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Understanding the most beautiful equation in Mathematics

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Re: Understanding the most beautiful equation in Mathematics

#51

Post by [deleted] »

neotrap wrote: post #35
senthil_rajasek wrote: post #32

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...

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

[deleted]

Re: Understanding the most beautiful equation in Mathematics

#52

Post by dnautics »

betterunix wrote: post #23
dnautics wrote: post #13

Earlier quoted context omitted.

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! + ...

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.

Re: Understanding the most beautiful equation in Mathematics

#53

Post by dnautics »

dbaupp wrote: post #50
dnautics wrote: post #12

Earlier quoted context omitted.

or you could use MacLaurin polynomial series (Taylor series at zero)

I'm probably missing something, but that is the Taylor series at 0.

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.

Re: Understanding the most beautiful equation in Mathematics

#54

Post by jhandl »

jessaustin wrote: post #34
Roedou wrote: post #29

Earlier quoted context omitted.

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.

Haha, fundamental constant counting fail. b^)

Well, these are the four constants: 0, 1, i, e and tau. Yes, that really is four constants.

Re: Understanding the most beautiful equation in Mathematics

#55

Post by pmiller2 »

dnautics wrote: post #52
betterunix wrote: post #23

Earlier quoted context omitted.

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! + ...

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.

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

Re: Understanding the most beautiful equation in Mathematics

#58

Post by senthil_rajasek »

neotrap wrote: post #35
senthil_rajasek wrote: post #32

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...

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

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

Re: Understanding the most beautiful equation in Mathematics

#59

Post by gohrt »

betterunix wrote: post #14
gohrt wrote: post #7

Earlier quoted context omitted.

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

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…

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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