underrated
Beautiful math theorems, equations, proofs
Mason Mitchell
Logan Morales
[math]sin^2x + cos^2x = 1^2[/math]
[math]x^2 + y^2 = r^2[/math]
[math]a^2 + b^2 = c^2[/math]
Cooper Rogers
This for me
Angel Ramirez
Minkowski's inequality is beautiful:
For all [math]d \,\in\, \mathbf N^*[/math], for all measurable subset [math]A[/math] of the Lebesgue measured set [math]\left( \mathbf R^d,\, \mathscr B,\, \ell\right)[/math], we have:
[eqn]\forall p \,\geqslant\, 1,\, \forall \left(f,\, g\right) \,\in\, \mathscr L^p \left( A \right),\, \left[ \int_A \left(f \,+\, g\right)^p\, \mathrm d\ell \right]^\frac{1}{p} \,\leqslant\, \left( \int_A f^p\, \mathrm d\ell \right)^\frac{1}{p} \,+\, \left( \int_A g^p\, \mathrm d\ell \right)^\frac{1}{p}[/eqn]
It exists in the [math]\ell^p[/math] spaces too:
[eqn]\forall p \,\geqslant\, 1,\, \forall \left( u,\, v\right) \,\in\, \left( \ell^p \right)^2,\, \left[ \sum_{n \,=\, 0}^\infty \left(u_n \,+\, v_n \right)^p \right]^\frac{1}{p} \,\leqslant\, \left( \sum_{n \,=\, 0}^\infty {u_n}^p \right)^\frac{1}{p} \,+\, \left( \sum_{n \,=\, 0}^\infty {v_n}^p \right)^\frac{1}{p}[/eqn]
This guarantees the existence of a normed vector space that has the same norm as [math]\left\_| \cdot \right\_|_2[/math] but in infinite dimension. It would be the most natural space structure extending the physical world to an infinite dimension (it would also be a reflexive Banach space which suggests physics could be done in it).
Luis Morris
I always thought the better Euler's identity to meme would have been the product form for the zeta function
It's has a similar "what the fuck" appeal when you first see it and it's probably the most elementary way to extract stuff about primes out of the zeta function so it's actually useful
Christian Powell
Poncelet's closure theorem.
According to Marcel Berger, the most beautiful result about conics.
James Rodriguez
sin(πx)=πx/[(x!)(-x!)]
Jacob Lopez
bump
Elijah Perry
scribd.com
Is there anything more beautiful than the Barnett identity?
Anthony Collins
The hierarchy of the Cantorian cardinalities -- the fact that there isn't just one infinity, there's actually an infinite number of infinities, each provably smaller than the next. They are all easily constructed by taking the set of integers and using it as the starting point for iterating the powerset function. Even nicer is the theorem (still unproven) that there are no levels of infinity that cannot be generated using that powerset iteration technique (a.k.a. the "generalized continuum hypothesis"), and the fact that nobody has yet found a counterexample.
>The irrationality of square root of 2.
I like the fact that an irrational number to the power an irrational number can be rational. The proof is extremely elegant:
If sqrt(2)^sqrt(2) is rational, then you're done.
If not, then take that irrational number to the power of sqrt(2):
(sqrt(2)^sqrt(2))^sqrt(2)
= sqrt(2)^(sqrt(2)*sqrt(2))
= sqrt(2)^2
= 2
which is rational.
Either way, you get a rational result, without needing to know which branch is actually true.