/sci/ - Science & Math

[Anonymous No.7093073]

>autists will defend this nonsense

[Anonymous No.7093076]

>>7093073
http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#Zeta_function_regularization

kthxbai

[Anonymous No.7093079]

>>7093073
Here's how your thought process should go unless you're an engineer.
> Hey, an equation.
> Wait, the left side is divergent.
> I wonder what the equality means.
> Ahh, it's analytic continuation.
> Cool, what can you do with it?

[Anonymous No.7093094]

>>7093073
Apparently I'm not autistic enough then. Fuck this pseudo-math.

[Anonymous No.7093107]

>>7093094
Zeta-function="Pseudo-math"?

[Anonymous No.7093120]

>>7093107
> naming a function after the symbol used to describe it
good job autists

[Anonymous No.7093122]

>>7093120
do you know your ABCs?

[Anonymous No.7093123]

>>7093120
Not like it is a usual thing
http://en.m.wikipedia.org/wiki/Gamma_function

[Anonymous No.7093124]

>>7093122
My alpha-beta-gamma?

[Anonymous No.7093126]

>>7093107
Analytic continuation is not real math. It's handwavy physicist "math", i.e. "it works in experiment so it must be true even though it makes no sense".

[Anonymous No.7093129]

>>7093126
Well how do you define real math?
Sure its implementation (in Sting theory if I am not mistaken) is based on your mentioned concept, yet the function itself is well-definend and proven

[Anonymous No.7093682]

>>7093126
it is not handwavy. topology and any analysis course will teach you otherwise. do you know what an analytic continuation is?

[Anonymous No.7093706]

>>7093682
don't get trolled, bro

[Anonymous No.7094063]

>>7093126
But that's wrong.
>>7093129
It's used in all QFT, not just stringy things.

[Anonymous No.7094068]

>>7093079
Thought process of an engineer
>What is this garbage
>Give me the answer, and if it's useful, I'll apply it to the real world.

[Anonymous No.7094084]

>>7094068
Real thought process of an engineer
>This isn't a dick
>Give me more dicks to suck

[Anonymous No.7094089]

>>7094068
I work with a lot of engineers and they really don't care for any math if they don't have to learn it for an exam.

[Anonymous No.7094198]

>>7094089

probably because engineers care about applying their craft. Anything higher than Calculus III isn't even worth knowing in the real world. Hell, most jobs just require a decent knowledge of Calculus.

[Anonymous No.7094205]

>>7093126
>Analytic continuation is not real math
AHAHAHAHAAHAHHAHAHAHAHAHAH

[Anonymous No.7094253]

>>7094205

[Anonymous No.7095205]

>>7093120
What would you propose calling it? A Riemann function?

[Anonymous No.7095214]

>>7093073
Just solve for variable.
... = -85/12

[Anonymous No.7095218]

>>7093073
this thread again. great.

[Anonymous No.7095269]

>>7094068
Thought process of an engineer
>Can I suck it?

[Anonymous No.7095628]

>>7093073
autists will make make a thread about it
autists will argue about it
people who didnt turn their brain off halway through school will facepalm and move on

[Anonymous No.7095645]

>>7095214

Actually the middle dot is a multiplication sign, which means . . . is actually . squared, so the answer is the square root of -85/12

[Anonymous No.7095657]

>>7093073

[Anonymous No.7095670]

What do those dots mean?

[Anonymous No.7095790]

>>7095218

[Anonymous No.7095797]

>>7094068
>Thought process of an engineer
>Give me the answer

Because god knows the engineer isn't going to figure it out on his own and needs the solution spoonfed to him like everything else in his pathetic "education."

[Anonymous No.7095839]

>>7095645
That's assuming A=A

[Anonymous No.7095863]

>>7095790
>if you sum up all POSITIVE NATURAL NUMBERS...
-1/12 is not the sum of all positive natural numbers, but of a series which contains them, and more.

>NO INFINITE SERIES IN NATURE
Infinity can not be grasped by language, which is why a word is used to describe its many concepts (imaginations) of "never ending."
Therefore there can't be infinite series in nature, as unfortunately as there can not be zeros in nature.
>It's a divergent series.
It's a series that resets every other term, but who never ends. 1,0,1,0. The sum (term 1 to term NOPE) can then only be described as 1 or 0. It does not increase nor does it converge.
The average of the two sums that are equally probable (although impossible) can be assumed to be the average (since even the idea of summing an infinity is an assumption)
(1+0)/2.

>This is when I stopped taking math seriously.
Good for you.
I assume you don't read books, talk to people, or read poetry either.

[Anonymous No.7095873]

Know-nothing here. If they come to the unavoidable conclusion that all that stuff equals -1/12, why don't they just take it as proof that all or some their previous theories were false?

[Anonymous No.7096006]

>>7095628
the one correct post

[Anonymous No.7096832]

>>7095670
triple multiplication

[Anonymous No.7096866]

>>7093073
And this kids, is why you never use the equality sign in the wrong context, because people just won't get you.

[Anonymous No.7096883]

>>7093079
>what the equality means
this is where most people go wrong
the series isn't *actually* equal to -1/12, it is a magical equal sign

[Anonymous No.7096884]

>>7093076
this is the best troll math i've ever seen

[Anonymous No.7096886]

>>7095863
found the autist

[Anonymous No.7096887]

>>7093076
lel how stupid you mathsfags are
>one sum that diverges is replaced by a different one that converges
>toy with some other bullshit
>wala
fuck off you idiot, don't believe everything you read on the internet

[Anonymous No.7096896]

I stopped caring about math when I was introduced to the concept of analytic continuation. What a crock of shit. If your function can only be extended by inventing values that you didn't know, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

[Anonymous No.7096898]

>>7093076
-3c would be -3-6-9-12...
this proof makes no sense

[Anonymous No.7096920]

>>7094089
this is true but as an electro engineer i like to play with math in spare time since even those math things that are not applicable and required in real world are still there , defining our universe and things we cant touch but we know they are there

[Anonymous No.7096934]

>>7096896
>analytic continuation
where is it even used? you sound very sure and i kind a believe this

[Anonymous No.7096944]

string theory uses this factoid to validate the 24 spatial dimensions out of the 26 total dimensions in our universe.....it's a bullshit string theory thing, no mathematician gives a shit about it, any mathematician doesnt care about the infinite partial sum of a diverging sum

[Anonymous No.7096972]

>>7096887
>wala

[Anonymous No.7096978]

>>7096934
It's pasta, but was originally about imaginary numbers.

[Anonymous No.7096991]

Engineeringfag here.

My understanding is that this isn't an arithmetic solution that you can perform algebraic operations on, rather an analytic one used to differentiate divergent series. For example, the sum of all natural numbers squared approaches infinity differently than the sum of all natural numbers, and this is represented by the two series giving different values using the analytic continuation method, which is more useful than just saying they both go to infinity.

Am I understanding it correctly?

[Anonymous No.7096999]

I've replaced the confusing symbol with a less ambiguous substitute. Please spread the word.

[Anonymous No.7097012]

>>7096999
Now it makes sense. Thank you.

[Anonymous No.7097014]

>>7096887
>>7096972
>>>/ck/

[Anonymous No.7097033]

>>7096883
The series isn't equal to anything, it diverges.

[Anonymous No.7097046]

>>7095863
Infinite series do exist in nature. Zeno's paradox is a rather rapidly converging infinite series.

[newt !!NLjz0OTzgIO No.7097075]

There's nothing clever about lying and pretending that that equals sign means the same thing as normal equality in Rn.

call it "can , in a sense, be assigned to" or better yet "magical equals".

[Anonymous No.7097080]

>>7095214
>>7095645
>-85/12
>not -121/12
>thinking 1 + 2 + 3 + 4 = 7

[Anonymous No.7097108]

>>7096991
Yes and no. Arithmetic doesn't allow to compute any infinite sum.
If you say
"Call <span class="math">\sum_{n=1}^\infty f(n)[/spoiler] the number which comes closer and closer (with respect to the distance function on R) to the partial sums <span class="math">\sum_{n=1}^m f(n)[/spoiler], as m goes towards infinity"
then this uses the limit and is hence also part of an analytic theory.
You're right in that it gives you more power to separate different expression from another.

-1/12 is the second Bernoulli number B2=1/6 times -1/2! and those often show up when you relate discrete quantities with their smooth interpolations.
For example, the linearization of the exponential function at 0 is "exp(0)·h" and the finite difference is "exp(0+h)-exp(0)". Now sure enough, you get the Taylor expansion

<span class="math">h/(exp(h)-1) = 1 - h/2 + h^2/12 + ...[/spoiler]

More dramatically, there is this equation relating sums and integrals due to Gauss:

<span class="math">\int_a^b f(n) dn = \sum_{n=a}^{b-1} f(n)+(lim_{x\to b}-lim_{x\to a}) (1/2\,f(x)\,-\,1/12\,f'(x)+...)[/spoiler]

For example, for a=2, b=4 and f(n)=n^2, you have

<span class="math">\int_2^4 n^2=(2^2+3^2)+(1/2)(4^2-2^2)-\frac{1}{12}2(4^1-2^1)[/spoiler]

Now naively

<span class="math">\int_0^\infty f(n) dn = \sum_{n=0}^\infty f(n)-(1/2\,f(0)\,-\,1/12\,f'(0)+...)+lim_{x\to b}(1/2\,f(x)\,-\,1/12\,f'(x)+...)[/spoiler]

In the standard theory, f(n)=n gives
"(undefined int) = (undefined sum) - (-1/12) + (undefined lim)".
Note that we speak of undefined expressions only in the meta-theory, limits are not functions.
The point is now that you get a consistent theory of infinite sums, in your sense, by dropping all other undefined terms. E.g. here
"(the sum over all n) = -1/12".
The theory isn't more or less inconsistent than the one that assigns 2 to
1+1/2+1/4+1/8+...,
but they are not compatible with one another.

>>7097075
You keep bringing that up, but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same.

[Anonymous No.7097127]

>>7097108
>You keep bringing that up, but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same.
Saying 1 + 2 + 3 + 4 + ... = 1 + 1 + 1 + 1 + ... would also be incredibly misleading (Get it, the + is set union, and equality is in terms of cardinalities of sets, where each integer is identified with the set containing that integer)
We have conventions, and when your notation breaks the normal conventions, you should specify what it means if it differs from the normal conventions. OP equation is just as misleading as saying 1 + 1 = 0 without metnioning you are working in Z/2Z

[Anonymous No.7097131]

>>7097108
Maybe I should add the full formula:
http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

And pic related is the -1/12 relation in Mathematica, when transferred to the standard theory of limits,
where you can't just cancel the undefined integral but instead subtract it form the sum.
After evaluation of the inner limits, it's this relation:

http://www.wolframalpha.com/input/?i=Limit[z+%28-1+%2B+z%29^-2+-+Log[z]^-2%2C+z+-%3E+1]

[Anonymous No.7097164]

>>7096920
eurphoria

[Anonymous No.7097235]

>>7093073
>autists will take this as a literal sum and get huffy

[Anonymous No.7097271]

>>7097235
what is a literal sum?

[Anonymous No.7097294]

[Anonymous No.7097307]

>>7097271
my dumb layman way of saying this isn't dumb layman addition

[Anonymous No.7097583]

>>7096999
thanks this really cleared things up for me

[Anonymous No.7097710]

>>7097033
the equal sign begs to differ
of course, in actuality it is not an equal sign, it is a "fantastical equal sign"

[Anonymous No.7097829]

>>7093079
Here's how your thought process should go unless you're an engineer.
> Hey, an equation.
> Wait, the left side is divergent.
> I wonder what the equality means.
> Ahh, it's analytic continuation.
> Cool, all this thinking makes me feel like sucking cocks

[Anonymous No.7097849]

What's the fastest way for a physics student to learn about this "analytic continuation" so these damn threads will finally make sense?

I'm familiar with general principles of groups and sets and some topology. It has become very apparent that the -1/12 has nothing to do with addition like I learned in 1st grade.

[Anonymous No.7097867]

>>7093073

What I love about this equation is what so many people seem to miss.

Its like the "Schrödinger's cat" thought experiement. The autistics and engineerers will argue about it.

They dont see the underlying deep thought behind it. ( Because they are autistics and Engineers).

What this equation so beautifully tells us is that the foundations of mathematics are totally fucked up. Yes, they work for shit like building bridges, sending probes to Mars, counting beans in a bean factory and other such mundane tasks. Just like Newtonian physics works for most common day applications. But no one would seriously suggest that Newtonian physics is superior compared to the Einstein's Relativity. Well, maybe Engineerers and Autistics would.

The crux of the matter is that modern mathematics do not describe reality. Nor is it a system of logical proofs. We have made some fundamental errors in our mathematical systems. Fundamental errors because this sort of bullshit would not happen in a perfect mathematical system. And no one knows what the errors are yet. And the guy who realises what the errors are and comes up with the new way of looking at mathematics will make Einstein look like an intellectual dwarf.

[Anonymous No.7097872]

>>7097867
Or you know... it's a divergent series.

[Anonymous No.7097975]

>>7097872
the fact the the series is divergent, i.e. no limit of it's partial sums exists, doesn't hold us from assigning -1/12 to it.
Of course, if, your notation is to denote the predicate D(s):=(the series S is divergent) by <span class="math">D(s):=(S = \infty)[/spoiler], then you're bunt to confusion.

[Anonymous No.7097985]

>>7097975
then why cant you math faggots accept evaluated at x=0 sinx/x EQUALS 0
why you gotta write lim and arrows and whatnot?

>doesn't hold us from assigning -1/12 to it.
it actually should
i have no problem with zeta(-1)=-1/12
but you cant write that the sum of all natural numbers EQUALS -1/12

[Anonymous No.7097993]

Look, it's very easy. You can't take the value at the end so you take the value at the beginning.

[Anonymous No.7098014]

>>7097993
which means you cant use the notation
1+2+3+4+... because if you were to add them up, youd reach the END not the beginning, which is infinity
>oh but infinity is not a number
fuck you

[Anonymous No.7098022]

>>7093126
>2015
>not analytically continuing your functions

kill yourself

[Anonymous No.7098027]

>>7097108
That you even bothered to typeset this bullshit is commendable.

[Anonymous No.7098045]

>>7098014
You can, try plotting the smoothed asymptote of a convergent sum and look at it.

Just define the sum as the y-intercept.

[Anonymous No.7098167]

>>7098045
>Just define the sum as the y-intercept
the most mathematician thing ever said
"just define it"

[Anonymous No.7098232]

>>7094063
You got me curious, where can I read something about it(what particular topic involves it)?

[Anonymous No.7098408]

>>7098232
Keywords are regularization and renormalization.

In field theories you often set up theories with parameters which yet have to be measured, like the mass or the charge of a particle.
For example, you might conjecture that there is some mechanical quantity Y associated with an energy <span class="math">E=\frac{1}{2}m_Y(x'(t))^2[/spoiler]. But that alone doesn't tell you what the value of <span class="math">m_X[/spoiler] is. There might be an experiment, say measuring the inertia of Y, which tells you it. Say the experiment says <span class="math">m_Y[/spoiler] should be about 7. If the predictions of the theories with <span class="math">m_Y=7[/spoiler] stand the test of further experiments, you might be on to something.

In the above case, we measured masses and adopted the values. A fundamental theory of elementary particles might actually predicts the masses (e.g. string theory, in principle), or at least correlates the masses/charges etc. strongly and that a priori (e.g. quantum field theories). Here the masses are determined in the way that the theory actually only works for particular values of it.

In field theories the energy terms are more complicated than just the velocity x'(t) squared, there are energy densities and whatnot.
Instead of doing actual renormalization theory here, consider the integral

<span class="math">E=\int_2^\infty \left(\frac{2}{x+2}+\frac{1}{(1+x)^2}\right)\,dx[/spoiler]

The sign <span class="math">\int_2^\infty[/spoiler] is defined as <span class="math">lim_{a\to\infty}\int_2^a[/spoiler].
The integral over 1/x to infinity is undefined, i.e. the integral is convergent.

[Anonymous No.7098412]

(cont.)

Say you have a physical situation for which no established theory works.
To explain what's going on, you might conjecture that maybe there is a yet undiscovered quantity Z, interacting with the system (historically: neutrinos, anti-particles) and you come up with a physical theory where the situation actually looks like

<span class="math">E=\int_2^\infty \left(\frac{2}{x+2}+\frac{1}{(1+x)^2}-\frac{1}{x}m_Z\right)\,dx[/spoiler]

And now the integral has the chance to be convergent, but only if <span class="math">m_Z=2[/spoiler], in this case. Not that it matters here, but in that case the integral has the value 1/4-log(4).
You’re in the situation now that you can say:
>Guys, guys, I predict further experiments will hint at E being -log(4)+1/4=-1.136…!
That value is kinda random, so if this happens they will be even convinced that your Z-quantity theory with mass equal to 2 is correct.
The idea that there is a higgs is from the 70’s, i.e. 40 year before it was possible to make that particle scattering measurement. It was just conjectured to be there because it was the simplest solution to complete the energy expression of the standard model.

http://en.wikipedia.org/wiki/Renormalization#Renormalized_and_bare_quantities

Assining -1.136… to an integral over a positive integrant might seem completely arbitrary without any context, of course.
But as math isn’t arbitrary, there aren’t so many theories to modify self-consistent theories.
At least if you want to keep things simple. Ramanujan came up with a theory assigning -1/12 to 1+2+3+… somewhere in India after having read age old books.
Here is a computation in quantum electrodynamics using that result

http://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension

[Anonymous No.7098427]

(cont.)
The Ramanujan sum can also seen as a counter term situation.

<span class="math">1+2+3+4+…=lim_{z\to 1}(1z+2z^2+3z^3+4z^4…)=lim_{z\to 1}\sum_{k=0}^\infty kz^k[/spoiler]

which for moderate z is

<span class="math">z\frac{d}{dz}\sum_{k=0}^\infty z^k=z\frac{d}{dz}\frac{1}{1-z}=\frac{z}{(1-z)^2}=\frac{1}{(1-z)^2}+\frac{1}{1-z}[/spoiler]

We have

<span class="math">\sum_{k=0}^\infty z^k=\frac{1}{1-z}[/spoiler]

and

<span class="math">\int_{k=0}^\infty z^k\,dk=-\frac{1}{\log(z)}[/spoiler]

The smooth counter term for the divergent sum is

<span class="math">-z\frac{d}{dz}\int_{k=0}^\infty z^k\,dk=-\frac{1}{\log(z)^2}[/spoiler],

which expands as

<span class="math">-\frac{1}{\log(z)^2}=-\frac{1}{(1-z)^2}-\frac{1}{1-z}-\frac{1}{12}-\frac{1}{240}(z-1)^2+…[/spoiler]

[Anonymous No.7098430]

<span class="math">z(d/dz)\sum_{k=0}^\infty z^k=z(d/dz)(1-z)^{-1}=z(1-z)^{-1}=(1-z)^{-2}+(1-z)^{-1}[/spoiler]

<span class="math">-\frac{1}{\log(z)^2}=-(1-z)^{-2}-(1-z)^{-1}-\frac{1}{12}-(1/240)(z-1)^2+…[/spoiler]

[Anonymous No.7098433]

>>7098408
isn't convergent*

>>7098427
<span class="math">z(d/dz)\sum_{k=0}^\infty z^k=z(d/dz)(1-z)^{-1}=z(1-z)^{-1}=(1-z)^{-2}+(1-z)^{-1}[/spoiler]

<span class="math">-\frac{1}{\log(z)^2}=-(1-z)^{-2}-(1-z)^{-1}-\frac{1}{12}-(1/240)(z-1)^2+…[/spoiler]

[newt !!NLjz0OTzgIO No.7098440]

>>7097108
those equals are all more similar to each other than magical dur hur sum of negative numbers can make a positive number equals.

[oldt No.7098463]

>>7098440
I'm not entirely sure what's your saying here, did you refer back to the post from yesterday on purpose?

[newt !!NLjz0OTzgIO No.7098473]

>>7098463
I responded to
>but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same

[Anonymous No.7098522]

>>7093073
>sum of positive integers is a negative fraction

when will people stop posting this meme

[Anonymous No.7098667]

>>7098408
Thanks very much!
Only wondering how E can be defined as your mentioned integral, the units don't make sense?

[Anonymous No.7098759]

>>7098667
np.
That particular integral is just the simplest divergent one I could come up with, the actual ones are more complex.

What you do in quantum- and also statistical field theory is to compute probabilities that one initial field configuration developed into others, essentially by looking at all possible developments.

In Newtonian physics, the development of the path x(t) is given by
<span class="math">mx''(t) = F[/spoiler]
or with the potential U(x),
<span class="math">(d/dt) (d/dx') (m/2) (x'(t))^2 = -(d/dx) U(x)[/spoiler]
I.e. it's determined by the energy function.

Now imagine a ballon in your room, filled with 100 kelvin hot gas, which shake wildly and then pin with a needle.
Upon explosion, the gas will mix with the atmosphere gas and on the microscopic level it will be a mess. You have a temperature density field temp(x), which characterizes the energy and the total energy might be related by the spatial integral over that function

int temp(x) dx.

In the statistical theory, you don't actually know any particular temp(x), but only the initial balloon state at time zero and probablistic time evolutions. Those include some quite chaotic ones, pic related.
In particular, the possible states include some where at come volume there might be a random hot temperature peak. And, and here comes the problem, on within that hot volume there might be finer random ups and downs. And one those too, and so on. This means if you care to integrate for ALL POSSIBLE field configurations, you have a situation like when you want to compute the boundary of a fractal.

[Anonymous No.7098771]

In the statistical theory, you don't actually know t(x), but only the initial balloon state t0 and probablistic time evolutions. In particular, the possible states include some where at come volume there might be a random hot temperature peak. And, and here comes the problem, on within that hot volume there might be finer random ups and downs. And one those too, and so on. This means if you care to integrate for ALL POSSIBLE field configurations, you have a situation like when you want to compute the boundary of a fractal.

Pic related: On the first scale, it has some (gold) length
L,
made up of 3 sections. Then, if you take into account the first finer details (brown), you it's not made up of 3 sections but of four, i.e. the length is actually
L·(4/3).
But going deeper the (red), you find the length is actually
L·(4/3)^2.
Really, the length is the limit of n to infinite of
L·(4/3)^n,
which is not sensible. Exponential divergence.
The problem is that the temperature bumps can not really be arbitrary fine, because at one point you’re at the atomic scale where the madness ends. But you don’t can’t or want to use the atomic theory for the fucking gas balloon. Hence you introduce the „right“ counter term, which gives the right cutoff.
See also
http://en.wikipedia.org/wiki/Regularization_%28physics%29

This way, by taking the values that work, you predict the size of atoms, or whatever. Or you come up with atoms in the first place.

The divergence of the integral is one of scaling down the system. Here I talked fractions of L, that’s the only unit.
The example was made up, but the good quantum field theories actually do have those 1/x divergences. You might have some unit in front of the integral for
„upper bound to infinite of int 1/x dx = log(upper bound)“
to make sense

[Anonymous No.7098777]

(sorry for the sloppy engrish)

[Anonymous No.7098807]

>>7098759
with reference to the "integrate over all field configurations" it's also easy to motivate why quantum gravity is hard and in what sense string theory turned form physics to math:
In general relativity, spacetime is dynamic, has it's own (metric) field. If you want to make a quantum field theory of spacetime, you want to average over all possible space times. But opposed to a temperature field
temp: R^3->R
position x \mapsto temp(x)
the spacetime can be very complicated, topologically, and there are also defects (if you change perspective within a room, it's still the same room).
As with gravity (as opposed to the nuclear interactions) we already know what the fields kinda must look like, we can't invent new particles. We must take differential geometry, classify different spaces, and average them. And so those physicists started to get into "how to classify different 4-dimensional geometries", and the mathematicians saw purpose and took over part of the business.

[Anonymous No.7099157]

>>7098771
Well, you can solve the boundary of a fractal problem by introducing fractal dimensions.
So can you say that QFT equivalent way of dealing with infinity is your mentioned method?

[Anonymous No.7099172]

>>7093073
>I think I know math after completing real anal
Read this dumbasses:

https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

[Anonymous No.7099203]

>>7099157
It's but one method for a simple example.
The whole subjects goes by renormalization theory and is about as old as Dirac himself. Since 40 years we have some idea what's going on.
If I want to shock (and confuse) you, I could point out methods like

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

Here, to solve a triple integral

<span class="math">\int(x)\,d^3x[/spoiler]

you regularize it to an integral of "dimension c"

<span class="math">\int f(x)\,d^cx[/spoiler]

where c is a complex number, and then you let c to against 3.

[Anonymous No.7099209]

>>7099157
It's but one method for a simple example. The whole subjects goes by renormalization theory and is about as old as Dirac himself. Since 40 years we have some idea what's going on.
To really say what integrals you need to solve we would have to go into functional integration,
en.wikipedia.org/wiki/Functional_integration
Feynman diagrams etc.
If I want to shock (and confuse) you, I could point out methods like
en.wikipedia.org/wiki/Dimensional_regularization

Here, to solve a triple integral

<span class="math">\int f(x)\,d^3x[/spoiler]

you regularize it to an integral of "dimension D"

<span class="math">\int f(x)\,d^Dx[/spoiler]

where c is a complex number, and then you let D to against 3.

[Anonymous No.7099215]

>>7099157
It's but one method for a simple example. The whole subjects goes by renormalization theory and is about as old as Dirac himself. Since 40 years we have some idea what's going on.
To really say what integrals you need to solve we would have to go into functional integration,
en.wikipedia.org/wiki/Functional_integration
Feynman diagrams etc.
If I want to shock (and confuse) you, I could point out methods like
en.wikipedia.org/wiki/Dimensional_regularization

Here, to solve a triple integral

<span class="math">\int f(x)\,d^3x[/spoiler]

you regularize it to an integral of "dimension D"

<span class="math">\int f(x)\,d^Dx[/spoiler]

where D is a complex number. You evaluate, and then you let D to against 3.
>that's what physicists actually believe

[Anonymous No.7099224]

>>7099215
alright, I think everything beyond this goes over my basic knowledge, is this regular physics-bachelor/master stuff or is it more deep?

[Anonymous No.7099229]

>>7099224
I guess, realistically, 7'th semester upwards, if you go into field theory.

[Anonymous No.7099232]

>>7097867
You are an idiot.

There is nothing wrong about the way we count things! There is nothing wrong about our laws of mathematics!

It is plain and simple. The sum of every positive cardinal number is EQUAL to negative 1/12.

Its EQUAL TO IT!

What is so hard to understand about that?

[Anonymous No.7099237]

>>7099229
ok, thanks then for the clarification(Einsicht)

[Anonymous No.7099256]

Excuse me, not a maths whizz kid, I have seen this thing before and I am puzzled.

What are we really dealing with here? Can someone explain this in layman's terms? I only took maths to high school so I dont know much about some of the high level "abstract" maths you guys often talk about.

Is this some sort of high level trick of maths? You know like that equation that makes it look like 1 = 2 but anyone who has a high school maths education can easily find the error if they follow the eqautions through? Is it something just to catch maths dummies out?

Or is this something different where the "=" sign means something different from what most of us expecty in maths equation? For example is this equation being used in some sort of high level maths field where you are dealing with multiple dimensions or whatnot?

I just want to find out if this is some sort of mathematical joke, troll or whatever, because obviously on the face of it, it really does not seem to make any sense.

Thank you to anyone who would take the time to explain it to a maths pleb like me.

[Anonymous No.7099262]

>>7099232
>Its EQUAL TO IT!
this post has to be a joke

[Anonymous No.7099311]

>>7098014
No, you'd reach the complete additive.
Infinity isn't a number that can be plotted on an image.

Are you retarded?

[Anonymous No.7099315]

>>7098522
>negative fraction

Now I know you're trolling.

[Anonymous No.7099329]

>>7099315
why exactly?
-1/12 is a negative fraction???

[Anonymous No.7099341]

>>7099256
https://www.youtube.com/watch?v=w-I6XTVZXww

[Anonymous No.7099408]

>>7099341
that explanation for 1-1+1-1 sucks...
this makes way more sense:
S=1-1+1-1+1...
1-S=S
1=2S
S=1/2

[Anonymous No.7099419]

>>7099408
To the layman all your doing is asserting 1-S=S by fiat. They need to understand why that is so in the first place.

[Anonymous No.7099497]

What kind of mathematics is this?
I'm just about to enter a major for electrical and computer engineering and a separate pure maths major, will I be doing this kind of work?

[Anonymous No.7099589]

>>7093076
>look at zeta function
>1^s+2^s+3^s+...=-1/12 for s=-1
>autists think adding integers sums to be -1/12
>autists dont realize this is just calc 3...

truth be told, the only reason i come on this board is to get a hardon from looking at all the stupid comments

[Anonymous No.7099976]

>>7099497
I'm sure a layman who wants to understand that shit can negate S and cancel the 1 at the beginning

[Anonymous No.7100000]

>>7093073
relies on 1-1+1-1+1-1+1...= 1/2

When I honestly feel it could equal zero just as easily with parenthesis.

[Anonymous No.7100003]

>>7095863
>Infinity can not be grasped by language

Wrong, on account of the fact that the word "Infinity" exists.

[Anonymous No.7100010]

>>7100000
>When I honestly feel it could equal zero just as easily with parenthesis.

I'd argue but there's too many zeros in your get.

[Anonymous No.7100021]

>>7100000
get of truth

[Anonymous No.7100057]

>some variable is bigger than 0 by defenition
>let's assume it's negative

[Anonymous No.7100059]

>>7100000
teh quints

[Anonymous No.7100103]

>>7100057
>some variable is bigger than 0 by definition
You're hereby challenged to formally prove that the limit partial sums, which are all positive, is positive. And independent of whether that limit exists or not.

[Anonymous No.7100118]

>>7100003
>are there words for things of which we have little understanding?
... like infinity, universal, perfection, certainty, grace, supernatural, satanic, etcetera?

[Anonymous No.7100119]

I like Emma-fag. I'm proud to be on one board with a near-genious. I'm jealous of him in good sense. I wish I were so smart. He helped me many times with type and cats theory. I wonder how he grasps so wide range of subjects.

[Anonymous No.7100158]

>>7100103
>formally prove
>mfw mathematifags actually care about such shit

[Anonymous No.7100183]

I honestly believe that in a few hundred years mathematicians will discover some new math that shows the way we currently use infinite series and sums is flawed. Of all the firmly cemented math, infinite series is not one of them.

[...and eventually find combina(...) No.7100190]

>>7100119
Not sure if trolling or just awkward.
Having the same discussion on semantics with newt the 5'th time on 4chan when I really set out to work on my PhD on the weekend is the last thing anyone should admire. The uselessness of it becomes apparent when new posters make a comment showing they didn't read the thread. But I'm glad I do category and analytic number theory for escapism, not vidya.

As for learning, my advise is to take something new, promising and odd (like pic related for me) and work towards it. As a kid, I heard on the radio how good people always write down a lot and it made sense to me. Some of my notes and research ideas could not be more vague. I redo them often and the math in it becomes formal. This guy (>>7098027) said on the Euler formula leading to -1/12
>That you even bothered to typeset this bullshit is commendable.
But I didn't, I cherry picked the LaTeX from my notebook.

Though I don't know to what extent /sci/ is a waste, it can be stimulating. Sometimes I stole a question that I couldn't solve, took it to MathSE and it has a 40+ rating now.
Or someone asked why "-1/12" in particular and I didn't knew. I read the book by Kac on quantum calculus, which has some formulas where it plays a role. To answer the 4chan question, I try to present it as concise as possible. Finally I reduce the source to the expansion

<span class="math">\frac{1}{1-e^{h}}=\frac{1}{h}+\frac{1}{2}-\frac{1}{12}+O(h^3)[/spoiler]

Okay, but "why"? I work around, and finally find that in general

<span class="math">f(z)=\sum_{k=0}^\infty a_k z^k[/spoiler]

implies

<span class="math">\frac{f(0)}{f(f(0)\,z)}=1-a_1\,z+(a_1\,a_1-a_0\,a_2)\,z^2-(a_1\,a_1\,a_1-2\,a_0\,a_1\,a_2+a_0\,a_0\,a_3)\,z^3+O(z^3).[/spoiler]

Of course, if in another thread make a computation and justify it with this formula you might go like
>wtf where does this come from?
But I didn't just have to typeset this either ;)
Okay, but why does it look like that?
So I read generatingfunctionology, a book someone once posted here,...

[Anonymous No.7100219]

>>7100183
I believe it lies in the way they develop HoTT and constructive/computable math in general. My personal favorite is Finite Nature. Btw, it's also another way of interpreting renormalization.

[Anonymous No.7100226]

>>7100190
May I thought you were like an associate prof. already. You should have written 5 theses with that skill. Really dunno what you're doing out there, but it feels way too much for a single dissertation.

[Anonymous No.7100251]

>>7100190
Btw. the last expansion was the result of a conjecture when I worked out that in the book

1 / (exp(h)-1)

really comes from it being

exp'(0) / (exp(0+h)-exp(0))

and so looking at the fraction between approximated and actual finite difference suggested itself, pic related.
You get -1/12 when you plug in a_{n-1}=1/n!. This was also the week where I came to terms with the "why" of why the Laplace transform involves the exponential function. With Fourier transforms, you can argue with Hilbert spaces and their bases, at least.
That being said, this sucker would be a good candidate to word towards, it's one for me at least:

http://ncatlab.org/nlab/show/Fourier-Mukai+transform

>>7100226
Yeah thx, but in actuality I'm somewhat clueless about where to take life atm.
Plot twist: The PhD is in computational combustion chemistry

>>7100219
It literally saddens me that the computational perspective, Curry-Howard etc., isn't part of what you learn when you learn about formal logic the first time. I wonder if we miss out, the way our brains are wired now I mean, compared to kids who'd grew up with some of the main connections being pointed out early.

[Anonymous No.7100267]

Believing this to be true has resulted in people also believing that they universe is made up of tiny strings spread out over eleven dimensions. I rest my case.

[Anonymous No.7100277]

Is this the new 0,9999... = 1 flavour of the month?

[Anonymous No.7100279]

>>7093073
the concept of infinity doesn't belong in mathematics and mathematics went to shit right around the time of "calculus"

p.s. Euclid

[Anonymous No.7100284]

>Believing in something that makes no sense irl
That is why I believe 0.9999...=1 but not this bullshit. The former isn't weird at all if you realize that 0.9999... is just another way or writing "1" exactly like how 0.3333... is another way of writing 1/3 but this is complete and utter tripe.

1 + 2 + 3 + 4 + ... = (1 + 1 + 1 + 1 + ...) + (1 + 2 + 3 + ...)

The first half equals infinity therefore if the whole thing is equal to - 1/12 then that must mean that infinity is less than - 1/12 i.e bullshit.

[Anonymous No.7100295]

>>7100284
It the sum doesn't converge, how or why do you do arithmetic with it?
Also
http://en.wikipedia.org/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF

[Anonymous No.7100298]

>>7095790
>implying the integers are closed under infinite summation

> 3 + 1/10 + 4/100 + 1/1000 +... = pi
> sum of rational numbers
> NOT A RATIONAL NUMBER?????
> FUCKING MATH AUTISTS AT IT AGAIN

[Anonymous No.7100307]

>>7100000
yeah. rearranging diverging (or even just non absolutely convergent) series leads to all kinds of different solutions, which is why those bullshit feelgood videos for people without mathematical background are shit.
For example you can get every real number you want by rearranging the terms of
<span class="math"> \sum_{n=1}^{\infty} \frac{(-1)^n}{n} [/spoiler]

[Anonymous No.7100312]

>>7100307
I'm pretty sure that series is -ln2 moite

[Anonymous No.7100317]

>>7100312
only if you don't rearrange terms.
http://en.wikipedia.org/wiki/Riemann_series_theorem#Examples

[Anonymous No.7100355]

>all these undergrads getting mad at regularization
why does this intimidate you so much?

[Anonymous No.7100362]

>>7100355
maybe they want to hold on to their sanity and are afraid of fully devoting themselves to the cult and its indoctrination

[Anonymous No.7100370]

>>7100251
Yeah, I remember that you were actually doing physical chemistry or something like that. I know you feelings, but we all are better to determine the directions since science turn out to be a dangerous, unpleasant and slippery place to reside. I say it being a fresh postdoc.

[Anonymous No.7100381]

Fazit: we all agree that there is a legit (mathematically speaking) mambo-jambo that the identity in OP holds, but in no way is it natural or beautiful. Actually, pretty much like the whole renormalization thing. And famous physicists never argued against that.

[Anonymous No.7100401]

>1000 post
>nobody know that equation is a troll, made by pseudomaths like string theorists and ramanujah
>nobody know that zeta funtion is defined differently when it's argment's real part is negative, positive under 1 and positive over 1

[Anonymous No.7100418]

>>7100401
>>nobody know that zeta funtion is defined differently when it's argment's real part is negative
many people here knew that including myself
we talked about that shit years ago
not everyone is baffled by undergrad shit
>>nobody know that equation is a troll
no, it's saying "if we need this to be a number and not divergent, the number -1/12 is the most consistent"
sorry if that bothers you :^)

[Anonymous No.7100525]

>>7100381
I wouldn’t agree about the naturally.
One important observation is that

<span class="math">f(x+h)=\sum_{k=0}^\infty\frac{1}{k!}f^{(k)}(x)·((x+h)-x)^k[/spoiler]

<span class="math">=\sum_{k=0}^\infty\frac{1}{k!}(h\frac{d}{dx})^k f(x)[/spoiler]

<span class="math">:=\exp(h\frac{d}{dx})\,f(x)[/spoiler]

and therefore

<span class="math">\frac{f(x+h)-f(x)}{h}=\frac{\exp(h\frac{d}{dx})-1}{h}f(x)[/spoiler]

<span class="math">=\frac{\exp(h\frac{d}{dx})-1}{h}\frac{d}{dx}\int^x f(y)dy[/spoiler]

If you set up a context where you can invert the operator, then the integral is determined by finite and infinitesimal differences, and finally

<span class="math">\frac{h\frac{d}{dx}}{\exp(h\frac{d}{dx})-1}=1-\frac{1}{2}(h\frac{d}{dx})+\frac{1}{12}(h\frac{d}{dx})^2+…[/spoiler]

That is to say these numbers really naturally appear in the finite differences business.

When Planck started quantum mechanics 120 years ago, he discretized the spectrum of the black body and derived his famous radiation law,

http://en.wikipedia.org/wiki/Planck%27s_law

the formula from the
>science. it works, bitches!
xkcd comic

<span class="math">B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{kT} - 1}[/spoiler]
Is it an accident that Plancks radiation formula is the Mellin transform of the Riemann zeta function?

<span class="math">\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx[/spoiler]

Spoiler: No.
The number 12 is not as random as 17, say.
1+1+1+1+... and 1+1-1+1-1+... wanting to be one over +-2 is just it being the expression from the second most simple formula for coeffients, and 1+2+3+... wanting to be one over 2·3 (times 1/2!) is just it being the third most simple formula for coeffients.

>>7100370
I'm moving back to Vienna in summer and for now it seems that I might be working, for some time, at a firm that specializes in lube and friction. But I can always sell that as non-equilibrium statistical physics.

[Anonymous No.7100536]

<span class="math">\frac{f(x+h)-f(x)}{h}=\left(\frac{h\frac{d}{dx}}{\exp(h\frac{d}{dx})-1}\right)^{-1}\int^x\, f(y)dy[/spoiler]

[Anonymous No.7100543]

<span class="math">\frac{f(x+h)-f(x)}{h}=\left(\frac{h}{\exp(h\frac{d}{dx})-1 }\right)^{-1}f(x)[/spoiler]

[Anonymous No.7100853]

>>7093073
>limit of a summation approaching to zero equals its total
ayy lmao

[Anonymous No.7100939]

>>7100525
>at a firm
And scientific career ends there ...

Also, do you at least move to a company after you complete PhD?

[Anonymous No.7101013]

>>7100525
The problem is not with the particular number. It's the nature of the method.

[Anonymous No.7101045]

>>7097849
find a function y defined over a domain D. now, define a new function x where x=y over D, however x is defined over the domain DUP where U is the union of two sets and P is another domain such that P!=D.

[Anonymous No.7101084]

>>7101013
The method of analytic continuation isn't beautiful? I feel the complex numbers are maybe the purest algebraic structure out there.

I now feel I'm becoming too much of a 1+2+3+...=-1/12 proponent here, a statement which I really do not care for. What bothers me though, is when people think they can rationally argue against it, "It's can't be true, because I haven't seen something like this in Calc 1.", as if claiming this proposition would represent a wrong statement about the world.
It's because somehow people seek to be Platonists. It's math! You can either choose just to learn it, or to do it. The first is equivalent to learning the canon established 150 years ago. The second is to find interesting problems and invent new techniques.
If you do you differential geometry of two-dimensional surfaces, an established field, and somewhere pops up an expression

<span class="math">u=2+\frac{3}{\sum_{k=0}^\infty k}[/spoiler]

then you're pretty much guaranteed that u=2 gives the right results for computing the curvature of a curve or whatnot. If instead you work on an unsolved problems about radiating neutrinos in a star, and you have an idea to model the system, then you don't want to drop the research at the point where your exhaustive literature search on previous math used for particle statistics doesn't tell you what to do anymore. There were no Feynman path integrals or Dirac deltas before those people cooked them up, and they didn't sit down and learned set theory for years before they got back to their problems. They -did- math.

>>7100939
What's the difference between firm and company?
When I decided not to do a PhD at a university, I already dropped officially doing quantum gravity and such. Besides, I don't really know the purpose of money besides getting pussy and assuring health at old age. I hit the gym 3 times a week and it seems I do better than the people around me.

[Anonymous No.7101424]

>>7100158
this post actually made me lough

[Anonymous No.7101711]

>>7101045
same guy, just wanted to clarify that if you do this post then x is an analytic continuation of y

[Anonymous No.7101863]

>>7094068
thought process of an engineer
>misspell something
>IM NOT AN ENGLISH MAJOR
>required to do any math past algebra 1
>IM NOT A MATH MAJOR
>someone uses an equation
>THATS JUST A MODEL FOR THE REAL WORLD
>someone talks about science
>ITS JUST A THEORY
>someone talks about dick
>harglebargle

[Anonymous No.7102081]

>>7099341

Yeah, nah, i watched the vid ( the main guy is a real dick btw, fagging about over how incredible it is then turning in to Captain Asshole soon as the camera guy asks him a couple of questions)

Way I see it is you can not add two infinite series together. Becasue if one series is infinite it precludes the existence of any other number. So the hocus pocus bullshit the guys in the vid do to prove OP's equation is just a maths trick to impress maths plebs.

Fuck, i mean like at school we are taught that 1/0 is undefined becasue there is no way to divide something into equally zero parts. That makes intuitive sense. well the same thing applies here, you can not add something to infinity. Therefore you can not manipulate an equation with two sets of infinity to get the answer in Op's equation.

As for those pompous gits with their "but it proves string theory and shit!". You know, I am reminded of those incrediably complex geometric proofs the Midieval mathematicians came up with to explain the motions of the planets and sun around the Earth.

The motions of the planets AND the Sun around the Earth.

Yeah, you guys are just like them.