/sci/ - Science & Math

[Anonymous No.7448052]

1 + 2 + 3 + 4 + ... = -1/12
1 + (1 + 1) + (1 + 1 + 1) + (1 + 1 + 1 + 1) + ... = -1/12
1 + 1 + 1 + ... = -1/12
∞ · 1 = -1/12
∞ = -1/12

∞ is defined as a number that is larger than any other number. However, for example 0 > -1/12. Please explain.

[Anonymous No.7448057]

>>7448052
>Please explain.
The first equation is wrong.

[Anonymous No.7448058]

>>7448052
See:

https://en.wikipedia.org/wiki/Zeta_function_regularization

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

https://en.wikipedia.org/wiki/Analytic_continuation

[Anonymous No.7448061]

>>7448052
>Infinity is a number

nah m8 stay in school.

[Anonymous No.7448062]

>>7448052
Infinity is not a number, just a concept.

[Anonymous No.7448069]

>>7448052
Sometimes I wish Numberphile never existed.

[Anonymous No.7448108]

>>7448052
Irrationals are only solvable by deducing the common factors of the 0.00..01 in regards to the rest of a given number line, at which point they are no longer irrational, the point being that when an infinite is encountered it takes on the properties of 1 when compared to itself under an array of conditions, this makes for many different varieties of number lines, hence the infinite stacking of digits making for a variety of 1; which is infinite.

[Anonymous No.7448136]

>>7448108

[Anonymous No.7448485]

>>7448058
Read the text
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF#/media/File:Sum1234Summary.svg

[Anonymous No.7448580]

If 1+2+3+...=-1/12 then do 1+1+1+1+...=-1/12 since 1+2+3+...=1+(1+1)+(1+1+1)+...=1+1+1+...?

[Anonymous No.7448586]

>>7448580
No, it doesn't work that way.

[Anonymous No.7448587]

>>7448586
Why?

[Anonymous No.7448612]

>>7448587

<span class="math">
1 + 2 + 3 + 4 + ... = \varsigma \left( { - 1} \right) = \frac{{ - 1}}{{12}}
[/spoiler]

<span class="math">
1 + 1 + 1 + 1 + ... = \varsigma \left( 0 \right) = \frac{{ - 1}}{2}
[/spoiler]

[Anonymous No.7449036]

>>7448052
1+2+3+4+...=-1/12 is not a true statement. The taylor series of the zeta function is the sum from n=1 to infinity of n^-s. This taylor series is only valid for Re(s) > 1. To evaluate the zeta function for Re(s) < 1 we need the analytic continuation of the zeta function which has a different taylor series for Re(s) > 1, so assigning the value of this analytic continuation to the value of the taylor series of the function in another region is invalid.

However, if we apply regularization techniques like those used in QFT then we can say that the regularized value of 1+2+3+4+... is -1/12. Putting an equals sign there is dishonest. We must always say that this is a regularized sum, not the actual sum. If you would like, I can give you a physical explanation of the regularization of this sum and when it might be valid to assign -1/12 to 1+2+3+4+... in QFT

[Anonymous No.7449055]

>>7448136
are you implying you'd buy me a drink to discuss this elegant and truthful and unconquerable argument?

Cuz I sure could use a drink
>

[Anonymous No.7449060]

>>7448052
1-2+3-4+5-6...

[Anonymous No.7449078]

>>7448052
>1 + 2 + 3 + 4 + ... = -1/12

u fockin wat m8?

[Anonymous No.7449195]

>>7449036
id like this explanatiom

[Anonymous No.7449243]

>>7449195
So short version:
in perturbation theory (a technique used in quantum field theory that allows us to essentially take a taylor expansion to make calculations) we often end up trying to take some finite quantity and try to express it as a taylor series but the methods we use to make the calculation often give us divergent series, so we need to regularize these series to cut out the divergent part (which truly was never there, it was only introduced through our flawed math) leaving us with a finite result.

Long version coming soon

[Anonymous No.7449266]

>>7449243
Long version, part 1 of 2: Background

I'll give you the simplest example I know of that requires the least background

You may or may not have heard of the Casimir effect. It's a 'force' that pushes two parallel conducting plates together if you put them very close to one another. Yes, it has been experimentally measured.

What causes the Casimir effect? A property of conductors is that the electrons on their surface are not tightly bound to the atoms and so they can move around the material so long as they don't stray too far from the atoms. If you apply an external electric field to a conductor the electrons being free to move on the surface are able to move in such a way that they cancel an external electric field.

So now for some QFT:
The vacuum of a quantum field is a roiling mess of particles being spawned into and out of existence. The electromagnetic field is no exception, hence between any two parallel conducting plates there are photons coming into and out of existence, but since our plates are conducting the electric component of the electromagnetic field must vanish at each plate, giving us a boundary condition that the component of our wave vector k, perpendicular to the field must take on values of π n / d where d is the plate separation and n is any integer.


What this means is that between our plates only a certain subset of photons are allowed to exist, so the energy of our vacuum is actually lower than outside and gets lower the closer together we push the plates. If we try and calculate the change in energy ∆E when we move the plates ∆d, we end up getting something that is proportional to E(d)=sum from n=1 to inf of n pi / 2d

[Anonymous No.7449314]

>>7449266
Long version part 2 of 3 Regularization

So we now are told our energy is proportional to the sum from n=1 to inf of n, which is obviously infinite but we know that physically that's bullshit. It's gotta be finite. If we think about it for a bit, we may think 'Wait, electrons can only move so fast and we only have so many! surely any physical conductor can't stop a photon of arbitrarily high energy from passing through!' We should introduce a factor of e^(-a k/π) where a is some conductor dependant parameter that tells us how good it is at killing electric fields. Hence photons where k >> π/a pass right through the conductor.

So now lets take a look at our series and see if we can't extract some physics:
E(d)=π/2d ∑ ne^-(a n/d)=-π/2 ∂/∂a ∑e^-(a n/d) =-π/2 ∂/∂a{1/[1-e^-(a/d)]}=-π/2d e^(a/d)[(e^(a/d)-1]^2

What happens when a is small so that our conductor is really really good at blocking photons? Let's take a taylor series.
E(d)=πd/2a^2 - π/24d + πa^2/480d^3 - ....
taking the limit as a->0 we get

E(d)=πd/a^2 - π/24d

the πd/a^2 part diverges, but we forgot about the rest of the universe outside our parallel plates. An easy way to not deal with it is to move to a system with three plates. The two outer plates are of a fixed distance L between them and we are free to move the inner plate from side to side as we like which changes the local vacuum energy (see pic related) and we don't have to worry about the outside universe (as long as we continue to confine our interest to the 1-D case)


Now our total energy E is the energy between the middle plate and the left plate plus the energy between the middle plate and the right plate,
E= [πd/a^2 - π/24d + ...] + [π(L-d)/a^2 - π/(24(L-d)) + ...]

and the force produced by the energy configuration is as always minus the gradient of the energy, hence

F=-∂E/∂d=-{[1/π a^2 + π/24d^2 + ... ] - [1/π a^2 + π/24d^2 + ...]

[Anonymous No.7449325]

>>7449314
now notice that our divergent terms proportional to 1/a^2 have cancelled out so we are free to take the limit as a->0 to find that in the limit where our plates are perfectly conducting,
F=-π/24[1/d^2-1/(L-d)]

If we make L very large so that L>>d we find that
F=-π/24d
=(-1/12)π/d
where that -1/12 is in fact the regularized value of 1+2+3+4+5+...

Tada!

Obviously I did this in a 1D universe to save us some work, but in 3D the answer comes out the same. Also my argument with the 3 plates becomes a universe with only 2 plates in the limit where L is infinite and d is finite.

This is a great example of us using this sort of regularization to make physical sense out of a divergent series without resorting to saying nonsense like '1+2+3+4+5+...=-1/12 is a mathematical fact'. We say we are forced to regularize the sum 1+2+3+4+5+... because the model we used was not physical enough in that it did not account for the finite ability for a conductor to cancel an electric field.

[Anonymous No.7449329]

>>7448069

but muh klein bottles.

[Anonymous No.7449354]

>>7448052
>∞ is defined as a number
Found your problem

[Anonymous No.7449419]

<div class="math">1+2+4+\cdots = 1\times(1+2+4+\cdots) = (2 - 1)(1+2+4+\codts) = 2+4+8+\cdots -1-2-4-\cdots = -1</div>
<div class="math">\infty = -1 = -\frac{1}{12}</div>
Checkmate OP

[Anonymous No.7449546]

>>7448052
Infinity is not a number

[Anonymous No.7449565]

>>7449329
The best damn thing numberphile has ever made.

[Anonymous No.7449599]

>>7448108
> 0.000...1
go back to school please. You can't even explain what you mean by that.

[Anonymous No.7449600]

>>7448052
That's it. I'm leaving /sci/ for good. Only high schoolers / popsci retards here

[Anonymous No.7449613]

>>7449243
>>7449266
>>7449314
>>7449325
Thank you so much, please make threads describing topics just like this.

[Anonymous No.7449615]

>>7449600
Bye.

[Anonymous No.7449865]

>>7449613
No problem, most of that argument was ripped out of Quantum Field Theory in a Nutshell by Zee, it's a really good book. I highly recommend it. He has an amazing gift for making QFT make sense and his writing style is quite chatty and funny making his book a pleasure to read.
I can maybe make some threads though. What would you like me to make?

[Anonymous No.7449947]

>>7449865
Explainers on string theory, dynamical systems, and chaos

[Anonymous No.7449956]

>>7448052
if you define infinity as a number, you have to define how addition, multiplication etc work with it
that's how it is with 0 too
>∞ · 1 = -1/12
that rule doesn't work out with other numbers' rules.

[Anonymous No.7450934]

>>7448612
fuck off

[Anonymous No.7450970]

>>7448052
Except the sun of positive integers doesn't really "equal" -1/12.

Read this to understand more:
https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/
You're welcome.