Understanding the most beautiful equation in Mathematics

Loved the article, but there was this big jump between<p><pre><code> 1 - x^2/2! + x^4/4! - ... </code></pre> and<p><pre><code> cos x </code></pre> (and similarly with sin x). Why exactly are these equal?<p>(Also, just a nitpick, shouldn't the addition be actually subtraction before both elippses to demonstrate the alternating sign?)

I think the actually remarkable equation is<p><pre><code> e^ix = cos x + i sin x </code></pre> The cliched "e^(i pi) + 1 = 0" is a fairly mundane consequence of the fact that pi was chosen to make this equation hold.

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 (<a href="http://en.wikipedia.org/wiki/Exponential_function" rel="nofollow">http://en.wikipedia.org/wiki/Exponential_function</a>) and <a href="http://en.wikipedia.org/wiki/Euler%27s_formula" rel="nofollow">http://en.wikipedia.org/wiki/Euler%27s_formula</a> which were much more clearer.<p>Sadly, this article did nothing for me. I will remember to lookup wikipedia first...

Personally, I prefer<p>e ^ i*tau = 1<p>But that's because I'm a tauist.

And after you read this you should read this: <a href="http://symbo1ics.com/blog/?p=1089" rel="nofollow">http://symbo1ics.com/blog/?p=1089</a> which was kind of fun as well.

Here's my favorite explanation of this formula:<p><a href="http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/" rel="nofollow">http://betterexplained.com/articles/intuitive-understanding-...</a>

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: <a href="http://us.metamath.org/mpegif/eulerid.html" rel="nofollow">http://us.metamath.org/mpegif/eulerid.html</a>

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.

And here I was hoping it would be aboke Stokes' Theorem ;-(

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.

Small typo:<p>Euler defined the function e^x in analysis as:<p><pre><code> e^x = lim(1+x/n)^n </code></pre> as x tends to infinity<p>Should be "as n tends to infinity".

I find this view of e^z far more beautiful than a bunch of symbols rearranged by someone who thinks definitions provide insight...<p><a href="http://acko.net/files/mathbox/MathBox.js/examples/ComplexExponentiation.html" rel="nofollow">http://acko.net/files/mathbox/MathBox.js/examples/ComplexExp...</a>

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

&#62;&#62; Euler's brilliant mathematical mind replaced the real variable x with ix<p>Is there any proof that the equation remains true when x -&#62; ix transformation is made? OK, I know there is formal proof for this; can someone explain please? :-)

<i>Euler defined the function e^x in analysis as: e^x=lim(1+x/n)^n as x tends to infinity. So, we get:</i><p>It should be as n tends to infinity.&#60;/pedantic&#62;

I've seen it taken to the i'th power:<p>e^(i*pi)i = 1^i<p><pre><code> or </code></pre> e^-pi = 1^i<p>which seems very strange - e and pi are real numbers, so 1 to the i'th power must also be real?

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

I have this tattooed on my leg :) <a href="http://imgur.com/LcIlm5L" rel="nofollow">http://imgur.com/LcIlm5L</a>

Guys, there's lot more here - <a href="http://functionspace.org/discussion/new" rel="nofollow">http://functionspace.org/discussion/new</a> :)