What are some of the most surprising mathematical facts or theorems you know?
Pic related.
What's surprising and what's not really depends on what you are comfortable with. For example the banach-tarski thing looks wierd as hell but it loses its luster once you study some simpler examples of the axiom of choice in action so you can get an idea of what's going on.
>>7919030
that an n-sphere's volume peaks when n=3 and then approaches 0 as the sphere moves into higher dimensions
>>7919045
Let's put it as surprising when you first learn about it.
>>7919030
2+2=4
EYPHKA! num = Δ+Δ+Δ
>learn about tensor products
>go to take [math]\mathbb Z /m \otimes_{\mathbb Z} \mathbb Z/n[/math] for the first time
>collapses to 0 when m and n are coprime
>>7919049
that is fucked up
does that mean that having 3 dimensions, R^3 isn't just special in what happens to be our physical universe, but is necessarily special even in mathematics.
And the fact that green's theorem only makes sense in R^3.
Why would R^3 happen to be more specia than R ^4 or R^5 or any other one?
that's fucked up right?
>>7919030
there exist a continuous curve that can fill the square [0;1]x[0;1]
>>7919123
piecewise continuous? in which case it is not surprising whatsoever
>>7919126
No, what he said is correct.
Tarski's undefinability theorem.
But maybe it is because I don't really understand it.
>>7919126
no there exist continuous surjection of [0;1]->[0;1]x[0;1]
>>7919049
This is a good one.
How do you compare the volumes when they have different units? Drop the r^n?
>>7919144
measure theory ?
>>7919030
1+2+3+4+5+6+7+8.....=-1/12
https://www.youtube.com/watch?v=w-I6XTVZXww
>>7919151
that's not an answer
curious about how things of different dimensions are compared.
>>7919187
It's a great answer. We assign a real number to every set and compare those real numbers. There are no units whatsoever.
Every prime which is 1 mod 4 can be written as the sum of two squares.
>>7919184
He keeps saying "all the way up to infinity" as though that were a number.
>>7919184
>numberphile
It's best not to put much trust on that channel
>>7919144
you don't really need to drop the r^n. I mean you can -- just make it a unit ball and r^n will always be 1 anyway. Wouldn't matter either way because the factorial in the denominator grows much faster than exponentials anyway.
>>7919194
A related surprise is that the ring of integers of [math]\mathbb Q[\sqrt d][/math] is [math]\mathbb Z[\sqrt d][/math], unless d is equivalent to 1 mod 4, in which case it is [math]\mathbb Z[\frac {1 + \sqrt d}{2}[/math].
>>7919202
>take number theory exam yesterday
>let d be equivalent to 1 mod 4
>work in [math]\mathbb{Q}(\sqrt{-d})[/math]
>d is equivalent to 1 mod 4 so my ring of integers is [math]\mathbb{Z}[\frac{1+\sqrt{-d}{2}][/math]
>realize something is wrong
>Takes me 5 minutes to realize d = 1 mod 4 so -d = 3 mod 4
I feel stupid now.
>>7919030
Oh yes. The continuous but not derivable function
. If a polynomial function from [math]\mathbb C^n \to \mathbb C^n[/math] is one-to-one, then it is onto. (actually, the proof is even more surprising than the result)
. (Found in a thread here a while ago, proof that /sci/ is not a lost cause): C* and U = {z in C | |z|=1} are isomorphic as groups.
. The central limit theorem. It is a very remarkable result. Not difficult, but remarkable.
A series that converges non-absolutely, such as [math] \displaystyle \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} [/math], can be rearranged to converge to anything.
For any real number r, you can add the terms of that series in a different order so that the rearranged series sums to r.
You can find an uncountable collection of subsets of [math]\mathbb{N}[/math], such that every two sets in the collection are comparable.
>>7919198
yes. ignore the trolls.
I'm no mathematician but the law of large numbers and the fact that picking a mean over and over from any distribution will give you a series that converge towards the normal has always amazed me.
I read somewhere that any closed 3-manifold is homeomorphic to a connected sum of spheres and toruses. Is that true?
>>7920426
Neither of those is a 3-manifold, so it can't be. If by "spheres" you mean projective planes, then that is true for closed 2-manifolds.
>>7920426
https://en.wikipedia.org/wiki/Surface#Classification_of_closed_surfaces
>>7919192
jesus fuck
>ITT a volume is bigger than a surface
might as well say i is bigger than 0.
>>7919030
Numberphile talked about a proof that the sum of all positive integers is -1/12 and its totally sound and it blows my mind every time i watch it. I keep a written proof in my wallet, because i think it's that cool
https://www.youtube.com/watch?v=w-I6XTVZXww
>>7920466
Just... what? How can positive numbers ever add up to a negative number
This is why I hate math.
>>7920460
i is bigger than 0 in the dictionary ordering
>>7920466
It's bullshit. I fucking hate this meme
>>7920460
We're comparing real numbers, though, so it's nothing like that.
The axiom of choice is equivalent to the well-ordering theorem.
Also, if the axiom of choice is false, there is a vector space without a basis.
>>7920568
>if the axiom of choice is false
It's provably not.
>>7920491
you can always scale the n-sphere to make its volume lower or greater than its surface.
>>7920488
you can't have taken that seriously, m8
>>7920580
I meant if it is taken to be false.
Also, it is not provable in ZF, but that does not mean it is not provable elsewhere.
I'm not a maths major or anything and I'm also pretty early on in my college career, currently in Calc II. We just started talking about infinite series and power series. I think the fact that difficult functions can be written as infinite polynomials is pretty amazing.
>>7919412
On Ax-Grothendieck, are you thinking of the proof from pure algebraic geometry, or from model theory? The model theoretic proof isn't really surprising in my opinion. It's one of those silly quick proofs that model theory does well (Prove theorem in a structure where it's trivially true, show that structure is somehow elementarily equivalent to a structure of actual interest.).
>>7919030
I was really surprised by Kunen's Inconsistency Theorem when I first learned it. It doesn't really seem at first glance like Reinhardt Cardinals are all that problematic, they seem pretty much like every other large cardinal, and then it turns out they're inconsistent.
>>7919184
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
>>7919030
>cosine Weierstrass functions
Pretty sure those are differentiable at x = 0
There is no general closed form expression for solutions to polynomials of degree 5 or higher.
>>7919049
It actually peaks at n=5.
Intuitively, you can think of the ratio between the volume of the sphere with unit radius and that of the unit n-box as decreasing as you move up in dimesion. That is, the sphere 'loses' more corners as the number of corners increase.
>>7921061
So, in n=1, the unit sphere is the unit box. In n=2, there are 4 unoccupied corners. In n, 2^(n-1).
Euler
>>7919030
Definitely the Gauss Bonet Theorem.
A manifold's total curvature is directly computable from its euler characteristic, which in turn is a topological invariant.
This implies that when you deform an object the increase or decrease in curvature at that point is made up somewhere else to preserve the overall curvature.
Obviously it is also interesting because it implies the total curvature is calculable from the number of topological holes.
shortest distance between 2 points is a line, i was like wow
![[Nutbladder]_Hidamari_Sketch_SP_-_01_[D0C77CD2].mkv_snapshot_03.16_[2015.06.18_22.49.41].jpg](https://i.imgur.com/x9LIxYx.jpg "[Nutbladder]_Hidamari_Sketch_SP_-_01_[D0C77CD2].mkv_snapshot_03.16_[2015.06.18_22.49.41].jpg")
>>7919096
>seeing the entirety of elementary number theory fall out of simple theorems in group theory when working with [math]\mathbb{Z}[/math]
>Chinese remainder theorem
>cyclic decomposition
>fundamental theorem of arithmetic
>Fermat's little theorem
>mfw I wasted an entire semester studying number theory when I could've learned all this shit in a single group theory lecture
>>7921168
if you learned number theory without knowing algebra, you didn't learn number theory
>>7920820
Well I did the proof in an algebraic geometry class but I think it might have been the model-theoretic proof in disguise.
Basically you write polynomial equations that express injectivity and non-surjectivity with the Nullstellensatz and construct a finite field on which these identities make sense, which gives the existence of an injective and non-surjective function from a finite set to itself, hence contradiction.
I thought that was really neat as we did not really do any work, just changed the context in which we looked at the equations and bam
>>7921187
Euler will be glad to hear that
Maxwell's addition to Ampere's Law is pretty sexy
\triangledown \times \textbf{B} = \left ( \mu _{0}\textbf{J} + \mu _{0}\epsilon _{0}\frac{\partial }{\partial t}\textbf{E} \right )
>>7921224
>you
[eqn]\triangledown \times \textbf{B} = \left ( \mu _{0}\textbf{J} + \mu _{0}\epsilon _{0}\frac{\partial }{\partial t}\textbf{E} \right )[/eqn]
>>7921227
Thanks anon... Do you have to preface and end the LaTeX with code as well?
>>7921230
Use [eqn] if you want it to be on a line by itself, and [math] if not.
[eqn]\oint_C {E \cdot d\ell = - \frac{d}{{dt}}} \int_S {B_n dA}[/eqn]
>>7921241
>mathematical theorem
>>7920777
Well, you know "any reasonable function" can be written as a series.
>>7921645
Bump functions are unreasonable?
>>7921637
It can be proven mathematically. The fact that you might be able to interpret it physically doesn't change that.
It's like saying that [math]\pi_3(G) = \mathbb{Z}[/math] for G a simple Lie group isn't a mathematical theorem because you can interpret it as a statement about the existence of large gauge transformations in a field theory with gauge group G.
>>7921688
No, Maxwell's equations are empirical. It is a mathematical description of a physical law, not a mathematical theorem.
The Curry-Howard correspondence blows my mind
>>7921711
https://en.wikipedia.org/wiki/Peano_curve
>>7921200
>hence contradiction.
>I thought that was really neat as we did not really do any work,
>proof by contradiction
>neat
>>7921702
>Maxwell's equations
are indeed a theorem proved by varying the action of EM.
>>7921725
Scientific theorems cannot be proven, only falsifiable.
>>7921736
the deductions to get the ME are valid in classical logic.
it hurts innit ?
>>7921748
The deductions are valid but it doesn't prove anything since it still relies on physical assumptions about the natural world.
This shit is so retarded, I am not biting anymore.
Take [math]y = \frac{1}{x}[/math] with the domain [math]x \geq 1[/math] and rotate it about the x-axis. The resulting solid will have infinite surface area but finite volume.
>>7919030
The Riemann Conditional Convergence theorem is pretty good but makes Barnett's identity pretty trivial.
>>7919030
There exists a group isomorphism between the complex numbers with zero removed under multiplication and the complex unit circle under multiplication.
>>7921130
Lmao
>>7921993
Yeah, circles with a point removed and lines are topologically equivalent. That's basically topology, even when you add the complex component.
>>7920488
Oh boy, babby's first analysis is going to suck for you when you learn the Riemann conditional convergence theorem :^)
>>7920777
Cosine can be written as an infinite sum of sine functions and vice versa.
Look up Fourier series for something even cooler than Taylor/Maclaurin series.
>>7921066
What happens if we use a different p-norm?
>>7922011
that's not what he said, but yea, a plane with a hole is a circle, that's still not surprising at all
>>7920620
yea, like in ZFC.
>>7921718
Based
>>7919030
C-differentiable complex functions are infinitely differentiable.
Still fucking shocking miracle after all those years...
>>7922011
Retard.
>>7922029
I took linear algebra before abstract algebra, so when I saw that a seemingly 2 dimensional and a 1 dimensional set were isomorphic, I was surprised.
>>7920488
Disproove incoming?
>>7921722
Be amish all you want, I chose technology (I also unapologetically use the axiom of choice)
>>7919104
>green's theorem only makes sense in R^3
thats because its the 3D version of a theorem that works in all dimensions
>>7919104
>Why would R^3 happen to be more specia than R ^4 or R^5 or any other one?
It has enough dimensions to make things a little complicated, but not enough to give enough degrees of freedom to make many questions trivial.
>>7922159
There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in dimension 4. some examples:
In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H4(M,Z/2Z) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countable infinite number of non-diffeomorphic smooth structures.
Four is the only dimension n for which Rn can have an exotic smooth structure. R4 has an uncountable number of exotic smooth structures; see exotic R4.
The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson). If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
>>7922199
There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. In 2013 Ciprian Manolescu posted a preprint on the ArXiv showing that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.
http://mathoverflow.net/questions/47569/what-makes-four-dimensions-special
Less interesting: You can express any polynomial as a sum of derivatives of products of the terms
More interesting: I derived the explicit summation notation for it last spring
Less interesting: this is next to worthless because it's basically the binomial theorem, only it uses partial derivatives instead of the binomial coefficient
>>7920820
I found that about Reinhardt cardinals as well. It is interesting that Reinhardt is the critical point at which it breaks.
>>7919184
t-this is a troll right?
>>7922207
>You can express any polynomial as a sum of derivatives of products of the terms
I don't see how expressing a polynomial in terms of monomials is interesting.
>>7922026
yo fuck fourier series
I didn't go to partial differential equations for one goddamn day and now I have no idea what's going on
>>7921688
>look how smart I am
You ask someone to come up with a polynomial [math]p[/math] with natural number coefficients. It's a polynomial of the order of his choice and he doesn't tell you his [math]p[/math]. (For example, he might think of the order [math]6[/math] polynomial [math]p(x) := 3 + 7 x + 5 x^2 + 2 x^3 + 3 x^6[/math])
You ask him for the value [math]p(a)[/math] at a particular value [math]a[/math] and he tells you.
Then you ask him for the value [math]p(b)[/math] at another value [math]b[/math] and he tells you.
You can now compute [math]p[/math].
>>7922457
bullshit
>>7922457
Just a guess: Evaluate it at 1 and then evaluate it at a prime number greater than the result?
>>7922471
Close.
>>7922463
Rustled?
The claim is that an arbitrary long list of data is encoded in two numbers. Hint:
https://en.wikipedia.org/wiki/G%C3%B6del's_%CE%B2_function
It is possible to estimate the value of pi by tossing needles on the floor with evenly spaced lines.
>>7922439
>I'm stupid, so people saying smart things must be showing off.
>>7921725
>theorem proved by varying the action of EM.
>varying the action of EM
>a proof
Mathematically the equations of motion for the U(1) Yang-Mills gauge field has nothing to do with EM, it just happens that the gauge freedom of EM is locally U(1). So no, you're incorrect in saying that Faraday's law is a mathematical theorem.
>>7923237
>it just happens that the gauge freedom of EM is locally U(1).
what do you mean ''it just happens'' ?
>>7919194
How the hell do mathematicians even begin to prove something like this?
>>7923351
for the inspiration, literally by trying to write numbers as sums of squares
a pattern like that wouldn't be too hard to notice after so many computations
to try and actually prove it, setting up and studying the diophantine equation x^2+y^2=d
>>7921761
He is say that the fundamental axioms that the proof relies on are by definition, unproven assumptions. How can the result be certain if the axioms are not? I will tell you straight up though I have no idea about the specifics of what you two are talking about.
>>7921761
>>7921761
>>7923237
In 2016, the notion of a **theorem** is taken seriously in a few languages for deduction, what is called **deductive formal languages**.
Given a language formalizing what people think is the **deductive reasoning**, let us say the **classical logic**, what is a **(mathematical) theorem** ?
-The notion of **theorem** appears only after you choose a **theory** that you decide to express in your logic (for deduction).
-This theory is explicitly a bunch of **statements** which are called **axioms**.
the requirement to be a theory is that **some humans** must be able to tell whether a given statement is an axiom or not. (in 2016, nobody knows what **some humans** means)
-what is a theorem inside a theory expressed in some logic of deduction ?
BY DEFINITION of a theorem, given a theory expressed in some deductive logic, a **mathematical theorem** is the last statement of a **finite sequence of statements**, where EACH statement is **valid**, which means that each statement of this sequence is either:
-an axiom of the theory OR
-a statement got from the application, on a statement in the sequence, of the inference rules of the logic that **the human** has chosen
THERE IS NO OTHER DEFINITION OF A THEOREM IN 2016.
Now that everybody had their first look at the deductive formal logic, let's look at the Maxwell equations in classical mechanics or even the quantized version.
-get up in the morning
-to deflect your fear of your suicide, before the vacuity of your existence and the failure of your choices to make you happy in finding a sustained relevancy of your puny life, choose to cling to some fantasy that doing physics is great because it connects you to the **secrets of the universe**
-look at what other people are doing
-see that they talk about a formalized result, which they call maxwell's equations, that they have got through **inductive reasoning**
>>7923802
-ask yourself ''how can I derive these equations in a valid way ?'' inside some mathematical theory
-choose to work with the inference rules and the syntactical rules of what is called the **classical logic of deduction**
-choose some axiomatization of the mathematical theory which is called **differential geometry**
-notice that your maxwell's equations [ME] are deduced, in a valid way, from some statements which are expressible in the theory of differential geometry in classical logic
like here http://physics.stackexchange.com/questions/119604/recovering-all-of-maxwells-equations-from-the-variational-principle
in a typical formalization that is here https://ncatlab.org/nlab/show/electromagnetic+field
-look whether this new statements can be theorems of the theory of differential geometry in classical logic
-amongst the new statements, turn into axioms the few statements, for which you fail to provide a demonstration [typically, the existence of the lagrangian and the existence of the action of the U[1] group]
-because you choose the classical logic, you choose to take seriously the principle of explosion and contradiction, check whether these new axioms render inconsistent the theory of differential geometry in classical logic
--if the inconsistency appears, you failed and you pray very hard to find a new explanation of the events that you have chosen to explain [or publish your work as a ''negative result'']
--if there is no inconsistency, you have created a new theory, that you can call ''bare differential geometry in classical logic+ME'', but since it is only an extension of a famous theory, stick to the claim that you invented few new axioms which are consistent with the usual model of differential geometry in classical logic
>>7923806
-your next task is to find fruther deductions to turn the few new axioms [typically, the existence of the lagrangian and the existence of the action of the U[1] group] into theorems; for this, you will likely need a new mathematical theory, possible even a new formalization of the deductive reasoning.
Why all these confusions by undergraduates and by the graduate above ?
Because nobody on earth has a clue on what physics is about and because none of these people have learned formal logic, because very few mathematicians learn logics and very few physicians learn mathematicians and logics.
** what does ''to explain'' means in 2016 ? **
In 2016, this question relates to the meme questions
-''is it possible to do physics without mathematics?''
-''is it possible to do physics without logics?''
-''is it possible to do mathematics without logics?''
A few people seek a deductive framework precisely because they cling to their fantasy of **necessity, truth, objectivity, reality, universality, universal agreement** once they live a little and notice that theey are miserable.
A few people think that deductions of statements are less personal than **personal opinions**, because the inferences rules are claimed, by their proponents, as **obvious**, or, even better, **necessary**.
To explain **an event** means, in 2016, to:
-formalize this event into some formal statement, first in a natural language
-get a FINITE stream of **deductions**, where the last statement of this stream is your choice of formalization of this event
-invent some deductive logic, which means **invent** some **inference rules** (and syntactical rules), for which the deductions in the aforementioned stream are **valid**
-formalize further your new deductive logic in formal deductive logic
-publish your results
-pray that somebody will look at them
>>7923808
-pray that people who will look at them will not laugh at you, to the point that you pray that people who pay you will continue to pay you
--if a few people adhere to your ''explanation'', enjoy being praised for your ''explanation of the event''
--if people do not adhere to your deductions, claim that this lack of acknowledgement does not matter since, after all, what you did is valid logically and therefore is not a waste of time (and not a waste of money for whoever chooses to pay you)
-get other theorems from your deductive framework
-try to get some strangers to ''validate inductively your mathematical theorems'' through the empty concept of **empirical proof**
-pray again that your deductions are again ''verified empirically''
a few people choose to have faith in the **scientific-mathematical realism**, like here >>7923237 where, still in 2016, they choose to conflate the various formalizations of events with the events themselves, which means that they choose to see as identical the **stream of events** and the various statements inside each of their favourite various deductive models. So for instance, with ME, they say that there is indeed some electric field floating in RL space, interacting with electronic fields, or electrons (it depends on what model you choose to explain the events), in what people call antennas and so on.
>>7922044
Why is this shocking?
Do I need to know complex analysis to appreciate this?
>>7923910
For real valued functions differentiability according to the usual limit formula does not imply any higher order derivatives exist. But for complex valued functions that same stupid formula (allowing h to vary in the complex plane as it goes to zero) results in a long list of nice properties real-differentiable functions only wish they had. And this isn't a case where you are reduced to just a few boring and trivial examples, see the riemann open mapping theorem for a demonstration that the space of complex-differentiable functions is big enough to be useful.
bunp
>>7919184
They even say it has no sum and that the limit is infinity. They say it's -1/12 using a different method.
>>7922199
How methods in topology can be so essentially depending on dimension?
>>7922457
[math]p_1(x)=1 + x,p_2(x)=1+x^2[/math]
[math]p_1(0)=p_2(0)[/math]
[math]p_1(1)=p_2(1)[/math]
>>7924796
With high numbers of dimensions there are more ways to deform things, and topology allows you to continuously deform things at will. For example, in four and more dimensions, the question of knot theory is resolved by that four dimensions are enough to untie any knot so that you would have to study knotted spheres to get an interesting theory. 4 is the hard number of dimensions greatly because 4 is not high enough to grant powerful tools, but not low enough to make things simple. Mathematicians wonder whether it is a coincidence that our world is 4-d when including time, since that makes things harder than the alternatives.
>>7921707
THIS
Also, the Y-combinator.
> λ (λ b (a a)) (λ b (a a))
I can't hunt down the actual quotes now, but some of the biggest names in theoretical computer science have referred to it as ‘the closest we come to doing *actual magic* in our field’ and ‘downright spooky’ and shit like that.
Abstract algebra is a terrible subject.
>>7919030
Spectral accuracy with the trapezoid rule when integrating trig functions.
>>7924848
He didn't say you could ask for p(0) and p(1). He said there are a and b such that p(a), p(b) completely determine p.
>>7925165
bait
>>7925188
He didn't say that either.
He said there are a and some function f of two variable such that p(a) and p(f(a,p(a))) determine p.
>>7919030
1+2+3+... = -1/12
Mind blowing and apparently used to calculate the casimir effect :o
>>7925248
You might define b := f(a,p(a)) and recover exactly what I said.
>>7925254
No. There are not a and b such that if you ask "Tell me p(a) and p(b)", you can determine p. But that is what you statement says.
>>7925363
Do you think I'm saying that one set of a and b will work for any polynomial?
That P=/=NP
>>7919030
there is an 100% chance OP is a faggot
>>7925354
Good thinking! However, your construction [math](x-a)·(x-b_1)[/math] makes it so that the second coefficient of [math]p_2(x)[/math] is negative, i.e. not a natural number.
>>7925363
As the guy below you tries to point out, b depends on p.
--------
Here's how it's done:
You ask the person to come up with any [math]p[/math] and not tell you. Say
[math] p(x) := 3 + 7x + 5x^2 + 2x^3 + 3x^6 [/math].
First you ask him to evaluate it at [math]a=1[/math]. This number [math]p(1)[/math] tells you the sum of all coefficients and is larger than each individual coefficient.
E.g. with the polynomial above,
[math] p(1) := 3 + 7 + 5 + 2 + 0 + 0 + 3 = 20 [/math].
Then you ask him to evaluate it at that result [math]p(1)[/math]. This number [math] p(p(1)) [/math] is some small number plus powers of [math]p(1)[/math].
Note that [math] p(1)^n=p(1) * p(1)^{n-1} [/math] is a multiple of [math]p(1)[/math], so that [math]p(1)^n[/math] mod [math]p(1)[/math] is zero.
E.g.
[math] p(p(1)) = p (20) = 3 + 7 * 20 + 5 * 20^2 + 2 * 20^3 + 3 * 20^6 [/math] mod [math]20[/math]
is the leading 3.
Now take [math]p(p(1))[/math], subtract the small number and divide by [math]p(1)[/math].
You get e.g. [math] 7 + 5 * 20 + 2 * 20^2 + 3 * 20^5[/math] and you can play the game again.
(remark: the above works better when adding [math]+1[/math] to [math]p(1)[/math] in some leading zero's cases)
0.99999999....=1
>>7923802
>spazes out entirely while missing the point
Fucking kek. Never change /sci/
>>7919045
I attribute that to the popsci memes implying that it somehow applies to physical objects.
>>7919030
http://math.stackexchange.com/a/63062
This was surprising to me desu.
>>7919104
Thinking about it the wrong way. R^3 is special which is why the universe seems to exist in R^3, ie sensibility only occurs in this dimension.
>>7921130
In euclidean space sure. But we don't live in euclidean space m8
>>7919030
>the definable real numbers are countable
>tfw almost every number literally cannot even be conceived of
>>7925674
You mean computable. All of R can be defined via Dedekind cuts or equivalence classes of Cauchy seuenes
>>7922457
I literally jumped out of my seat.
>>7925751
https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC
>>7925802
Yeah that frog-like creature is pretty startling.
>>7925751
Wrong. I'm not the person you're replying to.
A "definable real number" has a specific definition. It does not mean "definable" in the layman sense of the word.
See: https://en.wikipedia.org/wiki/Definable_real_number
You are referring to the fact that R can be CONSTRUCTED from Dedekind cuts or equivalence classes of Cauchy sequences.
He's right. The set of definable real numbers is countable.
Whether they are computable is unrelated.
The Veblen-Young theorem.
A few simple axioms imply a projective space over a division ring.
Let X be a K3 surface and [math] \Delta [/math] the modular discriminant. Then the number of g-nodal rational curves on X is the g-th coefficient of the quasi-modular form [math] \frac{q} {\Delta(q)} = 1+ 24q+324q^2+3200q^3+...[/math].
What are you trying to say with this picture
>>7922457
>you can get all coefficients of a polynomial with natural coefficients by testing at two points
very nice, and very constructive! if one were to allow real arguments to the polynomial however, then you'd be able to get all coefficients by testing in a single point even. only in a less constructive manner.
>>7926572
also, one might allow rational coefficients for that little trick.
>>7925479
so what is the number a and the number b in your illustration ?
>>7925674
That is a misconception. It is consistent that every real number can be definable without parameters, insofar as there are models of ZFC over which every real in that model is definable over that model without parameters.
>>7926572
Real argument x of the polynomial? Or coefficients.
Anyway, in the case you speak of the Taylor expansion process, this is not the same as asking for values at an argument, it's operating in a whole (topological) neighborhood of a point.
>>7927078
>every real number can be definable without parameters
Isn't that just a statement about definability, of the elements of some version of the "set of real numbers" (defined in some way) in some model of that set theory.
Can you clarify your statement in a way which tells us which language is used to define the set and which is used to speak from the outside of the set and also its element.
We'd want to fix a language, define the set of all reals in that language and then, still sticking to that language, look which of its individuals elements can be defined from within.
>>7922027
Huh, no clue. Let's find out.
>>7919045
Q: whats an anagram of Banach-Tarski
A: Banach-Tarski Banach-Tarski
hahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahahaha
hahahahahahaha
>>7919420
intution for 1)
there is a sequence of positive terms and negative terms which approach 0.
a rearrangement to approach m:
add enough positive terms to go above m,
then enough negative terms to go below m,
e.c.t
then prove that this does actually converge to m
>>7920474
doesnt.
but let zeta(s)= sum k=1 to infinity (1/k^s)
when we find the analytic continuation and evaluate at -1, we get -1/12
but thats the analytic continuation of the riemann zeta function is only sum k=1 to infinity of 1/k^s when s >1
funnily enough though, the result that the sum of natural numbers is -1/12 comes up a lot so there's SOMETHING in it.
also im told in physics if you replace 1+2+3... with -1/12 (you'd normally replace this with infinity) you get more accurate results.
idk.
A non-constant, complex-valued function defined on a closed region reaches its maximum/minimum value only on the edge of that region. I refused to believe it when I first heard it.
We should make these threads a regular /math/ general ongoing.
>>7921675
f= e^(1/x) ; x>0
f=0 ; x=0
>>7926966
Well a=1 and b=p(1).
>>7927364
The geometric series for z in (0,1) is
[math] \sum_{n=0}^\infty z^n = \dfrac {1} {1-z} [/math]
The smooth analogous is
[math] \int_0^\infty z^n \, d n = \int_0^\infty e^{ n \, \log (z) } \ d n = \dfrac {1} {-\log(z)} [/math]
Applying [math] z \dfrac {d} {dz} [/math] to the first yields
[math] z \dfrac {z} {dz} \dfrac {1} {1-z} = \sum_{n=0}^\infty n \, z^n [/math]
and applying it to the second yields
[math] z \dfrac {z} {dz} \dfrac{1} {-\log(z)} = \dfrac {1} { \log(z)^2 } [/math]
How do those two differ?
We have the Taylor expansion
[math] \log(1+r) = r - \dfrac {1} {2} r^2 + \dfrac {1} {3} r^3 + O (r^4) = r \left( 1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) \right) [/math]
Using the geometric series expansion, we get
[math] \dfrac {1} { \log (1+r) } = \dfrac {1} {r} \dfrac {1} {1 - \dfrac {1} {2} r + \dfrac {1} {3} r^2 + O (r^3) } = \dfrac {1} {r} \left( 1 + \dfrac {1} {2} r - \dfrac {1} { 2 \cdot 2 \cdot 3 } r^2 + O (r^3) \right) = \dfrac {1} {r} + \dfrac {1} {2} - \dfrac {1} {12} r + O (r^2) [/math]
With r=z-1 we see
[math] \dfrac {1} { \log(z) } = - \dfrac {1} {1-z} + \dfrac {1} {2} - \dfrac {1} {12} (z-1) + O ((z-1)^2) [/math]
and taking the derivative, we get
[math] \sum_{n=0}^\infty n \, z^n - \dfrac {1} { \log(z)^2 } = - \dfrac {1} {12} + O ((z-1)) [/math]
For z to 1 this says 1+2+3+4+... minus a 1/log^2 divergence is [math]- \dfrac {1} {12} [/math].
For gathering the data bits [math]z^n[/math], that limit doesn’t exist for neither the operation [math]\sum_{n=0}^\infty[/math] nor [math]\int_0^\infty dn[/math], but it does for [math]\sum_{n=0}^\infty - \int_0^\infty dn[/math].
The space(time) your physical field theories are defined on generally fuck with you, but there are often such systematic renormalizations of your physical expressions, and for the addition of integers it relates to that relational.
>>7927637
pic: the sum and the inverse log^2, which both diverge at z=1, and their difference that converges to -1/12
>>7927645
Fermat's last theorem has nothing to do with writing primes as sums of squares
>>7927754
>reading comprehension
The Gaussian integers were investigated in order to try to tackle Fermat's Last Theorem, and it turns out they tell us which primes are sums of squares.
>>7928007
find remain mod p(1) -> subtract it and divide by p(1) -> find remain mod p(1) of that -> subtract and divide -> ...
>>7927147
>Real argument x of the polynomial? Or coefficients.
>Anyway, in the case you speak of the Taylor expansion process [...]
this is what i get for not using concise language, my bad, sorry. let me claim something even stronger:
there is a real number, such that any polynomial with rational coefficients (even fom [math] \mathbb{Q}_{\[i\]} [/math]) can be evaluated at this point, and the function value determines all of the polynomial's coefficients. i.e.
[eqn] \exists x \elem \mathbb{R} \exists f : \left( \mathbb{Q}_{\[i\]} \right)_{\[x\]} \rightarrow \mathbb{R}^\mathbb{N} \forall p \elem \left( \mathbb{Q}_{\[i\]} \right)_{\[x\]} \forall n \elem \mathbb{N} : \pi_n \circ f(p(x)) = n^{th}-coeff(p) [/eqn]
(fingers crossed for no tex typos)
>>7929064
sorry, that abomination of an ugly mess should have read
[eqn] \exists x \in \mathbb{R}~ \exists f : \left( \mathbb{Q}_{[i]} \right)_{[x]} \rightarrow \mathbb{R}^\mathbb{N}~ \forall p \in \left( \mathbb{Q}_{[i]} \right)_{[x]}~ \forall n \in \mathbb{N} : \pi_n \circ f(p(x)) = n^{th} \! - \! coeff(p) [/eqn]
where [math]\pi_n [/math] is meant to be the projection onto coordinate [math]n [/math], of course.
every "it cant be done" theorem was surprising for me
>you cant have a bijection between N and R
>you cant trisect an angle
>you cant find formulas for solving polynomial equation of degree 5
at first i was always like
>but if you were really smart then u would find the solution right?
7929083
Looks like you're evaluating f at a number, namely the value p(x).
Anyway, give an example and we know what's up
>>7929064
There's "preview tex code" in reply form.
>>7929104
>but if you were really smart then u would find the solution right?
Doesn't seem like you understand the concept of mathematical proof.
>>7929359
he speaks of the past, why are you being mean?
>>7929359
>at first
>was
>>7920820
The most surprising set-theoretic result I've encountered — and perhaps the most surprising mathematical result I've encountered at all — is that it is consistent with ZF that [math] \omega_1 [/math] is singular.
Yes, it is consistent with ZF that the first uncountable cardinal is a countable union of countable sets.
>>7929115
alrighty, you think of some polynomial with rational or complex-rational coefficients. then i tell you to evaluate it at... say... [math]\pi [/math]. once you tell me [math]p(\pi) [/math], i tell you all the coefficients of your polynomial.
still don't see how it works? here's how:
[spoiler]the natural powers of any irrational number are linearly independent over the rationals, so using those powers as a base, you get unique coordinates to any linear combination of powers. those coordinates are the polynomial's coefficients.[/spoiler]
>>7929953
>the natural powers of any irrational number are linearly independent
True only for non-algebraic numbers.
>>7929999
yeah sorry, meant to say "transcendental", not "irrational". powers of irrationals might very well be dependent, as is quickly shown: [math]\sqrt{2}^2 = 2\sqrt{2}^0 [/math]
>>7921972
are you thinking of y = 1/x^2? because i'm pretty sure the integral of 1/x from 1 to inf does not converge.
>>7930071
https://en.wikipedia.org/wiki/Gabriel's_Horn
Surface area and volume turn out to behave fairly differently
>>7930071
>because i'm pretty sure the integral of 1/x from 1 to inf does not converge.
[math]\int \dfrac {1} {x} \, {\mathrm d} x[/math]
doesn't converge.
>>7929953
So you suggest finding the coefficients via an exhaustive search using trial and error?
Okay..
Question though, if the rationals can be negative, is there a prove that given a fixed precision you use for the test (which you have to set for meme numbers like R), you're search couldn't trigger 'yes' accidentally on a smaller polynomial?
>>7920460
i is bigger than 0, [math] z=a+bi, |z|= \sqrt { a^2+b^2 }, |i|=1>0 [/math]
a^2 + b^2 = c^2
x^2 + y^2 = r^2
sin(x)^2 + cos(x)^2 = 1
Also Taylor and Power series for e^x, sin(x) and cos(x) and how similar they are
>>7919184
Its much more logical when you consider that the riemann zeta function at -1 is -1/12. Numberphile shit is stupid as fuck though.
>>7921972
You are probably talking about the Gabriel's horn meme, and this is only true for [math] \dfrac{1}{x^n}[/math] where [math]n>1[/math] or any other general function that converges.
-1/12
>>7930635
Ignore what I said, I forgot that when finding volume you square the function, which would make it a converging integral. My bad.
Take any two non-zero elements from any field. Let one be a_0 and the other be a_1. Define successive terms in the following manner:
[eqn]a_{n}=\frac{\left ( 1+a_{n-1} \right )}{a_{n-2}}[/eqn]
Then if a_5 and a_6 exist, a_0 = a_5 and a_1 = a_6.
>>7919571
The central limit theorem is probably the best and most useful thing I've ever come across. Stats and econ just would not be possible without it.
>>7919030
Spanish hotel theorem
When I first heard it, I was like "nah there has to be a counter example".
https://proofwiki.org/wiki/Peak_Point_Lemma#Proof_2
I'm diggin some of these crazy names, like the drunkard and two policemen theorem
>>7930946
The best name for a mathematical concept is the Cox-Zucker machine.
>>7929628
Guys, I don't think you appreciate how unintuitive this is.
It is consistent with ZF, relative to large cardinals, that a countable union of countable sets can be uncountable!
>>7930490
true, just like -1 is bigger than 0, since [math] |-1| = 1 > 0 [/math]
There is another family of results that's odd at first, like
[math] \int_{-x}^x \, X^6 \, \left( \frac {1} {1 + e^{ - \tfrac {1} {12} \, X^7 } } \right) \, dX = \dfrac {x^7}{7} [/math]
>>7932058
I guess I just don't get it.
For one, it's set theory and accountability. We know it's fucked up from the start.
ZFC can't even prove that if a set B is bigger than another set S, the power set of B is bigger than the power set of S.
Thinking about your statement for a second, the numbers in (0,1) are uncontably many and while for the decimal expansion represent there isn't a map from N to all those reals, it is still something like a union of strings (the numbers) where you can count and observe each digit.
Even if that's not exactly it, I don't find your proposition so troubling. It would be something else if you said it's a countable union of finite sets. But since each set in this infinite countable union is allowed to be essentially a copy of N, that's still like
N x N x N x N x ...
I believe it can be big.
>>7932176
A countable union of [math] \omega [/math] copies of [math] \omega [/math] (maybe more familiar to you as "[math] \mathbb{N} [/math]") is certainly countable; just enumerate along the successive finite diagonals.
Choice implies that a countable union of countable sets is countable by the same reasoning as above: choose a bijection of each set with [math] \omega [/math], and we biject with the above countable grid.
>>7932176
>you will never look like her
feels bad man
>>7931949
But that's just their name. Not like the hairy ball theorem.
I was also surprised about the friendship paradox
https://en.wikipedia.org/wiki/Friendship_paradox
We could extend this surprising facts discussion to physics.
>>7932191
Not sure what successive finite diagonals means here, but at least since Skolems paradox one knows that all the naming for some intended properties can turn out fishy and this and that context (and don't be mad at me if I don't think it matters).
>>7932200
r u a grl?
>>7932271
>r u a grl?
I want to be
>>7932293
accept your body
>>7919196
I think he meant an infinite amount of terms
Take any four-digit number with at least 2 different digits. Arrange the digits in descending and then in ascending order to get two four-digit numbers, adding leading zeros if necessary. Subtract the smaller number from the bigger number. Repeat the steps with the new number.
This process will converge to 6174 in at most 7 iterations.
>>7933044
That's not going to stop anon trans feels
>>7933915
but this is just coincidence. Do it with 3 digits you'll get something else, probably.
>>7923351
Wow, something I have just covered in combinatorics! You can prove it via the pigeonhole theorem.
>>7932223
>hairy ball theorem
>you can't comb a hairy ball flat without creating a cowlick
So lewd
>>7919184
this is viable in physics but not mathmatics
>>7919030
If you put one US quarter on top of another on a table so the ridges mesh like gear teeth, and hold one quarter fixed, then roll the other around the circumference of the first one time, George Washington makes two revolutions.
>>7920488
The only reason the meme is funny is because you reply literally everytime.
>>7933927
it is just logic
>>7919030
-1 = e^(\pi * i)
it seems like bullshit if you don't go in-depth with complex numbers
>>7936202
>if you don't go in depth
you must only believe that the sin function has a complex derivative and once you got the Taylor expansion you're there
>>7921047
what!?
It was fuckin amazing when I realized that
[math]\lim{n \to \infty}\sum_{i=0}^{n}[/math]
has closed form solution.
I was so fukin amaze
>>7936660
[eqn]i^2[/eqn]*
https://www.youtube.com/watch?v=BVVfs4zKrgk
good infotainment for engineers imo
Draw any closed squiggle. (Formally, consider the image of any non-degenerate continuous map of the unit circle into [math]\mathbb{R}^2[/math].)
Then there are four points on that squiggle that are the vertices of a square.
>>7930231
Gd point sir
>>7921209
haheeey
>>7936859
If image is just a line segment?
>>7937313
I guess by non-degenerate, I meant that the image is non-contractible.
>>7936859
>Then there are four points on that squiggle that are the vertices of a square.
does this hold constructively ?
>>7937565
Constructivism is the stupidest.
Also, no. It implies the intermediate value theorem.
>>7919045
Honestly Banach Tarski isn't super interesting or surprising really. It's just an example of why AOC can be an issue. There is nothing deep or surprising, it is just an bug in mathematical formalism. There is no insight really.
>>7937735
AOC isn't the bad guy here, it's actually infinity together with non-constructive arguments (I'm not related to the two posts above btw., lel)
As soon as you restrict that things that are out there (no too big sets), the AOC becomes the conservative claim it ought to be.
reminder that the excluded middle is a axiom of choice in disguise
>>7919049
3-dimensional space becomes exceedingly more arbitrary as our dimensions increase. Interesting.
>>7936859
Is it possible to have an infinite number of triangles within that square?
An infinite number of infinitesimals... Over time, reaching this by cutting a square in half from vertices AD to BC, cutting it in half again from vertices DC to BA, cutting those again via the bisectors of lines AB, BC, CD and DA cutting again ad infinitum.
Which equates to an infinite number of infinitesimal angles at the center of a square equaling 360. HOW IN THE FUCK?
Yes, I'm aware of 1/2 + 1/4 +1/8...= 1, but how does infinity equal 360?
>>7938036
Don't understand your cutting thing at all, but you can find an infinite sum adding up to anything you want.
Note 1/2 + 1/4 + 1/8... = 1 as you know
so
360/2 + 360/4 + 360/8... = 360
Not saying that is the sum, it's just an example of one. This isn't the same as saying "infinity equals 360" though. Even though there are infinitely many elements in the sum, it never equaled infinity.
>>7938040
Thank you for clarifying, kind anon.
>>7936859
Isn't this an open conjecture?
>>7923930
It's not actually that amazing. To be complex differentiable is very restrictive, so having a bunch of properties flow out of that is nothing special.
>>7938077
In the most general case, yes. But any closed squiggle you can draw is locally given by a monotone continuous function y = f(x) in some translation of the coordinate plane, and it's proved for all squiggles of that form.
>>7938051
That... it looks like an official textbook, but...
IT'S WRONG
Completeness and compactness are known to be equivalent to the ultrafilter lemma. The ultrafilter lemma is strictly weaker than the axiom of choice. Therefore completeness does not imply AC.
The top answer in this link provides resources in which you can find this.
http://math.stackexchange.com/questions/402543/is-the-compactness-theorem-from-mathematical-logic-equivalent-to-the-axiom-of
>>7921707
Could you recommend a self-contained treatment of that correspondence? (cs, who also did mathematics)
13+24+36+48+... = 0
>>7936632
https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
I know of (I think online available) pdf's
"Lectures on Curry-Howard"
"Type Theory & Functional Programming"
(a beloeved book, in face, but maybe not emphasising Curry Howard)
"Intuitionistic Type Theory"
(the Per Martin-Löf book/paper which extended this spiel to quantified logics)
Maybe also
"Introduction to Lambda Calculus"
and
"Proofs and Types"
