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.

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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.

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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".

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unconed
I find this view of e^z far more beautiful than a bunch of symbols rearranged
by someone who thinks definitions provide insight...

[http://acko.net/files/mathbox/MathBox.js/examples/ComplexExp...](http://acko.net/files/mathbox/MathBox.js/examples/ComplexExponentiation.html)

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mohas
Can anyone name some of the actual uses of this equation in solving real world
problems?

~~~
thedufer
Well, this equation is really a consequence of the more general e^i _x =
cos(x) + i_ sin(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.

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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? :-)

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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>

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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?

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comub
There is nothing particularly beautiful in this, it's just a trivially obvious
identity (once you know the relevant theory, of course).

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togasystems
I have this tattooed on my leg :) <http://imgur.com/LcIlm5L>

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aditgupta
Guys, there's lot more here - <http://functionspace.org/discussion/new> :)

