Beautiful equations thread

>>7645099
So how does one derive that formula?

>>7645109
That's the beauty of it.

How to check if this is correct? Who derived it?

>>7645117
Where is the proof?

>>7645099
2+2=4

>>7645099
9/11=0.8181818181818182

Ramanujan


https://crypto.stanford.edu/pbc/notes/pi/ramanujan.html

>>7645234
>The proof is trivial and left to the reader.

>>7645234
>he doesn't know who ramanujan is

there is no proof

http://math.stackexchange.com/questions/14115/motivation-for-ramanujans-mysterious-pi-formula

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>>7645117
>>7645192
The formula in OP's pic converges really quickly. Two parts of the sum converge within ~20 decimal places. Six parts of the sum converge within ~50 decimal places.

>>7645099

x=x

https://sites.google.com/site/tpiezas/0013

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>>7645099

>>7645263
This lol

>>7645269
>tfw discovered a pi series but it converged slow as shit and wasn't massively different to the one you get from arctan.
How did he do it man

>>7645099
Beautiful? That's a clusterfuck of seemingly random terms.

>>7646145
>liter.jpg
>liter

>>7646145
>I love math, I find it beatiful
>I failed the calculus exam. The exams don't determine my intelligence.

>>7645099
Series i can't solve with my con/div flow chart for 200$, please.

Pi = (e^(BEi))/(Σn(e^(BEn))

It's mindblowing how such a simple application of statistics is so powerful.

>>7646318
>denominator has fourth order factorial and exponential
Jeez, I dunno man

>>7646317
/sci/ in a nutshell

>>7646322
can someone simplify this equation so any idiot with a preschool understanding of maths can learn.

>>7646398
B is the set of all functions in a certain space whose integrals satisfy a certain property

>>7646317
Finding something beautiful and being good at it are two different things

>>7645263
>i project my taste in antiquated hats onto people with different interests than me
back to reddit pls

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>>7646317
Not quite.

You got an A in Multivariable Calculus and post it in the thread about Math and Science.

So did everyone whose got half a brain.

>>7646405
Amazingly you've managed to explain it in a way that doesn't actually clarify anything, but also isn't understandable to a preschooler

>>7647486
>calc III

hello reddit

>>7648083
The guy said calc test. So I posted calc tests.

>>7647486
>I can inspect element

what's that, cuntboi?

>>7647841
Ok hotshot, you explain that shit to a preschooler.

[math]1^{1^{1^{1^{1^{\dots}}}}} = 1[/math]

>>7647447
Of course. But I meet retards that think science is " muh the space is beatiful".

>>7648665
Prove it

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>>7648761
[math]1^{1^{\dots}} = 1[/math]
[math]ln(1) = 0 = ln(1^{1^{\dots}}) = 1^{1^{\dots}}\cdot ln(1) = 0[/math]
>both sides are zero = true
>QED = Quickly Eat Dick

>>7648761
[math]a^{a^{a^{a^{a^{\dots}}}}} = 1[/math]
[math]a^{(a^{a^{a^{a^{a^{\dots}}}}})} = a^1[/math]
[math]1 = a[/math]

>>7648775
>>7648780
were you trying to give shitty proofs on purpose?

>>7648782
My proof is undeniable

[math]a^{a^{a^{a^{a^{\dots}}}}} = 1[/math]

[math]a^{(a^{a^{a^{a^{\dots}}}})} = a^1[/math]

[math]1 = a[/math]

>>7647486
>average 48.71/50
>median 50/50

Shit tier community college detected

>>7648787
even allowing that the steps are valid, which is not clear until youve defined exactly what is meant by a^a^a^..., you still havent proved that 1^1^1^... = 1.

what you proved is that if a^a^a^... = 1, then a must be 1. this is different.

e.g. if x is a real number such that x^2 = -1, then x^4 = (x^2)^2 = (-1)^2 = 1. so x must be either 1 or -1. but this does not prove that (1)^2 = -1 or (-1)^2 = -1

>>7648787
And so, by extension, is mine.

>>7648801
>e.g. if x is a real number such that x^2 = -1, then x^4 = (x^2)^2 = (-1)^2 = 1. so x must be either 1 or -1.
No, that's false. x = 1^(1/4) = 1,-1,i, or -i. This is true since x = i or -i

>>7646398
It's a collection of memes.
People on /sci/ always argue about 0.999... = 1, because they don't understand that the 9's go on forever. Hence the denominator is 12. the [math]e^{i \pi}[/math] is just -1, so the right hand side is -1/12, which is, in a certain sense, equal to the sum of all the positive integers - another favourite of /sci/.
The left hand side is a triple integral. This is like finding the area under a curve, or the volume under a surface, but one step up; like the 4D-volume under a 3-cycle. Triple integrals are often touted as an extremely difficult part of mathematics, but they are not. It all comes from a video about applying to grad school, where a stupid student says that they want to specialise in triple integrals.
Barnett is an autistic child who is studying physics at uni and thinks he will somehow "disprove" Einstein in some way.
Perelman is a sucessful mathematician who proved the Poincare conjecture; one of the Millennium prize problems. He turned down the $1m prize, because he thought it should be shared with Richard Hamilton, who invented the technique he used to solve the problem.

>>7648815
i assumed x was a real number

its certainly true that "if x is a real number s.t. x^2 = -1, then x=1 or x=-1" regardless of the fact that no such x exists. thats the point

>>7648838
>its certainly true that "if x is a real number s.t. x^2 = -1, then x=1 or x=-1"
Again it's not. You used a function with branch points, so the result needs to be checked. If you check the results then you get no real solutions. It's really not that hard to understand. And it's not analogous to what I did.

>>7648867
>Again it's not.
https://en.wikipedia.org/wiki/Material_conditional

>the result needs to be checked.
exactly

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OP, that is the ugliest fucking equation you can find anywhere. Too many discrete quantities slapped together. I wouldn't be able to tell the difference if the discrete values were adjusted. Pick related is a real beautiful equation, OP.

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[math](-\frac{1}{12})\cdot (-\frac{1}{12})^2\cdot (-\frac{1}{12})^3\cdot \dots = (-\frac{1}{12})^{-\frac{1}{12}}[/math]
[math]\approx 1.18816 - 0.318367 \cdot i[/math]

>mfw

>>7648877
The fact that it's random as fuck is what makes it beautiful. Total class of it's own.

>>7648876
No, you really don't get it. The function a^1 has no branch points, so there is nothing to check. Restricting a variable to the reals and then solving it with a branching function has literally nothing to do with my proof.

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>>7648891
>[math](-\frac{1}{12})\cdot (-\frac{1}{12})^2\cdot (-\frac{1}{12})^3\cdot \dots = (-\frac{1}{12})^{-\frac{1}{12}}[/math]
How the fuck do I [math]L^aT_ex[/math]

[math](-\frac{1}{12})* (-\frac{1}{12})^2* (-\frac{1}{12})^3* \dots = (-\frac{1}{12})^{-\frac{1}{12}}[/math]

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>>7649092
I give up
[math](-\frac{1}{12})\cdot (-\frac{1}{12})^2\cdot (-\frac{1}{12})^3\cdot \dots = (-\frac{1}{12})^{-\frac{1}{12}}[/math]

>>7649026
sorry you dont understand "if-then" statements

>>7645099
There was this one equation where this one indian guy magically found that calculates pi to a certain amount of digits. That's pretty amazing how he could see that.

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Latex is for [math]Fetishists[/math]

>>7649178
[math]A~~Y~~Y~~~~~~L~~M~~A~~O\\Y~~Y\\Y~~~~~~Y\\\ \\L~~~~~~~~~~~~~~~~L\\M~~~~~~~~~~~~~~~~~~~M\\A~~~~~~~~~~~~~~~~~~~~~~~~~~A\\O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~O[/math]

>>7649100
Sorry you don't understand how algebra works...

>>7648816
Can I get a link to that video on triple integrals?

>>7648791
Seriously, what class has over half the students getting perfect scores...

>>7649325
https://www.youtube.com/watch?v=FOZO0WWULLA

https://www.youtube.com/watch?v=w-I6XTVZXww

https://www.youtube.com/watch?v=n12bfWTw9Hk

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>>7649209
That's actually really cool, senipa

>>7648089
>is currently in calc 3
>does not refute the assertion that he is not from reddit
>thinks a screenshot of some grades mean anything
back to reddit

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It says a lot in a very small statement. That kinda shit is my jam.

[spoiler]the proof however, is ugly as hell. or, at least, the inductive step is. my teacher for some reason didn't want to prove or use Leibniz rule for exchanging the derivative and integral sign so showing it was an algebraic mess. I did it for homework this weekend and it took me hours.[/spoiler]

>>7649584
>is currently in calc 3
Check the dates, from this past summer.
>does not refute the assertion that he is not from reddit
I am not from reddit
>thinks a screenshot of some grades mean anything
No I do not think it means anything. Guy made it out to seem people on /sci/ can't even pass baby calc tests, so I posted to show otherwise.

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nice thread guys

>>7645099
People on /sci love to post Stokes' formula in those kind of thread, but I find something as simple as Cauchy's formula to be jst as beautiful if not more.
The expression of the Fourier transform of a Gaussian is also nice.

I also recently discovered Itô's Lemma. Ultra dank stuff.


Too tired for tex, but those four can be googled easily.

>>7647486
>bragging about passing calcIII

>>7649617
Nice, is it your first time through complex?

>>7649756
it isn't in the sense that I took a baby version of this class as an undergrad. now i'm in the graduate version and we're proving everything from scratch. it's pretty satisfying.

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The most beautiful equation.

>>7649092
>[math](-/frac{1}{12})* (-/frac{1}{12})^2* (-/frac{1}{12})^3* /dots = (-/frac{1}{12})^{-/frac{1}{12}}[/math]

>>7645099
fuck you never put a goddamn factorial in an equation and call it beautiful ever again you blubbering neanderthal what the fuck is wrong with you

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These
t.bh fâm

>>7650062
>he still thinks with non-factorial terms
This is why you'll always be plebeian.

>>7650057
[math](\frac{-1}{12})^{1} (\frac{-1}{12})^{2} \dots = (\frac{-1}{12})^{\frac{-1}{12}}[/math]

I honestly do not understand what is wrong with this formula. The infinite power of [math]- \frac{1}{12}[/math], maybe?

>>7650101
Cool 101 me
Testing...
[math](\frac{1}{12})[/math]
[math](- \frac{1}{12})[/math]
[math](\frac{-1}{12})[/math]
[math](\frac{-1}{12})^{2}[/math]
[math]{\frac{-1}{12}}^{2}[/math]
[math]\dots[/math]

>>7650062
>has a strong opinion on beautiful equations
>hasn't even take enough math to get through sequences and series in calc 2
>this is literally a freshman class if you're not smart, and a high school class if you are somewhat smart

epic

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Difference Equations are Pretty Amazing

>>7650136
>That definition
>A function
ayyyyyyyy lmao
f=2f, with that definition, so f=0. Wait a second...

>>7647486

>this much damage control

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>>7650161
Think of hitting a long metal bar with a hammer. The delta function says we can assume the hit occurs at one instant in time, and in this case the "bang."
The picture is a spring system hit by a hammer.

>>7646322
Why isn't it [math]dx^3[/math] instead of [math]d^3[/math]?

>>7650174
... because it's the third derivative, not the third power of the first derivative, 'tard.

>>7650174
I don't know what is going on in the meme picture, but usually [math]{\operatorname{d} ^n}x = \operatorname{d} {x_1}\operatorname{d} {x_2}...\operatorname{d} {x_n}[/math]

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>>7650806
sin(ab) = sin(a)sin(b)

>>7650806
Pls no the real answer follows from the geometric series

>>7649343
Good teaching /s

>>7649343
>>7648791
That first quiz was literally just on vector addition, subtraction, and the dot product. If the average wasn't that high it would imply the students were retarded. And yes it was at a CC, I hadn't started university yet.

>>7650161
In Physics and Engineering they often often call it function although it´s a irregular distribution, because they dont care about rigor that much. I mean just look at that definition: "inifinitiy" for t=0...

0.9999....=1

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>>7651031
>>7650950
I'm not saying that you can just blithely substitute x for sin(x) or vice versa; I'm saying that this equation is particularly beautiful because if you replace each term with sin of itself, you still get equality. That rarity makes it beautiful.

>>7647486
Q U I Z 4
U
I
Z
4

>>7645263
>>7645258
Has any of his hypothesis proven to be wrong?

>>7651296
Fucked up on a question that wanted us to setup a triple integral over a box in spherical coordinates.

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There is only one most beautiful formula...

>>7651373
One equation to rule them all

>>7651373
i^i=e^(-π/2) is cooler

>>7650065
Those contours look like they got the down's

>>7645235
Elegant, Minimal, I love it.

>>7651446
I don't think you understand what's so beautiful about the first one.

>>7645099
x^2-10x-8=0

>>7651585
Do you? (seriously, I never understood the beauty behind euler's equation, even after studying complex analysis)

>>7645272
>x=x
This has my vote for most beautiful equation. It is simple, elegant, and it is immediately obvious that it is true. Plus it has application in mathematics, logic, and philosophy.

I wish I knew how to type that damn upside-down "A" so that I could do the formal logic version.

For all x, x equals x.

>>7651373
literally reddit the formula

>>7646322
why the fuck would you write e^(i*pi) instead of -1? Looks kinda fake to me.

>>7646322
This is one of the most profound one.

>>7651698
It links e, complex numbers, pi, 1, 0, and arithmetic - arguably some of the most fascinating things of mathematics - all into one equation. It also provides a link between the complex plane and the real plane.

>>7651356
tard alert

>>7650023

Underrated

>>7648761
[math]1^{1} = 1 \Rightarrow 1^{1^{1}} = 1^{1} = 1 \Rightarrow 1^{1^{1^{1}}}= 1^{1^{1}} = 1^1 = 1[/math] etc

>>7646322
I fucking love this one. Barnett is truly THE genius of our time.

>>7651852
>why the fuck would you write e^(i*pi) instead of -1? Looks kinda fake to me.
The reasons are obvious if you take the complex logarithm of both sides and pick branches.

Obviously you're no Jacob Barnett but it becomes pretty obvious doesn't it.

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Relating the gradient of a relativistic scalar particle wave function and the sum of their antiderivatives.

(Derived from the Klein-Gordon equations)

>>7651585
i know what the beauty behind that is. but it seems amazing to me that the the imaginary power of an imaginary number is real

>>7652032
Have you ever seen a proof in your life?

i basically like anything thats very maths related. i dont like any maths equations relating to reality and physics. Sorry physic bros.

>>7651852
(e^(i*pi))/11.999... = -1/12

x=1

>>7651307
I can't remember where I read it but some of his stuff has been proven wrong, but that's only a few out of a thousand theorems to be proven true

>>7645235
5*
:^)

>>7646398
>>7650174
>>7651852
hello, college freshmen, welcome to /sci/

>>7645099
KISS

F(x1+x2)=x1+x2

/thread

>>7654367
*fuck that is stated wrongly

but superposition

>>7645109
From the theory of modular forms. Ramanujan gives this formula along with a few other examples in one of his papers. In typical Ramanujan style he gives no proof for any of them.

>>7649654
Bro, calm down. You're arguing with a fegit. Just let him shitpost because he hates his life working at McDonalds full time.

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>While on his death bed, the brilliant Indian mathematician Srinivasa Ramanujan cryptically wrote down functions he said came to him in dreams, with a hunch about how they behaved. Now 100 years later, researchers say they've proved he was right.
>Ramanujan believed that 17 new functions he discovered were "mock modular forms" that looked like theta functions when written out as an infinte sum (their coefficients get large in the same way), but weren't super-symmetric. Ramanujan, a devout Hindu, thought these patterns were revealed to him by the goddess Namagiri.

>>7649654
lolol

>>7645099
>if a statement is true, it can't be proved

kek math

>>7654754
I lost my social security number and couldnt get my passport renewed without it. I had been out of the states for almost 2 decades and it was as good as gone.

Stressed out the better part of a week tearing my shit apart trying to find it or proof of it or anything, nothing. Cant remember it to save my life.

Fall asleep one night, in my dream I realize my social security card is in my wallet. Take it out of my pocket, pull out the card, and read the number.

IMMEDIATELY I wake up. Vividly recall the number, write it down on my dresser, fuck paper. Get online, check it. Shit actually works, its mine.

MFW

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>>7654787
this with the keys to my house
>mfw i feel like rain man

>>7654754
/sci/, how would you rationalize that complex mathematical truths came to Ramanujan in his dreams out of nowhere and only the interpretation was left for himself to do?

>>7655058
What is there to "rationalize"?

>>7655059
It's a very unusual phenomenon and surely it's not a coincidence that those mock modular forms that appeared to him turned out to be the real deal.
Does everyone's brain use modular forms constantly, but only Indian savants can see this in their psychedelic dreams on their deathbed?

>>7655094
I see, I thought I might be misinterpreting you.

I think the brain is doing far more, and using means to do it, than we have direct access to or awareness of. I have visual snow for example, and although it's only happened twice and requires severe physical exhaustion and prolonged sleep deprivation, I've closed my eyes and finally saw the pattern. It wasn't so much my field of vision as it was my mind's eye, something more on the mental plane. I saw a very complex semi-symmetrical pattern, and it was both rotating and constantly changing. There was some grand sense of balance and order to it. And I know with near certainty I can't consciously generate something like that wilfully, or perform those sorts of spatial transforms.

I'll also see other types of cyclical geometry occasionally. Chunks coming together to form 3d shapes that layer on top of each other into a structure, then dissolve only for it to loop again. Billions of red lights within objects swirling around, but it isn't random. It isn't strictly context based either, there are patterns to be resolved.

It's not quite the same, but what you said made me think of this. It leads me to wonder if this is happening all the time, and it's not a hallucination so much as a temporary bridging. Spillover where you have access to the backend functions.

I've also had dreams where I've designed machines. I still have the schematics for these goggles I made packed into my head, though I wrote them down just in case. They would work, though I don't have the means to actually make them.

Don't know. Dreams are a curious thing, and the mind is likely doing far, far, far more than people may or even could imagine. Perhaps in his case, he had just the right arrangement of functions for it to all occasionally align.

>>7646322
my sides

>>7646322
>Jacob Barnett
>14 year old autist
>11.9999....=12
keking

>>7646134
lol, the beauty lies in the fact that from the "cluster fuck" a simple, reciprocal of Pi emerges.

>>7654816
>that picture
ouch my sides

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>>7651373
The [math]\pi[/math] in the equation is only arbitrary, though. You could just as well write it as
[math]e^{i \cdot 180 °} = -1[/math].

>>7647486
What's up wot Dat quiz 4 brah

>>7645249
9/11 = .8181818181818181
A is 1st letter of the alphabet, H is 8th.
9/11 = hahahahahahahaha

>>7654754
Is it possible that Ramanujan's brain had developed an autonomous way of rendering these "functions"? Perhaps "forms" is better put.
Since he had the kind of intuition about the functions that he did, surely this must have been correlated with specific geometric imagery and/or geometric synaesthesia.
Isn't it almost every time you ask those autistic savant supercalculators how they're viewing the process, they just see the answer. They must have some sort of autonomous geometry mill in their brain that does the work for them. And a very natural learned association with what they're "seeing", this must necessarily be in close relation to the linguistic centers of the brain, no?
>hear/see/decide/etc what to calculate
>some moments
>ding, the brain's done, now just express the answer

>>7645234
Its in the beauty

>>7651647
Oh no. Dont bring out quadratics. The /sci/entists can't handle that much

I fucking hated my precal professor because he would always bring up how beautiful equations are in the most toolish manner. I hope I never become a pretentious shit like that. This thread sucks.

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