Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Understanding the most beautiful equation in Mathematics

functionspace.org

Post reply on HN
Print view View original

11–20 of 59 posts

Re: Understanding the most beautiful equation in Mathematics

#11

Post by lmm »

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?)

In advanced mathematics it's common to define cos and sin by these series (and pi is defined as the smallest strictly positive x with sin x = 0). (Of course that just reduces the question to "why do certain geometrical identities match this sin function")

Re: Understanding the most beautiful equation in Mathematics

#12

Post by dnautics »

lmm wrote: post #11
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?)

In advanced mathematics it's common to define cos and sin by these series (and pi is defined as the smallest strictly positive x with sin x = 0). (Of course that just reduces the question to "why do certain geometrical identities match this sin function")

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

Re: Understanding the most beautiful equation in Mathematics

#13

Post by dnautics »

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?)

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?

Re: Understanding the most beautiful equation in Mathematics

#14

Post by betterunix »

gohrt wrote: post #7
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 ?

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.

Re: Understanding the most beautiful equation in Mathematics

#15

Post by jckt »

gohrt wrote: post #7
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 ?

Good question. Though I suppose it's not too difficult to state in terms of the Abel-Ruffini theorem, but then again that watered down version would probably fail to mention the wide-reaching consequences of Galois...

Re: Understanding the most beautiful equation in Mathematics

#17

Post by iTrollFreely »

dnautics wrote: post #13
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?)

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?

exactly. the dude has no idea what he's talking about, or didn't bother with even a cursory proofread. e^x as "x tends to infinity" is infinity.
Post reply on HN