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/sci/ - Science & Math

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4667077 No.4667077 [Reply] [Original]

Infinite series in general
Fermat's last theorem
The fundamental theorem of arithmetic (prime factorization)

>> No.4667082

Riemann Zeta Function

>> No.4667092

Galois criterion

>> No.4667094
File: 13 KB, 268x268, my-brain-is-full-of-fuck.jpg [View same] [iqdb] [saucenao] [google]
4667094

<span class="math">\frac{\sqrt{2}}{2}[/spoiler] being the same as <span class="math">\frac{1}{\sqrt{2}}[/spoiler]

>> No.4667098

Banach-Tarski, Zorn's lemma

>> No.4667104

Dichotomy paradox and Achilles and the tortoise.

>> No.4667103
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4667103

this

>> No.4667109

most obvious one is euler's identity

>> No.4667120

derivative of e^x
and generally the usefulness of calculus

>> No.4667122

(1/2)! until my professor showed us the gamma function. Still impressive I guess since no one before that had ever said anything about it besides "just multiply all the previous blah blah. It sure gets big fast huh kids"

>> No.4667129

Using the Riemann sphere to integrate real functions with singularities at x=0. Seriously, to me, this is the most incredible thing I've learnt in mathematics.

>> No.4667503

Banach-Tarski Theorem
<span class="math">e^{i \pi} = -1[/spoiler]
In a discrete metric space, for <span class="math">0 < \epsilon \seq \frac{1}{2}[/spoiler] we have <span class="math">B_{\epsilon}(x) = \{x\}[/spoiler]

That last one might seem trivial to most of you, but as someone who's only just started doing topology, thinking about non-Euclidean metric spaces is hilarious.

By the way, is the LaTeX preview script no longer working due to some kind of update?

>> No.4667504

>>4667503

>Uncontrolled blah blah

That was supposed to say <span class="math"> 0 < \epsilon \leq \frac{1}{2}[/spoiler]

>Hope it came out right this time.

>> No.4667507

fundamental theorem of calculus

irrational numbers existing

cantors diagonal argument

construction of the reals

>> No.4667509

>>4667504
epsilon strictly less than 1 is enough

>> No.4667512

I remember back in highschool e^x being it's own derivative through my brain for a swirly

>> No.4667519
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4667519

>>4667509

Oh, you're right. Still, my mind's blown, because whenever I think about it I imagine drawing a circle with a non-zero, <1 radius, but there's nothing else inside the circle except for its center. Not a single other coordinate... so does the circle still have a radius, then?

>> No.4667525
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4667525

Cauchy's residue theorem.

The topological version of the Galois correspondence still blows my mind.

<span class="math">\sum_1^\infty \frac{1}{n^2} = \pi^2 / 6[/spoiler]

>> No.4667538

http://en.wikipedia.org/wiki/Borwein_integral

>> No.4667539
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4667539

>mfw I understood how the Mandelbrot set is plotted

>> No.4667548

0.999... <= 1

>> No.4667549

I'm only a first year pleb, but finally realizing what the "you can just derive it from first principles" fags mean when they say that. For instance, being able to derive the definition of a composite number by negating the definition of a prime number blew my mind at the time, as it made me realize I didn't have to remember the definition of composite from the book, as I could do it quickly in my head through logic.

>> No.4667550

>Yoneda lemma

Why haven't I come up with this?!

>Derived functors

Holy fuck it actually all pans out...oooh that universal coefficient theorem

>Poincaré–Hopf theorem and other relations between analysis and topology

WHY

>Knot genus behaves well w.r.t. connected sums

Sooooo pretty

>Acyclic models

FUCK YEAH

>> No.4667551

>>4667539
This. It made me cry.

>> No.4667554

>>4667519
in taxicab metric a ball is a square

>> No.4667558

>>4667550
http://en.wikipedia.org/wiki/Derived_functor
teh fuck man???

>> No.4667560

>>4667539
mfw proof that mandelbrot set is comoact subset of C is easy, but that it's connected is very difficult

>> No.4667565

cmon everyone admit it......

a2 + b2 = c2

>> No.4667568

http://www.relativitybook.com/CoolStuff/julia_set_4d.html
>headexplode.gif

>> No.4667572

deriving ax2 + bx + c = 0

idgaf

>> No.4667574

Incompleteness theorem by far is the most profound insight ever made.

>> No.4667600

Riemann re-ordering theorem
Cantor's set

>> No.4667676

can't remember the name of this function, but it was defined like that:
<span class="math"> f:]0,1]\mapsto [0,1] [/spoiler]
<span class="math">if \ x=\frac{p}{q} \ \ \ p,q\in \mathbb{N}\ and \ gcd(p,q)=1 \ \ f(x)=\frac{1}{q}[/spoiler]
<span class="math"> else \ \ f(x)=0[/spoiler]

this function is continuous on the irrational numbers but discontinuous on the rational numbers.

>> No.4667701

>>4667676
are you sure? it doesn't seem possible to me

>> No.4667717

>>4667701
not same anon
but it's a well known function

http://en.wikipedia.org/wiki/Thomae%27s_function

>> No.4667745

>>4667701
yeah, basically it works because a rational number P/Q , with gcd(P,Q)=1 , "close" to an irrational number have a very big P and Q
So the closer you get to an irrational , the closer to 0 f(x) go.
you can verify by yourself the proof is not very hard. I used the fact that every rational numbers admit a cycle (of length < Q ) in the decimal representation. and an irrational doesn't admit any cycle in the decimal representation.

>> No.4667791

eulers` shit.

>> No.4667840

>>4667503
i thought that e^ipi was DeMoivre?

>> No.4667859

e^ix=cosx+isinx

>> No.4667863

(sinx)/n=6

>> No.4667865

Kayley's Theorem of groups

The fact that every group is really, at its core, a matrix (and vice versa) really made Algebra much more intuitive for me.

>> No.4667889

Probably already done in the thread but:
e^(i*pi)=-1

Woahwoah nigger, you tellin me that you get an irrational number , to the power of FUCKING PI of all numbers times by a number THAT DOESN'T EVEN EXIST and from all this you get MINUS ONE

>> No.4667904

>>4667865

At least spell it correctly, man.
>Cayley

And yea, that theorem has much better value for me now that I know more about algebra. It's application to group classification is absolutely invaluable.

>> No.4667953

>>4667904

Yeah, i have a bit of dyslexia so hard C's I often write as K's.

It really is a wonderful theorem, and it is often overlooked as trivial, when in fact it's quite deep and far reaching.

>> No.4667976

>>4667953
you know it is pronounced Sayley?

>> No.4667984

>>4667865
What?! How does Cayley's theorem even suggest that every group is a matrix?!! This sounds like nonsense to me.

>> No.4668007

So all you people saying euler's identity, are you just saying you're amazed how seemingly disparate constants can come together without actually understanding it? (the very first time you saw it I mean)

Or were you actually able to visualize e^ipi going in a semi circle in the complex plane and then going back to zero after adding one?

I remember Gauss saying that you'll never be a great mathematician if you can't intuitively picture euler's identity the first time you see it.
;_;

>> No.4668016

>>4668007
>The mathematician Carl Friedrich Gauss was reported to have commented that if this formula was not immediately apparent to a student upon being told it, that student would never be a first-class mathematician

This is bullshit, the first time I was told this I didn't even know what a number to an imaginary power means.

>> No.4668021

>>4667976

No, it's a hard C, not a soft one. You're thinking Sylow.

>>4667984

Here's how it suggests that every group can be represented as a matrix:

Cayley Theorem states that *every* group is isomorphic to a subset of the symmetric(permutation) group acting on G.

The permutations group has a very natural representation in NxN matricies.

Therefore, after you perform an appropriate isomorphism, any group can be represented as an NxN invertible matrix, as a result of the Cayley theorem.

Pretty neat huh?

>> No.4668024

>>4668021
lol i didnt even understand a word you said you fucking nerd faggot

get a life dweeb

>> No.4668036

>>4668007
Gauss supposedly said that you should understand it immediately when you see it, but that's only sourced from one single book. Basically, it makes sense from the taylor series expansions of sine and cosine, but there isn't any further intuition than that; imaginary powers don't make any sense in terms of real powers so you have to compute them analytically.

>> No.4668044
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4668044

>>4668024

>> No.4668048

How to derive polynomials

>> No.4668060

>>4668036
I thought part of the intuition is that multiplying by i is going in a counter-clockwise circle in the complex plane?

>> No.4668066

>>4668036
Visual Complex Analysis gave a pretty intuitive geometric derivation of taylor series

>> No.4668076

>>4667865
>>4667904
>>4667953
Cayley's theorem IS trivial and it isn't anywhere near deep! Maybe you're confusing it with the Cayley-Hamilton theorem which is often overlooked and incredibly useful (e.g. in algebraic geometry through the Nakayama lemma), but Cayley's theorem is about as deep as the Yoneda lemma (which is to say not at all) and not even used that much.

>> No.4668082

>>4668021
>The permutations group has a very natural representation in NxN matricies.
How? I don't see this being "natural" in any sensible way, and additionally it surely doesn't make sense for infinite groups. This is the first time I hear about someone thinking of a group as a matrix and I'm pretty damn sure that it's neither helpful nor canonical in any way. Please elaborate or stop thinking of groups as matrices.

>> No.4668083

The fact that there are no irrational or transcendental numbers in the base e numbering system.

>> No.4668084

generalized stokes
eulers formula
projective geometry
hopf fibration
that ramanujan series for pi

>> No.4668087

>>4668076

/shrug

If you associate a strong theorem with being deep or profound, you wont agree with my statement.

If you associate a far reaching, fundamental, and all inclusive theorem with being deep, you will agree.

Difference of opinion honestly.

>> No.4668089

>>4668083
Yeah that blew my mind as well.

>> No.4668092

>>4668082

http://en.wikipedia.org/wiki/Generalized_permutation_matrix

Look under the Subgroups tab, first entry. That is your natural group representation of the symmetric(permutation) group, which every group is isomorphic to.

>> No.4668093

>>4668087
>If you associate a far reaching, fundamental, and all inclusive theorem with being deep, you will agree.
Precisely not! Cayley's theorem might be fundamental in the way it's alsmot unavoidable and it might be very generous in it's assumptions making if "all inclusive", but it's in no practical way far reaching! To get any further interesting results from Cayley's theorem you have to apply a lot of additional techniques and knowledge, that sounds like the complete opposite of "far reaching" to me.

>> No.4668095

No one else was surprised that the NSA knew P = NP all this time and kept it a secret?

I mean the fact that P = NP, and not that the government kept secrets of course.

>> No.4668097

>>4668093

Perhaps I used the wrong word in far reaching. I meant that it applies to every group, which I suppose would belong more under the "all-inclusive" heading. My mistake.

>> No.4668100

>>4668092
Sure there is an isomorphism but it's in no way "natural" as you claimed. And as I already said it doesn't make much sense for infinite groups. In any case, I was right and thinking of groups as matrices as you described is neither useful nor enlightening. Every finite dimensional vector space is isomorphic to the appropriate R^n, that doesn't mean I'll replace every instance of a finite dimensional vector space with the corresponding Euclidean one! Soon enough you'll learn what it means for something to be natural and why this is really important.

>> No.4668102

>>4668100
But that's fucking wrong you idiot.

>> No.4668117

>>4668095

What? Source on his?

>> No.4668119

>>4668116
>gets told

>hurr durr isomorph dur eculidean

go kill yourself

>> No.4668116

>>4668102
Elaborate or remain silent.

>> No.4668123

>>4668117
wikileaks

>> No.4668126

>>4668100

Your statement regarding being "natural" was in regard to my "natural" representation of the symmetric group, not to the isomorphism. The isomorphism can be rather difficult, but that's not the spirit of the Cayley theorem.

There is, in fact, a very natural representation of the symmetric group. That fact is indisputable.

Your later statements are rather nonsensical.

>> No.4668128

>>4668117
It was leaked in Wikileaks last week. Unfortunately it could not be verified in polynomial time

>> No.4668139

>>4668119

He learned something and he's too embarrassed to admit it.

>> No.4668149

>>4668126
I will have to reiterate because there seems to be a lot of confusion:

The isomorphism that the Cayley theorem provides is what one would call "natural". Every element of the group is sent to the automorphism given by multiplying with that element.

The isomorphism between the symmetric group acting on N elements and the group of NxN permuation matrices is in no way natural: it depends on an ordering of the N elements! Furthermore, for an arbitrary group G, this would only yield a (non-natural) isomorphism between G and a subgroup of some group of permutation matrices. You on the other hand claimed that every group could be identified with a single matrix and I don't see this happening in any natural way nor being helpful. And here's the quote in case you're wondering:
>>4668021
>any group can be represented as an NxN invertible matrix

>> No.4668236
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4668236

That 0! = 1

>> No.4668252

>>4668126
Do you mean natural in the categorical sense?

>> No.4668251

>>4667574

Profound insight into the lack of insight :P

>> No.4668262

Euler's identity for sure.

>> No.4668275

Induction. Simple but infinitely powerfull. Its fucking magic.

>> No.4668291

>>4668149
>automorphism given by multiplying with that element.
Elements are not sent to an automorphism, for example <span class="math"> g \cdot 1 \not = 1[/spoiler]. Its just a permutation representation.

>> No.4668294

sin^2(x)+cos^2(x)=1 can be rearranged to get pythagoraas theorem

>> No.4668298

>>4668291
I meant automorphisms in the category of sets ;)

>> No.4668299

Hell, it might have been a better idea to ask "what /isn't/ mind blowing in mathematics?". If you think about it, almost everything is amazing about the whole science.

>> No.4668318

BBP formula for pi. generates digits of pi in base 16. "what?" and yet we still don't know if digits of pi are normally distributed in base 10!

<div class="math">\pi=\sum_{k=0}^{\infty}\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac{1}{8k+5}-\frac{1}{8
k+6}\right)</div>

>> No.4668320

>>4668318
>we still don't know if digits of pi are normally distributed in base 10
We know they aren't. They are degenerate distributed.

>> No.4669208

>>4668128
>>4668123
link?

>> No.4669383

Crazy-ass triple integrals:
<div class="math">\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\pi}\frac{\mathbf{d}u \mathbf{d}v \mathbf{d}w}{3-\cos u-\cos v-\cos w}=\frac{\sqrt{6}}{96}\Gamma \right ( \frac{1}{24}\left ) \Gamma \right ( \frac{5}{24}\left ) \Gamma \right ( \frac{7}{24}\left ) \Gamma \right ( \frac{11}{24}\left )</div>