Understanding the most beautiful equation in Mathematics

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.

Re: Understanding the most beautiful equation in Mathematics

#5

Post by pestaa » Thu, Jun 06, 2013, 4:55 PM UTC

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?)

Re: Understanding the most beautiful equation in Mathematics

#6

Post by gohrt » Thu, Jun 06, 2013, 4:57 PM UTC

pestaa wrote: post #5

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?)

https://en.wikipedia.org/wiki/Taylors_theorem

Re: Understanding the most beautiful equation in Mathematics

#7

Post by gohrt » Thu, Jun 06, 2013, 4:57 PM UTC

betterunix wrote: post #4

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.

What is Fundamental Theorem of Galois Theory in the form of an equation?

Re: Understanding the most beautiful equation in Mathematics

#8

Post by a-nikolaev » Thu, Jun 06, 2013, 5:00 PM UTC

cmvkk wrote: post #3

Here's my favorite explanation of this formula: http://betterexplained.com/articles/intuitive-understanding-...

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

Re: Understanding the most beautiful equation in Mathematics

#9

Post by aditgupta » Thu, Jun 06, 2013, 5:03 PM UTC

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

Re: Understanding the most beautiful equation in Mathematics

#10

Post by Permit » Thu, Jun 06, 2013, 5:11 PM UTC

pestaa wrote: post #5

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?)

gohrt linked an article, and although you could derive the identity for yourself using that information, the article doesn't contain the series you're looking for.

You can find it here if you'd like: https://en.wikipedia.org/wiki/Taylor_series#List_of_Maclauri...