whats your fav formula/thing in math? i think my fav gotta be

My absolute favorite algebraic construct is the set of anticommutative operators acting on orthonormal Barnett spaces.

that no one can figure out if there are any odd perfect numbers

a perfect number being one equal to the sum of its proper divisors

e.g. 6 because 6=1+2+3 but not 8 because 8 =! 1+2+4

>>7866605
r=c
θ∈[0,2π)

>>7866605
Klien bottles.

>>7866605
Eclipse is more better coz it combines both

x = 0

>>7866933
You mean an ellipse and no, in some respects an ellipse isn't as pure as a circle since its eccentricity isn't the "perfect" 1.

>>7866970
Yeah,ellipse.
But circle is a kind of an ellipse. like how square is a kind of rectangle

>>7866974
as in... it isn't?

>>7866974
Yes, a circle is a special case of an ellipse, but a circle has an eccentricity of 1 whereas an ellipse has an eccentricity of less than 1.
So in some sense, a circle is more "perfect" than an ellipse.

>>7867043
Eqn of ellipse:
(x/a)^2+(y/a)^2=1
Now tell me what happens if a=b=r

>>7867043
A circle is an ellipse and a square is a rectangle, but an ellipse is not necessarily a circle and a rectangle is not necessarily a square.

The circle is a special case of the ellipse which is a special case of the ellipsoid which is a special case of the n-dimensional hyperellipsoid, etc...

>>7866974
Being more general doesn't always make an equation more beautiful.

>>7867080
See:Perspective

>>7867080
This. An example being Euler's identity.

>>7867098
fuck pop-math shitter

>>7867132
Pop-math or not, it's easily agreeable that it's more "beautiful" than the general exponential/cis/complex number relationship.

Currently my favorite is, where [math]D[/math] is the canonical ultrafilter on the measurable cardinal [math]\kappa[/math] generated by a nontrivial elementary embedding [math]j: V \rightarrow M [/math] of the universe of sets into an inner model, Ult is the corresponding (mostowski collapse) ultrapower of the universe, and [math]j_{Ult}[/math] is the canonical embedding of V into Ult, then where [math][f][/math] is the equivalence class of [math] f: \kappa \rightarrow V [/math] in the ultrapower,

[math] [f] = (j_{Ult} (f))( \kappa) [/math]

Remarkable.

>>7867210
no. it's absolutely not. your concept of beautiful comes from popsci shitters than don't actually do math

>>7867333
trips for truth

>>7867333
Why is it not? Disregard this "pop" whatever nonsense for a moment even though I'm sure you'd love to talk lots more about it.
Care to share something that you consider beautiful then?
I'm not really sure what I'd consider beautiful, but the topological criterion for continuity is certainly nice (we have a great base for continuity without epsilon-delta!)

>>7867333
Just because it's popsci doesn't imply it can't be beautiful.

>>7867343
"a mathematicians apology" by hardy explains quite well what I consider to be a beautiful mathematical result.

eulers identity is far from what I'd consider beautiful.

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>>7866605
Holy shit, is this the same guy?

>>7867351
Just because it's both doesn't mean I think one implies the other.

>>7867343
A definition is hardly beautiful. The fact that a natural concept extends to more general things is nice, but unless you prove anything about it then it doesn't mean much.

Something beautiful I guess would be a fundamental theorem of a field. Maybe the fundamental theorem of Galois Theory, amazing relationships between field extensions and normal subgroups.

Apparently you guys can't read critically.

>>7867361
Read that a couple of years ago but I don't have immediate access to it, could you recap?
>eulers identity is far from what I'd consider beautiful
I didn't say that it was beautiful.
>it's more "beautiful"
if there were such a thing as a beautiful scale.

>>7867377
>A definition is hardly beautiful
Yes, see the part of the post that says
>certainly nice

>>7867373
Has to be. Where did you get that?

>>7867384
a beautiful theorem is simple in statement/proof but generalises to a wide variety of problems and should use mathematics in a new and creative way. it should make you do a mental "double take" as you realise how and why it is true. often with theorems I consider elegant I'm left with a feeling that the author is "cheating".

I remember when I first started analysis and grasped the definition of real numbers from dedekind cuts I was immediately taken aback. it left me both excited and slightly shocked.

>>7867395
Someone texted it to me. https://newfaculty.uchicago.edu/page/dam-thanh-son

>>7866605
>>7867373
That's pretty amazing.

Someone make an image compiling these two.

>>7867210
Kill yourself

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>>7867210
>cis

>>7868765
It's much faster to type than cos(x) + isin(x).

>>7867048
>>7866970
Umm... guys... The conic with an eccentricity of 1 is a parabola...
>Circle - e=0
>Ellipse - 0<e<1
>Parabola - e=1
>Hyperbola - e>1
Fuck, year 11 maths taught me that.

>>7869211
You're absolutely correct, this is an error on my behalf.
I don't think I've done too badly however, not having used eccentricity since FP1 or 2 or 3 or whatever it was in for over 2 years.
If I recall correctly it had a relationship with the directrix and foci of these conics.

Wilson's theorem.

p is prime

if and only if

(p-1)! = -1 (mod p)

Even with the other more involved shit I always find this the cutest.

>>7869926
Sorry m8, too trivial.

>>7866605
Hmm maybe the generalized stokes theorem.

[math] \int _{\Omega} d \omega = \int _{\partial \Omega} \omega [/math]

It's also amazing because the proof is quite trivial once one has developed the necessary theory of differential forms, integration on chains, pullbacks, partitions of unity, etc.

e + i pi = 1

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>>7866605

[math] {\Omega ^k}M = d{\Omega ^{k - 1}}M \oplus \delta {\Omega ^{k + 1}}M \oplus {\mathcal{H}^k}M[/math]

>>7869959
Is Newton's "fundamental theorem of calculus" just a special case of the generalized stokes theorem?

>>7870156
Explain this.

>>7870178
https://en.wikipedia.org/wiki/De_Rham_cohomology#Hodge_decomposition

>>7870172
Yes, let [math]\Omega = \left[ {a,b} \right][/math] and [math] \omega = f\left( x \right) [/math].

>>7870178
Hodge Decomposition. Every k-form on a compact manifold can be decomposed into the sum of exact, co-exact, and harmonic forms.

>>7870143
All infinite series are OK to except continuing fractions. Fuck I hate continuing fractions so much.

>>7870186
With respect to the stoke's theorem:

Ha, I thought so! I'm a physics student, and I extensively use Divergent theorem, and stokes theorem in three dimensions. And I've noticed a pattern:

If a derivative of a function [math] f' [/math] is described on a set, the integral of the function can be expressed in terms of [math] f [/math] on the boundary of the set.

>>7870247
Where [math] f' [/math] is the derivative of [math] f [/math].

Collatz conjecture, but

>>7869959
Stokes is very interesting.

>>7870247
https://proofwiki.org/wiki/Implications_of_Stokes%27_Theorem

Pythagoras's Theorem.

No really. It establishes the connection between geometry and arithmetic.

[math] \operatorname{div} {\mathbf{F}} = \star \operatorname{d} \star {{\mathbf{F}}^\flat } [/math]

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>>7866605
I'm a simpleton and a numbers guy, so it would either be pi (I know the first ~40 digits), or the thought of all the different kinds of numbers in existence (they teach this in Pre-Calc, pic related).

>>7870923
Eh, not up to par comaried with the Hodge Laplacian as far as classical formulae expressed in modern language.
Obviously because of the implications of the Hodge Laplacian (harmonic forms, Hodge theory, blah blah blah).

>>7867210
Euler's identity isn't beautiful
It's like saying 1 = 1 is beautiful

>>7866605
a^2 + b^2 = c^2

sin^2(x) + cos^2(x) = 1^2

>>7871353
Weak :/

>>7870963
>Real
wew lad

>>7871350
>Euler's identity isn't beautiful
Did you even read the post
>it's more "beautiful"
It's like saying you're straighter than OP, but it doesn't mean you're straight you faggot.

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u wot m8

>>7870963
The best numbers are those that can't even be proven to exist, e.g. [math]0^{\#}[/math].

>>7866605
I'd say:
[math]|x+y| \leq |x| + |y|[/math]
Or the proof of the irrationality of [math]\sqrt{2}[\math].

My favorite mathematical object is the space of Barnett integrable functions
[eqn]B = \bigg \{ f \in C ( \mathbb{R} ) : \int \int \int f ( x ) \, \mathrm{d} ^ 3 x < \frac{e^{i\pi}}{11.999 \ldots} \bigg \} \ .[/eqn]