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...
Understanding the most beautiful equation in Mathematics
21–30 of 59 posts
Re: Understanding the most beautiful equation in Mathematics
#22Loved 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?)
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?
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.
Re: Understanding the most beautiful equation in Mathematics
#23Loved 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?)
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?
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! + ...
Re: Understanding the most beautiful equation in Mathematics
#24There'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.
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----...
Re: Understanding the most beautiful equation in Mathematics
#25Personally, I prefer
e ^ i*tau = 1
But that's because I'm a tauist.
Re: Understanding the most beautiful equation in Mathematics
#26And here I was hoping it would be aboke Stokes' Theorem ;-(
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.
Re: Understanding the most beautiful equation in Mathematics
#27And after you read this you should read this: http://symbo1ics.com/blog/?p=1089 which was kind of fun as well.
Re: Understanding the most beautiful equation in Mathematics
#28Can anyone name some of the actual uses of this equation in solving real world problems?
Re: Understanding the most beautiful equation in Mathematics
#29Personally, I prefer e ^ i*tau = 1 But that's because I'm a tauist.
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.
Re: Understanding the most beautiful equation in Mathematics
#30I 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.
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.