What's the most surprising math result?

420 = weed lmao

>>8493091
some infinities are bigger than other infinities
okay hazel grace?

>>8493091
That categories are a lot more useful than just as organizational formalism.

>>8493100
>>8493101
Apparently Cantor is a master at this

e^(i*pi)=-1

>>8493091
Euler's Identity:
e+2pi i=-1+0

>>8493091
Every symmetry leads to a conservation law.

>>8493091
1+2+3+...=-1/12 ayy imao

>>8493091
.999... != 1

>>8493130
What do I need to know to get noethers theorem.

>>8493150
Group theory?
I'm not sure, I just apply it, haven't proved it

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>>8493091
Euler's identity is def the coolest, but there's other unexpected ones too.

Like the golden ratio. It's the ratio of sides of a rectange such that the rectangle can be divided into a square and a rectangle with the same ratio as the larger rectangle. This can therefore be repeated infintely.

That's only mildly interesting. More interesting is that the golden ratio appears in all kinds of other geometry as well. On a regular pentagon where each side's length is 1, the distance between opposite angles is the golden ratio. Weird that a number involving rectangles also extends to pentagons. It's also related to the Fibonacci sequence.

Unfortunately this cool number has been kidnapped and raped by new age hippies because unlike pi or e, it isn't taught in schools so they are left with no education about it and ascribe it all kinds of retarded magical voodoo powers.

>>8493150
Nothing. Just read the theorem and then look up any words you don't know. Repeat until you arrive at something you do know.

>>8493177
what kind of "magical voodoo powers"?

>>8493177
> still excited about [math] \phi [/math]
> hasn't yet seen the wonders of [math] \digamma [/math]

>>8493091
>What's the most surprising math result?
to be really honest: none.
im not boasting its just that every math result that initially looked very strange turned out to be kind of not surpring after letting it sink in for a good amount of time.

harmonics are divergent

probably one of the better none meme proofs

>>8493188
Anything a new age hippy would fall for. Go onto youtube and look up "golden ratio frequency", that shit is endless.
>Cure any disease
>Alleve you of worldly needs (yes, some of them think that they can live without food and water with their voodoo- though I'd love to see them stope eating a drinking permanently)
>Cause eternal happiness and elightenment
>Find inner peace
>Ascend to a higher state of being
>Become a prophet of Ba'al the Destroyer
>Leave the matrix
>Wake up surrounded by hookers and blow
>Score some spaghetti
The multiverse is the limit, man

>>8493091
[math]sin(\pi x) = \frac{\pi x}{(x!)(-x!)}[/math]

>>8493205
That's a disease we need to stop

>>8493193
I'm interested, please elaborate

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The fundamental theorem of calculus

>tfw only in calc 2

>>8493210
There actually is one frequency that can directly alter your health. It needs to be made loud enough that a certain part of your heart resonates at it. Once that happens, it stops signalling to your brain stem and continues beating with the frequency. When the frequency is taken off, your heart stops beating and you die.

Yes, this really exists.
I should make one of those binaural beats videos of that. That'd solve it.

>>8493229
So you're telling me that if you took the frequency off, you would die?

>>8493228
I think so. Instead other surpirsing theorems i read in this post, it really changed math hystory.

>>8493233
Yes. How I described it may not be entirely accurate, but it boils down to a specific sound frequency that overrides your heart's clock so that when the frequency shuts down, the heart does too.

Rare azaghal King of oceania
nice pic OP

>>8493242
The rarest of them all

>>8493240
Even if you were a big guy?

>>8493091

There are lots of surprising math results. Often, surprising results are not merely surprising, but on top of that, /useful/ toward further math results. These are the ones that become famous to the point of being memes, and these are the ones that real mathematicians love.

One particuarly surprising math result to me, personally, was that

[eqn] \int\limits_{0}^{ \pi} sin(x) \; dx = 2 [/eqn]

The object lesson here does not matter exactly which definite integral is considered, as long as its graph has some subtle curvature bounding a non-obvious area. And yes, the above area so bounded is non-obvious.

This is personally an item that brought it home for me, as I learned integration (and my personal interest, the geometric interpretation of same) for the first time: going forward, it would often happen while integrating over the requisite elementary/rational/"nice" functions, that the result from such-and-such to such-and-such would be a completely straightforward rational number of some kind, perhaps with a pi or an e sprinkled in. but very nice, compact results. That this is true for integration over elementary functions, /which yet have complex curves for their graphs/, really impressed me and continues to do so.

Brouwer's fixed point theorem is also quite surprising.

Lagrange's theorem is also slightly surprising.

0=0.9999!

>>8493207
this looks like bullshit
proof?

>>8493207
>>8493266
You bet that's shit! RHS would be complex

>>8493125
e^(tau*i)=1

>>8493255
for you

>>8493279
Or more obviously [eqn] \sin ( \pi x ) = \frac { \pi x } { (x! ) (-x!) } \\ \implies \sin ( \pi ) = \pi [/eqn]

>>8493433
provided that you know the value of (-1)!

>>8493279
https://en.wikipedia.org/w/index.php?title=Euler%27s_reflection_formula

The proof is combining these two formulas
https://en.wikipedia.org/wiki/Gamma_function#Weierstrass.27s_definition
https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem#Examples_of_factorization

>>8493484
I'm sorry, you're right.

>>8493195
This this this.

Whenever you think a math result is surprising it's probably just because you haven't understood the background properly. Once you are over that, everything will appear quite straight-forward.

Anyway, most results from complex analysis were pretty baffling to me at first.

>>8493177
I once had a task where me and my work partner had to divide something by 0.6, but my partner insisted that it's multiplying by 1.6 (if your original amount is 60% of x, what is x?). The result for multiplying by 1.6 was close though (but not precisely so) and then he said "see, it's no difference, it's the same". I figured that there must be some kind of point where 1+x = 1/x and accidentally found the result for -1.618.. and 0.618, it was related to the golden ratio phi.

>>8493504
yeah the cauchy integral formula blew my mind when i first saw it

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For me it's the Lefschetz principle. It allows to use topological methods (in the euclidean sense) to study algebraic varieties over an arbitrary algebraic closed field.

>>8493197

But this is totally trivial, pic related.

>>8493239
The theorem is a corollary of the concepts of derivative and integral. It's an importante result but I think that is to simple to make a historical distinction between it and the developing of the general theory of calculus.

>>8493091
4.0(perfect score!)

For me, the most surprising results when I first learned them were the Dold-Kan correspondence (connective chain complexes in an Abelian category are equivalent to simplicial abelian groups there), Tannaka duality (the automorphisms of the forgetful functor from a category of modules over some algebra arrange themselves into that algebra), Pontrjagin duality (discrete abelian groups are dual to topological abelian groups), the geometric Langlands correspondence (number fields behave a lot like function fields, geometrically), arithmetic topology (links in the 3-sphere behave much like ideals in Q), and the Goodwillie calculus (pointed homotopy types behave quite similarly to formal power series).

>>8493514
nifty!

>>8493229
there are multiple neuron systems that each generate particular rhythms so that one won't go as easily m7

also this video is balllllllls
https://www.youtube.com/watch?v=ZyFRHW9OnHY

>>8494015

>goodwillie calculus

lal, just say calculus of functors

though, functor is still a stupid word...

>>8494816
Yeah, Goodwillie calculus is also a bit ambiguous, as it could also refer to the version pertaining to manifolds. Nonetheless, I really don't care for "x of y" names, like calculus of variations.

>>8493975
What a beautiful proof. We proved it with sequences, but this one, truly elegant.

[eqn]\sum_{n=0}^{\infty}2^n=-1[/eqn]

>>8493177
>His number isn't even trascendental

You call it phi because (1+√5)/2 makes it too obvious that it is a useless lame number?

>>8493188
『magical voodoo powers』

Liouville's theorem is both surprising and useful

>>8494015
why the fuck do you even post this shit

>>8493100
I love how reals are more infinite than rationals

>>8494842
I assume that by 'elegant' you mean wrong.

>>8494910
Please elaborate?

>>8494925
literally everything you post is just a flood of namedropping jargon because you think if you talk like a published paper people will think you're smart

I can guarantee you there are <5 people here who even understand more than 30-40% of your posts and they can all see what you're doing

>>8493975
This proof isn't valid. Sums aren't just convergent or divergent and nothing else, the infinite sum of (-1)^n is neither, in your proof you assumed that the infinite sum of 1/n is either convergent or divergent, which isn't valid.

>>8493091
that humans are to dumb to invent an axiomatic system without paradoxes

>>8495000
that guys pic shows it is not convergent

would stating it is monotone/strictly increasing be sufficient for it to be divergent?

>>8495060
Yes. The sequence of partial sums is non- decreasing, so it either diverges to infinity or converges

When i learned that a positive number minus a negative number is the same as addition. Shit's crazy man.

>>8493133
>>>/wsr/
>>>/hm/

>>8493091
Quaternions because they are so pretty

>>8494932
OP asked for the most surprising results to us, and I gave my answer. Plus, none of those are things someone needs to be particularly advanced to know; people start to learn Pontrjagin duality in basic group theory, and Dold-Kan is a widely-used tool in homological algebra (I'm guessing this is touched on by most early graduate students). Goodwillie calculus and the MKR analogy are perhaps more obscure, but that shouldn't matter. I answered this question sincerely, and I figure maybe others will have their interests piqued as well.

I don't come here to jerk off and get attention, but I guess I can't say anything to convince you of that.

>>8495142
don't worry bro i sort of understand you
though the pontryagin duality makes me want to vomit desu senpai

>>8493228
>the sum of the differences between numbers in a sequence gives you the numbers of the sequence

This is obvious though. Discovered it all by myself even before taking calc.

>>8495103
How do I understand quaternions. It doesn't seem like they follow naturally frombthe usual description of complex numbers. I read something about how multiplying by a quaternion actually represents a 180 degree rotation rather than a 90 degree and can't wrap my head around where this comes from.

>>8493091

Coexistent soundness and completeness of axiomatic systems of abstract logic; the resulting implications of the fact that the ontological statement of God's existence has been shown to be valid.

1+2+3+4+5+... = -1/12

111111111*111111111=12345678987654321

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>>8493091
i^i being real?
>>8495255
Ebin mene my dude, check this
S=1+1+1+1+...=1+(1+1+1+1+...)=1+S=>S-S=1=>0=1 xDdDDD

>>8495000
Fucking retard, it's wrong just because the algebraic manipulation in the first equality is wrong

>>8493091
idk but the least suprising one is when people just give up

>>8493133
>factorial(0,9999)=1

>>8495772
what are significant figure?

>>8493098
>>8493131
>>8493133
>>8493229

ayy lmao

>>8493091
Math here is trivial, it's the result that surprises.

1) tie a string around a tennis ball
2) add length to the string so its distance from the surface is, say, 10 cm
3) call the extra string length X

Now do the same with something huge like the earth's orbit.

Same X.

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>>8494015
>links in the 3-sphere behave much like ideals in Q
you want ideals in Z here, or else links would be pretty boring

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>>8494932
believe it or not, not everyone on this board is a brainlet like you, and that's hardly 'talking like a published paper'

i'm familiar, to varying degrees, with dold-kan correspondence, pontryagin duality, arithmetic topology and geometric langlands and i'm barely out of my undergrad

>>8493125
e^(i*0)=1
really makes you think, huh

>>8493193
isn't that that old joke where F=1

>>8495835
I apologize, I refer to ideals in a field's ring of algebraic integers as ideals in the field. [math] \mathcal{O}_{\mathbb{Q}}\simeq \mathbb{Z} [/math], so you are of course correct. Viewing it as the ring of algebraic integers is important since [math] \mathbb{Q} [/math] has algebraic properties which correspond to topological properties of [math] S^{3} [/math].

>>8493091


That your moms vagina has a capacity for 8 square inches of my dick lmao.

>>8493100


>some transfinites are bigger than other transfinites

Ftfy

>>8496193
>transfinites
Fuck this PC bullshit, they're either born finite or they're not. You don't just get to choose.

>>8497584
jej

>>8493091
for n≠4, there is only one smooth structure on R^n up to diffeomorphism.

For n=4, there are uncountably many non-diffeomorphic smooth structures on R^n.

>>8497767
No fucking way

>>8497770
yes fucking way https://en.wikipedia.org/wiki/Exotic_R4

Any field that is complete with respect to an archimedean absolute value is isomorphic to either the real or the complex numbers.

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>>8493188
listen johnny. You have to SPIN it.
With a "golden rotation". And while riding a horse; that way your GOLDEN SPIN will be powerful enough to hax trough infinite dimensions because muh gravity and muh golden ratio = infinite energy.

Ok johnny? Trust me bro, my family is a secret organization of italian executioners with balls of steel and has been doing this for generations now.

The Riemann rearrangement theorem: A conditionally convergent series can be rearranged to make it converge to any real number, and can even be rearranged to a divergent series

>>8493091
Banach Tarski

>>8493130
at least if it's a continuous symmetry

>>8493167
Lol, group theory and QFT are the magical places where results come from

>>8493091
TREE(3)

>>8495663
>S=1+1+1+1+...=1+(1+1+1+1+...)=1+S=>S-S=1=>0=1 xDdDDD

MY TEACHER LIED TO ME

>>8493133
What

theres a bijection between the set of seventuples of trees and the set of trees

>>8493975
Please speak plain plebbian, the what on the what???

>>8498238
Thatt isn't really surprising when you read the proof. Its actually pretty straightforward.

>>8498738
it's surprising because (as far as I remember) until you see it, there's no other results that suggest the order of the sum really matters

>>8498238
Was just about to post this.

>>8498741
There are a billion counter examples. It's just that most series you work with are absolutely convergent so it doesn't matter. However it's pretty neat though.

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what is the most surprising math result?......
good analysis of the causes and consequences of this occurrence :)

>>8495000
The proof is false, but it's not false because of what you said. The sum from n=1 to N of 1/n is increasing with N, so it either converges, or diverges towards infinity. THe reason the proof is wrong is because taking a limit does not conserve strict inequalities.

[math]\frac{\mathrm{d} }{\mathrm{d} x}e^x = e^x[/math]

>>8493091
Personally, I find surprising the existence of exotic differential structures on R^4.

>>8499315
This isn't really surprising in the least when you look at e's power series.

>>8496170
>>8493221

It's a digamma not an F.
https://www.youtube.com/watch?v=GFLkou8NvJo

>>8499667
but dont you need to know the derivative to find its power series first? like, ist that what a power series derived from?
( not same anon, but still) I remember you infinitely take its derivative at a point and add it together, so this result is rather intresting, only 1 other functnon has this result, of all the infinite functions