/sci/ - Science & Math

>>11399851
graduate high school first pls

>>11399851
>Is there a version of mathematics that doesn't have this garbage
Uhh, normal math...

>>11399861

>>11399851
just eradicate theoretical physicists from the world

Is it possible to explain in laymen's terms how adding an infinite series of natural numbers, somehow sums to a tiny, negative fractional value? It really does seem ridiculous.

>>11399872
Mathematician: there exists a finite shape, with infinite area.
Physicist: how is that possible?
Mathematician: well, imagine a horn with an outline that is the graph of y = 1 / x rotated through space, you see, because the shape... blah blah blah
Physicist: but that's not a real shape, atoms don't let you do that.
Mathematician: nooo you can't tell mathematics what to do, the numbers check out!

>>11399874
OP here, it's actually pretty simple. It's all based on the premise that 1 - 1 + 1 - 1 + ... = 1/2. If you believe that, you can derive -1/12. But it's still ridiculous and I can't accept the premise.

>>11399884
>physicist: our equations diverge to infinity, how can we fix it? 1+2+3+...=-1/12

>>11399885
The premise is wrong. Neither series converges, at least with respect to standard definitions of convergence. To assign it a value is incorrect.

1+1+1+1+1... = -1/2
1-1+1-1+1-1... = 1/2
1-2+3-4+5-6... = 1/4
1+2+4+8... = -1
1+2+3+4+5... = -1/12
hmmm

>>11399895
>to assign it a value is incorrect.
No, you can extend the definition of convergence to include all series then assign arbitrary (but preferably preserving usual algebra) values to previously divergent series. Which is what is done here.

>>11399851
It all hinges on how you approach infinite addition. The sum in the left hand sign of your equation actually diverges to infinity.

However you can also define a certain function (the Riemann zeta function) such that this left-hand term would be the value of that function at 1 (so zeta(1) = your left-hand term).

There is still an issue precisely because you can't define a standard function at a point where its value would be infinity. However you can define it in the standard way anywhere around that point.

Now analysis has plenty of way to understand the behavior of a function at a point given its behavior all around that point. In particular it allows you to "complete" the value of a function at a point where it isn't traditionally defined, provided it is properly defined all around that point, and provided certains conditions are met. This is called analytic continuation.

In the case of the Riemann zeta function at 1, the completion yields -1/12.

So it's not easy to explain in layman terms, but basically the equality you've written in your OP rests on an abuse of language and is a bit misleading. However it would be in essence correct if you added the proper qualifications about what the sign "=" means here. Namely, it's not an "=" in the traditional equality between numbers, it denotes an analytic continuation.

>>11399951
hate to nitpick, but you're interested in zeta(-1) here.
I hate pointing out sign errors after a professor made a big deal about it once desu.

>>11399874
It's really just applicable in a certain area of complex analysis. The statement is derived from generally unrigorous work involving infinite sums by Ramanujan.

The sum mentioned doesn't actually converge. It isn't -1/12 for all intents and purposes.

There's a good mathologer video about it that effectively explains much of the hoopla.

>>11399896
the viring divergence vs the chad renormalisations

check mate masoymatiscians

>>11399884
>misunderstood both basic math and contemporary physics

>>11399851
>>11399874

When they write -1/12 they simply omit the asymptotic terms. Someone who actually knows what they are doing would first justify their equal sign '=' next to the series to NOT have standard meaning (which is limit of partial sum). And since they were not talking about the limit of partial sum so they can ignore the divergent asymptotic term. In physics, the competent physicists would say the divergent term is cancelled by physical constraints (e.g zero point energies). This result is then misinterpret by some of their less intelligent students.

>>11401506
>Someone who actually knows what they are doing would first justify their equal sign '=' next to the series to NOT have standard meaning
The way this idea is presented is usually "the sum of the natural numbers is -1/12". Look all over YouTube and Google. You can forgive somebody for not learning that it's not a real sum.