ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

[@numberphile]

A blog with more links and info - www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help

[@w4rd44]

wtf a new comment from the channel after 10 years

[@dario16776]

​@@w4rd44 It's an auto comment. They can program to make any comment on all videos. That's generally used for sharing stuff like sponsors etc.

[@w4rd44]

@@dario16776 I doubt because it's not a link to one of their videos or an advertisement, it's an article about the topic

[@w4rd44]

@@dario16776 but this article is not sponsored, it is on the topic

[@jojogothic]

so guys lesson today is if someone offers to give you 1 dollar today 2 dollars tomorrow ect ect dont take the deal since he is obviously trying to steal you

[@clockworkkirlia7475]

Luckily our finite lifespans tell us that his plan is doomed to failure and you will die a rich person.

[@TrackpadProductions]

The logic immediately falls apart upon comparing infinity to a quantitative idea. Infinity is not a number - trying to disprove this by treating it like one is immediately self-invalidating.

[@jojogothic]

TrackpadProductions this video just proves that math is broken at some points not that the sum of all natural numbers is negative

[@TrackpadProductions]

Anonb8 Math isn't broken at all. All this video proves is that infinity does not function like a number, and when you try to treat it like a number, weird stuff happens. As it should.

[@lukejames4981]

That's an interesting interpretation of the Incompleteness Theorem.... Godel said that *if* a mathematical system is complete, *then* it is inconsistent. To interpret this as saying that our incomplete mathematical system is inconsistent seems just... wrong.

[@sempertard]

i always multiply both sides by zero. Seems to fix things up pretty well.

[@diinobambino822]

Just like my future! :D

[@achyuthamunimakula8212]

I try differentiating both sides always....funnily I get the same result as multiplying with zero

[@youme1414]

Zero is not really a value.

[@alice_in_wonderland42]

@@youme1414 it is but u don't get the point

[@elementalneil7967]

I mean, at this point, it seems to be a more logical way to go about it than whatever that was.

[@almircampos]

After having watched this video for infinite times, I realized that my knowledge had increased by a -1/12 factor every time I watched it.

[@matejalmasi6533]

After having watched once, then having read the comments with all the controversy, then having read an article explaining real maths behind this, then understanding the problem was with what they didn't say, my knowledge actually increased well beyond what I am expecting from watching a video on KZbin.

[@divergentmaths]

If you are interested to learn more about divergent series and want to understand why and how 1+2+3+4+5+6+... = -1/12, I recommend the online course “Introduction to Divergent Series of Integers” on the Thinkific online learning platform.

[@rajatgoswami2000]

Means decreased 😓

[@keirblank4870]

I really hope you didn't watch this 12 times

[@tiny_toilet]

Huh, shoulda only happened once but never.

[@perseusgeorgiadis7821]

My IQ increased by -1/12 after watching this

[@YT7mc]

Infinite intelligence!!!!

[@bartekordek]

​@@YT7mcquite the opposite. This is why most people here belive on this video despite ot having numerous errors.

[@N269]

@@bartekordektend to agree here.... I reckon this is a series that politicians are forced to believe in before takeing office.

[@asuuki2048]

@@bartekordekWhat errors?

[@ChrisJohnson-ww4vs]

⁠@@N269Could either of you point out a single error? Considering there are numerous, a single one shouldn’t be too hard.

[@bilbo_gamers6417]

A simple stack overflow bug. God will patch it in the next update.

[@lavishkumar5062]

Wow! You're the forefather of Albert Einstein.

[@LilMissMurder3409]

Quality.

[@waitweightwhite793]

Still no updates, support is clearly messing

[@Gruntled2001]

@@waitweightwhite793 Just hope they don't wipe the drive and do a fresh OS install...

[@kangkanlahkar9045]

But you won't get negative of irrational numbers. SO error in java would be some integers

[@gorillaman283]

Top 10 pranks that went too far.

[@Evoconic_design]

Brother this formula is find out by greatest mathematicians S.N ramanujan this formula also use in string theory. I understand u can't respect to him this habits is in your blood but don't comments with out any information

[@piotrstanczak8319]

@@Evoconic_design This is not true ramanujan provided note that it could be interested when you try to do this on divergent string. Which by the way normally is wrong math.

[@harishkumaar9085]

@@Evoconic_design Don't embarrass Ramanujan by your stupidity !! If you had any formal education, you would know alternate series properties are completely misrepresented here. Don't go about spreading fake math around without knowing wtf you are talking about.

[@piyushjain9913]

@@harishkumaar9085 these western people are just assholes don't waste energy to argue with them they copy our Indian culture and nothing much

[@harishkumaar9085]

@@piyushjain9913 What are you even talking about? I am saying that the procedure adopted here in this video is completely wrong. Don't be an idiot and claim it was derived by Ramunajan and insult his intellect. In fact the person who is claiming this, is an Indian.

[@UnknownRager96]

Me before watching this video: liar Me after watching this video: cheater

[@xkilla911]

S2 =/= 1/4 because S1 =/= 1/2 because you can't add itself at a different order in a sequence and expect a correct result after adding a specific number of times.

[@LapisCyborg]

why not?

[@cpotisch]

Jax Infinite series are often defined by their order. When there is no set end of the sequence, you can’t just reorder things.

[@GurmeetSingh-qv7dp]

This can explained by reimann hypothesis. If you don't understand it doesn't make it wrong. Explanation may be wrong but result is true.

[@zankhanabose6625]

Its ramanujan infinity sum

[@rupertolababwe5973]

The analytic continuation of the Riemann Zeta function does indeed map -1 to -1/12, however this does not mean that the sum of all positive integers is -1/12. The whole point of analytic continuation is to extend the function to the domain where the original function is divergent, and after doing that u CANNOT say that the original function maps the analytically continued domain to all these extended points

[@draganradosavljevic8982]

Thank you! Guys are easily deceived by their ignorance.

[@reyrayo2502]

Can we say it's a negative number ?

[@FakenameStevens]

You can do three proofs by contradiction that adding integers will always give something positive, integer and rational.

[@wierdocean]

Please explain in 100 iq terms. I do not understand.

[@Planetyyyy]

Unless mathematics has invented a new definition for equivalence, 1+2+3+... is not equal to -1/12, that would be ridiculous. I don't care what applications it has

[@ludvigpio9605]

But S1 and S2 are divergent series, they can't be assigned a value. This video just shows that if you try to assign a value to divergent series you can prove nonsense such as sum of all positive numbers equal -1/12

[@PacketWrangler]

It's kind of like how you can prove 1=2 if you divide by zero, or you can prove 0=1 if you ignore that the square root of a positive number has two answers.

[@1willFALL]

Exactly, this seems very contrived

[@user-fl2hl8qg7s]

this comment should be pinned

[@EaglePicking]

I agree 100%. I love math, but when it does hocus pocus with infinity and then tells me that by adding all positive numbers the outcome is a negative number, then that tells me that the hocus pocus with infinity must be wrong. Another fine example of this is when they tried to convince me that two parallel lines meet at infinity, to which my answer was: "No. Your logic must be wrong because it goes against the definition of parallel lines".

[@vallarisharma7391]

Hey,I've actually seen the proof of it ..I've also read the book but the person who proved the value of infinity himself -The Indian mathematician Ramanujan.Its not as easy as the proof shown in this video,but there's a more complex algebra involved, which can make the impossibility of getting a negative value out of adding all positives, a possibility.

[@jerrysteffens4540]

This video represents negative knowledge; if you watch it, you will know less about mathematics than when you started.

[@saratoga123321]

Jerry Steffens video represents more knowledge than you can comprehend

[@GurisaYudistira]

@@saratoga123321 you're negatively missing the jokes.

[@craneology]

It does, since it is false.

[@dropdatabase2569]

@@saratoga123321 It has error in the very first line...

[@UnknownRager96]

@@dropdatabase2569 r/woooosh

[@divergentmaths]

A reminder of the golden rules to be adhered to when dealing with divergent series: 1) Do not use brackets. 2) Do not remove any zero (unless you have proven that the divergent series is stable). 3) Do not shuffle around more than a finite number of terms. Not adhering to these rules yields incorrect sums.

[@mantejsingh5438]

I think they don't read the comments

[@divergentmaths]

@@harry_page The correct sums for the following divergent series mentioned in the blackpenredpen video "Not -1/12" are: 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12 1 + 9 + 18 + 27 + 36 + 45 + ... = 19/4 3 + 25 + 50 + 75 + 100 + 125 + ... = 161/12

[@Anchor9Studios]

Username checks out

[@gamester2495]

but you can prove 1-1+1-1..... = 1/2 by using binomial theorem if you use n=-1 and x=1, then on left side u get 1/2 and on right side u have 1-1+1-1....

[@harry_page]

@@gamester2495 I think that formula only works when -1

[@carpaltullar]

We were allowed to make an intuitive conclusion about 1-1+1-1…, but weren’t allowed to make a much more intuitive conclusion about 1+2+3…

[@marcop1563]

You can show it by using a trick similar of that used for S2. You sum S1 to itself and you compute the sum by shifting one of the two series one to the right so you have 1+(-1+1)+(1-1)+...=1+0+0+...=1 so 2*S1=1 and that means S1=1/2. Of course all of this is arbitrary since these sums don't converge so they are actually undefined.

[@lensenstark9819]

@@marcop1563No you can’t do that. Using this reasoning, you could practically prove anything. This is a logical flaw, like the rest of this video.

[@jh-ec7si]

​@@lensenstark9819 You can do anything you like so long as it's consistent in the system you're doing it in

[@le_croc]

@@marcop1563 If i am not wrong, you prove it by 1-S1 = 1-(1-1+1-1+...), so 1-S1 = 1-1+1-1+1-1+...., and that means that 1-S1 = S1 => 1 = S1*2 => S1 = 1/2

[@beardjoe11]

Watching this makes me think of the mathematician who, after watching two people go into a house and then later seeing three people come out, declares that if one more person goes into the house it will be empty.

[@hemangabaruah2486]

This is not for the calculation in a universe of 3 dimension, but for more that that which is totally out of our reach till date So have some sense not to comply things to everything

[@emmynoether5878]

What if one of the person that goes in was pregnant

[@ncrohawk]

no, they would declare the house has as many people as it had before

[@acudaican]

That's just bad practice as a burglar, assuming the house is empty.

[@CAMohitShah]

What of that one person is a serial killer with suicide mentality

[@stormysamreen7062]

Tony: "The answer can be either 1 or 0, so we take the average 1/2 Me: "Ok, now that's where you screwed up"

[@RaRa-eu9mw]

No. This is the universal value given to 1-1+1-1... Whenever it is given a value, the only one that makes sense and ends up being internally consistent is 1/2.

[@brandonklein1]

@@RaRa-eu9mw what are you taking about? This is a discrete function, not a continuous one. To assign any value to this other than 1 or 0 depending on the nth term is absurd.

[@robertdarcy6210]

WHY does everyone say this? He literally states at that point in the video they have another video going into detail about why that sum is 1/2. Go watch that one

[@brandonklein1]

@@robertdarcy6210 dude I've watched the video, they even say in that video that they are using a formula outside of it's radius of convergence. There are more rigorous methods for describing series like that, all of which, still do not converge. I know it is tempting to assign a value to such an object, but in doing so, you not only encounter absurdities like presented in this video, as well as others, you *never* can"assign" a value to a sum just so it looks nice, such series explicitly remain defined on their nth term.

[@efeyzee]

@@RaRa-eu9mw So the thing I don't understand about that is sum((-1)^n) from 0 to infinity fails the geometric series test (r=-1) and therefore does not converge. How is anyone claiming both this and the geometric series test is correct?

[@paulzapodeanu9407]

To quote a math teacher from my uni: "It's extremely unpleasant to approximate solutions that don't exist."

[@aeroscience9834]

paul zapodeanu unpleasant. But not always useless

[@wangdave5574]

Aeroscience in this case very useless

[@aeroscience9834]

Wang Dave not at all. Eventually this stuff lead into the Riemann zeta function. Which is very useful.

[@pankajmundhra7421]

Such a boring maths teacher you got

[@achyuththouta6957]

@@huhun23 There are other ways to prove it using basic arithmetic such that a 5th class student can understand. No need of zeta functions

[@GamingBlake2002]

It seems like there's all kinds of tricks you can pull to get whatever result you want, once you throw rigor out the window. For example, he took the average of 1 + 1 - 1 + ... to get 1/2. You could also do this: 1 - 1 + 1 - 1 ... = (1 + 1 + 1 ...) + (-1 - 1 - 1 ...) = (1 + 1 + 1...) - (1 + 1 + 1...) = 0

[@AchtungBaby77]

This should be a pinned commercial, well said 👏

[@aryan_verma_1729]

No bro...we do not know whether bith series have equal no. Of terms or not same condition is therebin this one also...i.e 1-1+1-1+1-1......if its ending with a 1 then result will be 1 is ends with -1 then 0 ...therefore we cannot say anything because its diverging... But we can use zeta function concept Let 1-1+1-1....=S Take minus as common after first 1-(1-1+1-1...) =S Means 1-S=S Hence S=1/2 These physicists ...idk after what logic they said take out the average...which is just logicless...this which i have given is real explanation..

[@chromiumbook-marx4417]

I dont understand the shifting tho, is it arbitrary? And could you just start S1 at -1 instead and end up with its value being -1/2? This seems like fishy logic

[@REALSLIK]

This is correct. The sum of 1-1+1-1+1... is 0 because omega is even, and so this sum converges to 0 at infinity.

[@garkeinbock7103]

If we continue doing bad maths, we could say: 1-1+1-1+1-1... = (1-1)+(1-1)+(1-1)+... = 0+0+0... = 0 and also: 1-1+1-1+1-1... = 1+(-1+1)+(-1+1)... = 1 + 0 +0 ... =1 Thereby: 0 = 1 Q.E.D Maths are fun

[@DreIsGoneFission]

One of the angriest KZbin comment sections since the incident with the forest in Japan

[@RWBHere]

You'll have to explain that reference to me.

[@KahraLoding]

@@RWBHere Logan Paul incident

[@NOAH-hu6be]

That analogy is inaccurate because this was here before then

[@culwin]

A little different, because nobody liked Logan Paul in the first place.

[@MikehMike01]

Because this video is spreading lies and making people stupider and less interested in math. It’s immoral.

[@ToothBrush531]

“So now do you believe me?” Me: *No*

[@jamesgrist1101]

numberphile is disseminating wrong maths and false claims. This vid should have been an april fool. But its still up after 7 years.

[@gurkiratsingh7tha993]

@@jamesgrist1101 I agree

[@TacticusPrime]

@@jamesgrist1101 You can also prove it with Rieman Zeta function.

[@tarunsidambaram6579]

@@jamesgrist1101 maybe numberphile did not explain the topic so well but THAT DOES NOT mean that the equation is wrong, kid.

[@tarunsidambaram6579]

see, Ramanujan's problem is hard to believe but does not mean it's wrong, infinity is big and you can not imagine and you just can't say your OPINIONS on it, instead go find out more on this problem, go and study this properly

[@giacomoverardo6446]

Yesterday I solved an equation and got 2 solutions: 0 and 1. However, I wanted to save time and only wrote that there was only one solution and that was the average 1/2 . Dunno why, I got a bad mark

[@arshawitoelar7675]

Ikr, this video feels like a scam

[@nycolasfelix8828]

I mean, there is another way to prove it

[@nycolasfelix8828]

You see, we have: S1 = 1-1+1-1+1.... Taking 1- out , we have: S1 = 1-( 1+1-1+1-1...) Which is the same thing as : S1= 1 - S1 Therefore... : S1 + S1 = 1. 2S1 = 1 S1 = 1/2

[@giacomoverardo6446]

@@nycolasfelix8828 Come on man, 2S1=2 -2 2 -2 ..., that doesn't converge to any value just like S1

[@shivamsahu]

@@giacomoverardo6446 I absolutely agree , you just put a 1 there

[@1BitLearning-dev]

I think the biggest assumption is that S1 is 1/2 which I think is the reason why we got all the natural numbers sum to -1/12

[@QuantSpazar]

The assumption was the = sign between S1 and the literal mathematical gibberish on the right. If you have an ellipsis (...), then there is a pattern we didn't write in full, but understand what it means. That part is ok but if you have an infinite sum, the value is takes is the limit of partial sums. For S1, we look for the number that partials sums of S1 approach, but those partial sums alternate between 0 and 1, a divergent sequence, so no sum. S1 doesn't exist, and nothing makes sense after. Same can be said of all other sums here

[@NotBamOrBing]

​@@QuantSpazarThe explanation given as to why S1=1/2 wasn't great, but the answer is still right. For a better explanation, if you take 1-S1, that evaluates to S1, and the only number that works for is S1=1/2

[@QuantSpazar]

@@NotBamOrBing Unfortunately the standard framework of analysis does not give a value to the sum S1. If we want to assign it a value we must use another system (like Ramanujan summation) that extends what kinds of sums actually have a value. But there's multiple systems that extend summing in different ways, so we must explicit what system we used to compute S1. What they did with S1 was not a rigorous calculation, because there are ways to compute the same sum in different ways using that system that will give you different answers

[@NotBamOrBing]

@@QuantSpazar but have you considered that the sum of all natural numbers is -1/12

[@r.sgopalan6404]

No need to assume. S = 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 +... Now, write it again but shift it by one place... S = 1 - 1 + 1 - 1 + 1 - 1 + 1 -... Adding these two, all the terms in the upper series get cancelled by all the terms in the lower series, except the first term in the upper series. That is, 2S = 1 Therefore S = 1/2.

[@joshboone33]

I think this video perfectly illustrates Proof by Contradiction: Start with nonsense, end with nonsense.

[@x_theandrey9614]

it all started when they used infinity as a number

[@dannyboy12244]

How did they start with non sense?

[@midknight1339]

@@x_theandrey9614 Where?

[@Jahus]

Exactly.

[@hecticfreeze]

@F a Except even in basic maths this kind of thing is done all the time. Pi might be an infinitely long string of numbers but we can still assign it a finite symbol (the letter pi) to represent it and then use it to perform useful calculations. It's also possible to sum an infinite series and get a finite value, like 1+1/2+1/4 etc equals 2 Theres a reason the "rational" numbers are a very small subset of all numbers. Because most numbers behave irrationally.

[@craigburkhart1616]

After watching this I have some idea why string theory went off the rails.

[@jeffw8218]

Yup, in science we need to be able to test hypothesis. And if strings are too small to be observed, then we can’t gather anything scientific from them.

[@katyab92]

Hahaha!! Only physicists...😓

[@nich8244]

This is a result that explains the Casimir Effect...physically. The analytical continuation of the Reimann Zeta function.

[@RWBHere]

String Theory is exactly what it's name says; a theory. It has never been proven to be valid. Mathematicians are not normal people. It seems to me that every mathematician I've met or read about has been eccentric in one way or another. Erdös, Einstein, Turing, Gauss, Feynman, Gödel,..... read about any of them, and it becomes clear that their minds were not in the same world as the minds of ordinary people.

[@microsoftword213]

@Brandon Neifert dont get excited that 44 is out of 200

[@rozeboosje]

"So we take the average of the two" - [raises eyebrow]

[@viola_case]

Mathematics when KZbin removes the dislike button:

[@user-vr2rq5hl6l]

I’ll give this a dislike just so someone may or may not see it.

[@iammaxhailme]

This whole "astounding" fact sums from the fact that people are mistaking Grandi's series for the ACTUAL sum. The sum of alternating ones is not a half, it SHOULD be a half. A half is an approximation, not the actual answer. The actual answer is that there is no defined sum. There's a big difference..

[@matthiash.3749]

That is true. Saying that it IS one half is just like saying that the sequence (1,0,1,0,1,...) tends towards 1/2 which is just complete rubbish.

[@dangerlahori9058]

4-4/4-4=1/2 prouf this question solved

[@manun7105]

Maa H. > He never said that this "sum" is the limit of partial sum. It is an other algebraic operation with sum properties, that's why it is correct to say that the sum of this alternating serie *is* 1/2. (1,0,1,0,1,0,1,...) does not converge in the usual sense but with a generalized notion of limit, it is correct to say that it tends toward 1/2.

[@mikeharpes7573]

No Manu N. It is not, like most of the content of this video it is pure nonsense. The basic error they are making is assigning arbitrary 'sum' values to series that are non-convergent and as anyone with a basic familiarity with mathematics knows, by appropriate use of brackets you can 'make' a non-convergent series 'sum' to pretty much anything you like, if you are an idiot. For example, their chosen series 1+(-1)+1+(-1)+.... can be bracketed as (1+(-1))+(1+(-1))+..... which = 0+0+0+0+..... which clearly sums to 0, but they proved it 'sums' to 1/2 => I've just proved 0 = 1/2, quick call the news papers, I'm a genius, NOT. They are just hiding their specific use of brackets by taking the series and 'shifting' them which is equivalent to adding brackets, but because the brackets aren't explicitly added the weak minded (like yourself) mightn't notice. Bottom line, the series being considered here are non-convergent and => you cannot perform algebraic manipulations on them. The only thing that converges to -1/12 is the analytic continuation of the Riemann Zeta Function evaluated at z=-1 and this is NOT equal to the sum of the natural numbers, if it was then there would be no need for analytic continuation in the first place.

[@cenowador]

Mike Harpes I'm waiting for your paper debunking mathematical theories. You know that a lot of mathematical institutions would be very glad to give you 1 million dollars for that, right? The incentive is there. Go for it, big boy.

[@lchen1144]

The key thing to note is that one should never add or minus with infinite on each side of equation. For example, 5 + oo = oo and 10 + oo = oo. Therefore, 5 = 10. That is how the mathmaticians trick our ordinary folks.

[@ammaleslie509]

You can't add or subtract infinities for the same reason you can't divide by zero. It's too easy to end up with 0 = 1

[@jaysukhbhaivaddoriya5621]

You can't compare two infinity if you want to compare two Infinity you need limit to compare infonity

[@Ray25689]

Physicians* Mathematicians care about this and don't trick people with false calculations

[@prashanthadepu3013]

@Tom Petitdidier it's just hard coded subjectivity induced by scientists to make things less complicated and more useful.

[@prashanthadepu3013]

@Ray becoz physicians work in lab, mathematicians work on paper. everybody can do math until u go to the lab. physicians don't trick its just necessity. physics is a superset of math. maths is just a tool to support and build physics concept, sometimes u run out of tools so does the tricks

[@ShinyStarOfDeath]

The issue started when you assumed S1 = 1/2 when you divided (1+0)/2. All the points after that make sense but they are built on a questionable foundation. S1 does not end, simple as that.

[@pierrecurie]

There are more shenanigans later - with divergent sums, you can't shuffle terms around willy nilly, etc.

[@Ennello]

I'd say the S1 assumption is actually quite logical for physicists. But as soon as they start adding up series, they forget they're actually dealing with infinity and they screw up...

[@xeroxsaw1303]

The problem is these are all divergent series and thus do not converge, even tho with a Césaro Sum the first two series can converge, the other two don’t This all should have been explained as values of the Rieman Zeta function

[@TacticusPrime]

You sound like someone claiming that you can't take a square root of a negative number, therefore math with i doesn't make sense.

[@ShinyStarOfDeath]

@@TacticusPrime We call them Imaginary Numbers for a reason. Not 1/2.

[@raenfox]

Mathematician: **calculates something, result doesn't make any sense.** Mathematician: "I define this as correct."

[@Sadnessiuseless]

it does make sense?

[@_bleck]

@@Sadnessiuseless stay in school kids

[@bardofhighrenown]

He's a Physicist. You'd be surprised how rough and sloppy their math skills actually are. I know I was when I took my first physics class.

[@SuperRaidriar]

You have to realize the reason they assign the value -1/12 to this sum is because it is useful in some way.

[@scoobydoofan3275]

@@SuperRaidriar I feel like there's a logical physical explanation for that which doesn't include abusing analysis

[@km1dash6]

You can reject the claim that 1-1+1-1... = 0.5 and instead say that it has no solution, or an indeterminate value. If you do that, the entire system falls apart. The thing is, this uses a different summation method than what most people are used to.

[@mutt8553]

This whole video is extremely nit-picky and circumstantial. Sure it’s -1/12, but when you manipulate all of the factors to your bidding it can be anything

[@cottoncandycloudsart]

@@mutt8553 It seems like a bit of a gimmick to me so not all that serious

[@TheScienceNerd100]

The whole series S1 = 1 - 1 + 1 ... is like any Supertask explained in Vsauces video. Say you take S1 and sum up the next turn, decreasing the time interval by a half each time. Say you start with 1, then after a min you get 0, then half a min 1, then a quarter min you get 0... After 2 mins you'll have the answer, but what would you get? After ever time you get a 1, you take 1 away, but after every time you take 1 away, you add 1 back. Its a paradox. Randomly making it 1/2, you can basically do anything you want now and make it *look* like it works. But it doesn't work like that, this is why String Theory failed.

[@Ray25689]

@@mutt8553 "it" isn't really -1/12. You can make sense of it when you change the meaning of the + symbol or talk about holomorphic continuation of the zeta function, but assigning the series a value doesn't make sense when dealing with the usual addition

[@JoeThomas-lu6fy]

You can also claim that 2+2=-1/12 if you want, but without evidence it doesn't really matter.

[@numberphile]

Obviously I cannot reply to all the comments and questions, but I did write a general blog: periodicvideos.blogspot.co.uk/2014/01/thanks.html

[@rhomaioscomrade]

To clear some misconceptions in the comments, the sum of all natural positive integers is not -1/12. -1/12 is not a value where the series converges to, but a "title". It is kind of a "name" for the series, a way to represent it as a valid finite number without having to use infinity in your calculations since it behaves badly and infinities don't exist in nature. People who complain that "this is not something you can find in real life and it can't be true" have little understanding of maths and physics. Complex numbers do not exist in nature either in of themselves, but using them in maths and applying Euler identities to real functions like sines and cosines is perfectly valid and one of the most common things in both maths and physics. And lastly, as for the sum of 1-1+1-1+..., the reason why it's 1/2 is the same reason as the sum of all natural positive integers. It does not converge anywhere therefore it takes a value based on mathematical extract. The explanation of taking the half due to the 2 equally possible answers (1 and 0) is the physical reason why this exists. This principle of the average value is especially applied to Fourier series which are not only irrefutable mathematically, but also all over classical mechanics in Physics as well as quantum mechanics due to the applications of those in wave functions.

[@Arkalius80]

+Eutychius Raptor Yes, I think what you said reflects something that wasn't adequately explained in the video. Saying that the sum of all natural numbers is -1/12 is a naïve way of expressing what is really going on. the value -1/12 is merely a characteristic that can be extracted from the divergent sum. The fact that there are multiple very different ways to arrive at this result suggests it is a meaningful one. In many ways, even convergent sums are the same. Technically, 1 + 1/2 + 1/4 + ... doesn't "equal" 2. At no point do you ever finish adding values and get 2. However, there is a rigorous and consistent way to extract the value of 2 from this series. Since the series doesn't grow infinitely, and converges on a limit, people are more comfortable saying the series "sums" to 2, and we denote it that way out of convenience. The way in which we arrive at -1/12 with the sum of natural numbers I think feels a bit more tricky, and the result, when looked at as an "equality", seems counterintuitive, so people resist it.

[@douggwyn9656]

+Arkalius80 The sum of the nonpositive integer powers of 2 does converge to 2, meaning that the sequence of partial sums becomes closer to 2 than any positive tolerance you can name, past a certain number of terms which depends on the tolerance (and can be readily calculated). No such property applies to the series of natural numbers and -1/12. The claim made in the video is simply wrong. There is a more remote connection between this series and -1/12, but it is not equivalence.

[@Arkalius80]

Doug Gwyn That's the point. It's not equality. -1/12 is, as I said, a characteristic, non-arbitrary value associated with this series, not it's actual sum.

[@AlcyonEldara]

+Arkalius80 : I4m sorry, but you don't even understand what "equals" means. Here is an example : 1/2 + 1/2 = 1 isn't correct from a set point of view. The set "1/2" "+" the set"1/2" isn't the set "1". So ? The equality is correct if you give the "rules" for equality. They haven't, because the rules are a bit complicated, but you are using a similar set of rules when you write 0.999999999..... = 1.

[@MrAkashvj96]

+David Sbabo 1/2 & 1/2 are numbers not sets. Addition of sets is not defined in set theory. Arkalius80 is actually spot on. This summation is a special case of the Riemann Zeta function. It's divergent, there's no 2 ways about it. But if you pretend that the sum behaves nicely & converges to some finite value which obeys the laws of arithmetic (which is precisely what you're assuming with S1) then you can "associate" a real number to this summation which happens to -1/12. But that doesn't mean that the summation in its entirety is equal to -1/12. There's a huge difference.

[@Prs722]

Trolley Problem: A trolley is on a track headed towards one person, and after this one person is two people, and after that is 3 people, and so on. You can flip a lever to send the trolley onto an empty track. Do you flip the lever?

[@mwzngd1679]

lol this is the greatest trick question of all time if you don't flip the lever than -1/12 people will die so you will save more people than if you do flip, in which 0 people will die

[@Prs722]

@@mwzngd1679 But, say you didn't flip the lever and there was an actual trolley headed towards people. Would you truly be saving a 12th of a person, or would you be killing an infinite number of people. I think the true answer is similar to dividing by zero. It is undefined. You can define it in various ways that can potentially have use, but the true answer is undefined. Likewise, 1-1+1-1+1... is undefined. Yes you can define it as 1/2, but you will never truly get an answer, so it is undefined. It will never equal 1, it will never equal 0, and it will never equal 1/2.

[@quasiZote]

Now this is brilliant

[@aayushpokhrel2892]

Of course not because if you let the trolley be, you can save 1/12 people. I mean who wouldn't opt for that option duh

[@francoiso]

The golden rules to be adhered to when dealing with divergent series are: 1) Do not use brackets 2) Do not remove any zero 3) Do not shuffle around more than a finite number of terms

[@rhythml6229]

Francois O yah this video is definitely miss leading

[@AbCd-zo5tb]

Thankuee sir

[@Mecal21]

Yeah, I came here from another video saying that they're wrong so I came here to see if people know

[@aadithyahrudhay2269]

I love you so much right now.

[@iPlayDotaReligiously]

But we need to converge this divergent series into concrete number so it can be used in string theory..that's why that result came up. I mean, jokes aside, dont take this video "mathematically". What Numberphile did in this video is explaining things about number in Physic fields, not Mathematic. Because in mathematic, u have infinity as a concept, while in Physic, u dont know about infinity.

[@jiggybau]

Let me prove that 1 = 0, using this premise: S1 = 1 + 2 + 3 + 4 + 5 ... = - 1/12 S1 - S1 = 1 + 2 + 3 + 4 + 5 + 6 ... - 1 - 2 - 3 - 4 - 5 ... = 1 + 1 + 1 + 1 + 1 + 1 ... Since S1 - S1 = - 1/12 - (- 1/12) = 0 It follows that 1 + 1 + 1 + 1 + 1 .... = 0 Let's name this sequence S2: S2 = 1 + 1 + 1 + 1 + 1 ... = 0 Now let's subtract it from itself: S2 - S2 = 1 + 1 + 1 + 1 + 1 ... - 1 - 1 - 1 - 1 .... = 1 Given that S2 equals 0, we can also write this as: 0 - 0 = 1 Which implies that 1 = 0.

[@SomeRandomFellow]

+Enzo Molinari *claps*

[@SomeRandomFellow]

Gregery Barton what a buzzkill

[@AkshayAradhya]

+Enzo Molinari Im pretty sure you can prove anything with this premise. Event that Kim Kardashin is smart

[@AkshayAradhya]

+Enzo Molinari Im pretty sure you can prove anything with this premise. Event that Kim Kardashin is smart

[@AkshayAradhya]

***** His proof is absolutely correct. Btw limits dont even come into the picture here. They used a sequence of numbers in the video. Not limits, which are completely different. What are you even talking about. ... ?

[@resistance7538]

According to riemann's rearrangement theorem; Infinity - infinity = (*any number*) Its just the way you rearrange the series...

[@raudh1]

According to the Riemann-Dini theorem *

[@edbear94]

It's also commonly infinity

[@JorgetePanete]

Adeel all it's*

[@prim16]

Exactly.

[@JorgetePanete]

Adeel all It's*

[@shreyansh2244]

If you're from 11th-12th science, and you got some amazing Professor who sometimes taught you this type of curious and out of the syllabus problem , just to keep you hooked to the wonder of science and Mathematics, you're lucky.

[@canyoupoop]

Yeah teaching false assumptions and statements without specifying in what branch we are actually talking about.... I don't think so

[@ifeanachosonia5787]

​@@canyoupoop😅

[@AnasArfeen]

​No, but ​@@canyoupoop

[@Grgrqr]

Yeah my math teacher taught me that when a sequence approaches infinity as its limit, the series will be divergent

[@umariqbal1031]

@@Grgrqryeah unless |r|

[@anothergol]

I prefer to see this as a demonstration that 1-1+1-1+1-1... does NOT equal 1/2

[@CyrusBeaman]

Yeah because in reality, the actual answer would be a superposition of both zero and one, so basically there is no answer, it's like trying to say if infinity is either odd or even, its neither. So to use that to answer so many other things is ridiculous

[@Marco-zv8xm]

@@CyrusBeaman i would prefer saying s = 0;1 at the same time. It has 2 possible answers so that would be the way to go i think

[@_Nibi]

@Sari Çizmeli Mehmet Ağa infinity is equal to two times infinity plus 1. Infinity is odd.

[@BUE687]

You are so right, showing that the limit does not exist is quite simple

[@pain10227]

S=1-1+1-1+1-... S=1-(1-1+1-1+1...) S=1-S 2S=1 S=1/2

[@michaelosborne9279]

Is this how the financial crisis happened? Add together ever-stacking credit risk to get no credit risk? Note to investment bank CEOs, do not hire physicists.

[@hdjdco5428]

No, they were making more money each day and ended up with negative money xd

[@externaldriver]

It's funny because they actually love hiring physics majors

[@brianmacker1288]

Does -1 - 2 - 3 - 4 - 5 - 6 ... equal to 1/12?

[@jvcmarc]

no, but [-1+2-3+4-5+6...] does

[@brianmacker1288]

Thanks, OK so what does it equal? Also what does this equal? ... -3 - 2 -1 + 0 + 1 + 2 + 3 ...

[@jvcmarc]

Brian Macker [-1+2-3+4-5+6...] equals 1/12 [. -3-2-1+0+1+2+3 ...], just equals zero

[@SussyBacca]

Brian Macker what does 58008 equal?

[@brianmacker1288]

Nick Steele The only thought between your ears.

[@akshit5363]

The sum of 1 to infinity is given by Indian Mathematician 'Srinivasa Ramanujan'

[@sidharthcs2110]

@@akshit5363 His summation is not the same as the normal summation function that we regularly use. It's different

[@vollsuessaba_9190]

"The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. By using them, one may draw any conclusion he pleases and that is why these series have produced so many fallacies and so many paradoxes …" - Niels Henrik Abel 1820

[@mr.coconut2310]

fukn rekt

[@wohdinhel]

Vollsuessaba _ because calling something “of the devil” is truly scientifically infallible

[@zoltankurti]

@@wohdinhel you clearly didn't get the point. They are abusing infinity in this video without clear definitions. Without definitions they twist the rules as they want, and hence get what they want. This is the meaning of the quote, unpacked for the weaker minds.

[@HL-iw1du]

Zoltán Kürti You clearly didn’t get the point. There is no answer that mathematicians “want”. They are simply trying to further their understanding of mathematics and of the Universe (hence the video’s mention of string theory). Also, by many different methods, only one sum is derived for each traditionally infinite series. So they can’t get “anything” that they want, if they want anything at all, since there is only one option. This is the meaning of the video, unpacked for weaker minds.

[@zoltankurti]

@@HL-iw1du alright, I will try again since some people are truely resistant to criticism. The video contains false information, the end. They never mentioned that they are not using the standard notion of summation. And they are not matjematicians in the video, they are physicists who communicate science in a very shameful way. They didn't specify what definitions they are using, and they could have arrived at a different answer very easily.

[@johnperry3000]

You can get whatever number you want if you get to pick your own rules and varibles.

[@daddymuggle]

Exactly, choose the appropriate axioms and logical rules, and we can develop a mathematics in which 0=1. Yay, woohoo, we're awesome. Unfortunately that mathematics is unlike to have any useful applications even in theoretical mathematics, so fame and fortune continue to elude us.

[@stefanuifalean5803]

@@daddymuggle this result is used in quantum physics

[@CreationWorkbench]

This is mathematically true. It's basically assuming that the summation of all natural numbers is a finite series (which it isn't). However, when you treat is as such you get this summation which is where this video tricks people without explaining itself properly.

[@juliobasulto4883]

@@daddymuggle if you get a nonsensical result such as 0=1 well you probably havnt used the mathematical logic or axioms correctly... this summation 1+2+3....=-1/12 only makes sense when we talk about infinity, this idea is used in Calculus alot to describe limits and such... 0=1 is just literally saying that 0 is directly related to 1, or 0 is the same as 1, which we can use math to prove its not true. Just like we can use math logic and axioms you mention to prove 1+1=2 or the sqrt(2) is irrational

[@silviavalentine3812]

That's literally how math and thus physics works. I can't count the number of times my professor has pulled out the "magical hat" in the middle of a derivation.

[@rjbse]

I don't know whether numberphile has heard this or not but, "Some infinities are larger than others" Side shifting sums or taking recurring terms as sums is definitely "debatable" as you are essentially equating a smaller infinite series with a larger one.

[@Tetrahelix]

***** The whole point is that infinity isn't actually a single concept -- you have the "infinity" that refers to the cardinality of the set of all natural numbers, but you also have the infinity that refers to the cardinality of the set of all real numbers. The latter refers to a concept that describes something bigger/larger than what the former concept describes, so it's not really that odd to say that "some infinities are bigger than others."

[@Crazy_Diamond_75]

Richard Coleman But you're missing his point, which is that "side shifting" does nothing to affect the size, or cardinality, of the infinity in question.

[@Tetrahelix]

AMGwtfBBQsauce I'd already made that point. In the comment immediately before his.

[@Crazy_Diamond_75]

Richard Coleman I'm sorry I think I might not have followed the conversation correctly the first time I read through. Please excuse my correction :P

[@gmalvaralonso]

AMGwtfBBQsauce But, just asking, by the same reasoning, isn't true that if: 1-2+3-4+5.... = S2 1-2+3-4+5..=S2 Then you can also say (2S2 = 1 ± infinite or n), beeing "infinite or n" the number or thing that is missing by side shifting?

[@shafaque1390]

The statement that the sum of all natural numbers equals -1/12 is correct within the specific mathematical context of zeta function regularization used in theoretical physics and certain areas of number theory. However, it's important to emphasize that this result should not be interpreted as the sum of natural numbers in the traditional sense, which is a divergent series. In everyday arithmetic, the sum of all natural numbers is not -1/12. This concept is a result of mathematical manipulation and regularization techniques used in specific mathematical and physics contexts.

[@RaRa-eu9mw]

This concept that youtube commenters have of "the traditional sense" needs to stop. Every context where the sum of the naturals appears, it is always taken to be equal to -1/12. It's useless talking about what the sum is "in everyday arithmetic" (whatever that is) when the sum never appears in everyday arithmetic.

[@SkyWKing]

I'm no professional mathematician but I figured out why it is wrong. At least the method used here. You cannot do a shift addition or subtraction with a divergent infinite series. Remember how they get Grandi's Series to be 1/2? They manipulated the second row so it's shifted by one place, and assume the second row to be the same as the first row. In fact it's not: Grandi's series: 1-1+1-1+1-1+1...... "Shifted" Grandi's series: 0+1-1+1-1+1-1...... But you will say "well anything plus zero is itself isn't it?". No, it's not in this case. The Grandi's series follows the pattern 1,0,1,0,1,0; the "Shifted" series is 0,1,0,1,0,1, now every term in the "Shifted" series is different from the original. Therefore by adding a zero to the beginning you get a different series. So now you cannot use 2S1=1. Try this at home: dilute the Grandi's series with 0 after each negative one, and do shift addition with three rows, you will get a number. Then dilute it by placing the zero after each positive one and do shift addition again. Compare the results (I will not spoil your fun of doing this). This happened at the end of the proof where he assumed 4+8+12+16......=4*(1+2+3+4......). Well it's right, but this is not the series appeared here. It's in fact 0+4+0+8+0+12+......You will figure out why they are different if you do the dilute Grandi's series experiment. In conclusion the method of shift addition to sum a divergent infinite series is inherently flawed. I cannot comment on the average partial sum method though. But I guess partial sum method is not enough to prove 1+2+3+4+......=-1/12.

[@sivad1025]

Simple answer: the numbers are infinite. The numbers will never end. You can shift it because of that. Why wouldn't you?

[@Dilandau3000]

So... the universe is susceptible to integer overflow?

[@linusherr8257]

Dilandau3000 Omg Dilandau.... I didn't know you were interested in math ! ! ! Your 'Let's Play's' are awesome!

[@MrKeotan]

You have nailed it.

[@EmdrGreg]

The bit bucket sprung a leak.

[@benthemusicalchemist]

This joke is so specific and wonderful. I know I'm late commenting on this, but well done.

[@ravenlord4]

The US government is trying this with the national debt.

[@gangulic]

the best comment ever!

[@ayushdhakal333]

@@gangulic you guys still alive? Just curious 😁

[@gangulic]

@@ayushdhakal333 allo allo zis iz night hawk can you ear mi?

[@paulmccartney8293]

@@gangulic wow he's alive

[@shiva4774]

​@@gangulic you guys still alive? Just curious 😁

[@jessenelson8106]

I’ve watched this video almost -1/12 times, and it never gets old.

[@LfeinYT]

This is not an astounding result, it is simply a false one. Whatever result this leads to cannot be the sum in the sense of the result you get when you add all the positive integers together. if you start with 1, the sum of all the integers must be greater than 1 because there are other integers to add. If you take the first 2 integers, the sum of all of them must be greater than three, because there are other integers left to add. And so on, potentially forever.. Since I'm not a mathematician, I can't deny that there may be a relation between the positive integer set and -1/12. You could call it the Riemann zeta function sum or the Rajamujan sum or some such (pun intended), but it clearly cannot be the sum in the first-grade sense of that term. To claim it is only tends undermine the integrity of mathematics. The men's demonstration is not at all convincing. They simply changed the subject, and never even addressed how these other series have anything to do with the original problem.

[@numberphile]

Here you go: www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help

[@janwollert1559]

Brance Finger Thanks for clearing it up, imo the video should get removed from KZbin.

[@SamVidovich]

Brance Finger "Guys, I've never studied infinite series or any math, really, but this is totally not true."

[@janwollert1559]

Sam Vidovich Why your quotation marks? This is simply not true, 1+2=positive, 1+x, while x>0, sums up to a positive term no matter how you look at it.

[@SamVidovich]

Jan Wollert It is indeed, because you can make assumptions about sums tending toward infinity by comparing them to other, similar sums. An example of this is the direct comparison test - en.wikipedia.org/wiki/Direct_comparison_test

[@HotslutGG]

The problem/flaw of all this begins at the assumption that the "average" of 1-1+1-1+1.... equals 1/2

[@sivad1025]

Watch their proof. It's in depth and makes sense. Additionally: s=1-1+1-1+1... 1-s=1-(1-1+1-1+1) =1-1+1-1+1 therefore: 1-s=s 1=2s s=1/2

[@maljamin]

but saying 1-s = s when you're dealing with this infinitely "oscillating" thing means: "0,1,0,1..." = "1,0,1,0...". It does and doesn't. 1-s doesn't mean what it would mean if s were a number. To my mind, S something unresolved, a superposition of answers. 1-S is a similar "unresolved" but it is "out of phase". Any moment you stop it is 0 when S is 1, and 1 when S is 0. So the best way I could make it seem less resolved is maybe to change the claim that 1-S=S (which is merely a guess based on what it "looks like"). Let's revise that claim to this instead: -S + 1 "=" S. In my view, adding 1 to a superposition merely "resembles" another superposition.

[@spaghetti4896]

Well, this didn’t come out on April 1st, so I’m confused?

[@GabrielTravelerVideos]

You're asking us if you're confused?

[@chiiing8288]

????? are you??????

[@bigbigx2250]

@@GabrielTravelerVideos They're asking if this is a joke because of how it uses bad maths :^)

[@GabrielTravelerVideos]

Yep, I got that. I was poking fun because Spaghetti 489 used a question mark instead of a period. Seems like that should be a statement, not a question.

[@darkseid856]

This series actually exists ! It's Ramanujan's Infinite sum .

[@Tobiasberger]

Combining two positive numbers will always be a positive number, no matter how far you go. It’s like a function thats trending towards infinity. It just cannot be negative

[@xenqor5438]

Wow you’re so smart 😮 😱

[@MarcusRost-m8x]

@@xenqor5438 That's not nice at all, he's trying his best, let him arrive to the conclusion O_O

[@j2kun]

I have now had multiple friends ask me to explain to them why this video is wrong. I don't care much that you want to keep things informal and allow for casual fun maths. What bothers me about the video is that you're claiming this is unconditionally true (by the fallacy of authority), and that there's nothing deeper going on for people to read about when in fact there is and the particular proof given in this video is flat out wrong. It doesn't matter that the "result" is used in physics (physicists are well known to abuse mathematics because the "results" are interesting), or that there is a second video explaining things in more detail (though I don't think it goes far enough to make it clear where the line between truth and falsity was blurred in this video). What matters is that this video, standing by itself, is spreading massive amounts of misinformation. This is numberphile's blessing and its curse: it's so popular now, and has gained so much influence, that the majority of ignorant viewers (which is the vast majority of all viewers) take what is presented as gospel. You might say that's their problem for being ignorant and not questioning things, but I think it's also seriously dishonest to knowingly do such a thing. To think that mathematicians, who so rarely get as wide an audience as numberphile has, would knowingly lie about mathematics! I can hope it was more of a misunderstanding on the editor's part, but until I see evidence of that, this video has made me lose a lot of respect for numberphile.

[@SpTh2]

They never claimed it is the undeniable truth. Numberphile isn't suitable for teaching people math, it's suitable for getting people to get interested in math. They did show the textbook that claims this and this series clearly has use when it comes to specific context and limits. You are being overly dramatic here by claiming they knowingly try to "fool" their viewers as the description of the channel itself simply states "videos about numbers", not "free PhD online, sign up now".

[@AnimationByDylan]

So is the *real* answer to a math test question simply the average of all of the test takers' answers?

[@rohitgejje3717]

You'll actually get a pretty accurate answer

[@jtm8514]

@@rohitgejje3717 I was about to say that

[@satyamtekriwal7376]

This phenomenon exists and is called 'wisdom of the crowd' (you can search that), this trick is used in game shows like who wants to be a millionaire (audience poll). And, S=1-1+1-1+1.....=1-(1-1+1-1+1-1.....) Then, S=1-S

[@jtm8514]

@@satyamtekriwal7376 Yeah, only when you have an infinite sum, you can't do that. 1+1-1+1-1... is divergent and therefore there is no sum. That would be true if the sum was convergent. It's just not a correct method

[@clouddistrictgang24]

I think he is referring to sum(n=0, Infinity)((A)^n) = 1/(1-A) so if you let A=-1 you get the thing but this only convergence for -1

[@VecheslavNovikov]

Dear God, I'd like to file a bug report (see attached video) Amen.

[@TTIOttio]

My thoughts exactly XD

[@declanpeters3305]

I don't get it

[@nickjohn2051]

+Declan Peters It means even infinite sum number could result in -1/12. That odd especially infinite is larger than -1/12.

[@MhD39]

+Vecheslav Novikov looooooooooooool

[@spinzed]

The most correct comment I've ever seen (this one is a little broken).

[@godbroccoli11]

Pretty sure shifting the second set of S2 over one space is why this shouldn't work. While each sum of numbers is infinite, by adding them all together in such a way you technically leave out the very last number of the second sum which was added to the first sum. So instead of 2(S2) = 1/2, I think 2(S2) = 1/2 + (last number added in sum), which thus would make everything else inaccurate due to the last number being undefined and now making the set undefined. Just my guess though

[@-entr0pY]

There is no last number tho.

[@godbroccoli11]

@@-entr0pY That’s why I said technically. At any point in time there is one less number in the second set than the first

[@-entr0pY]

@@godbroccoli11 But that logic doesnt work because it implies the series end at some point where one other number can be left out.

[@vickm7761]

@@-entr0pY I agree with Entropy--the sliding of the numbers is merely a strategy for organizing the infinite list into a more easily-understandable sequence. "One less number in the second set" doesn't apply here, as these sets have infinite elements. It's not something intuitive at all, but neither is the concept of infinity.

[@mikosoft]

Just an astounding leap of logic. How can you say that a sum is an average? Average is a sum divided by the size of the data pool. A sum is a sum. Your sum 1+1-1+1 ... is divergent and cannot be solved. The case is closed.

[@numberphile]

mikosoft One minus one plus one minus one - Numberphile

[@mikosoft]

Numberphile Yes, that's nice that you use Cesaro summation, however, this summation is not a strict sum. It is still an average. Just because it has summation in the name doesn't mean you can use it as a sum. On the other hand, if you in your videos consider "=" to be something else than standard equal sign then it's all right but you have to define your operators first. But considering your "=" is not equality than your arithmetic gymnastic has no practical application anyway.

[@numberphile]

mikosoft Tony's article is also good - bit.ly/TonyResponse - I am not really having arguments with people, and certainly when we start saying "this summation is different to this one" that is important stuff, but starting to move away from the realm of a quirky, smiling KZbin video. Don't get me wrong, a section at the start of the video defining operators sounds fun and all, but... :)

[@AndreaRoll]

i think the problem is not the first sum S1. Even if you don't do the average you still have that the result is 1 or 0 depending on where you stop. this leads the second sum to be equal to 1/2 or 0 and in the end you still have a finite number to handle. i think the problem is that he handles the S2 in the wrong way. He basically usues normal algebra to handle the infinite order. Therefore he would for example say that infinite divided by infinite ( oo/oo ) equals one. ( in the specific case of the demonstration he will say that at some point infinite minus infinite equals one )

[@zacharyst0ne]

Numberphile Let Z=1-1+1-1... then Z+Z=(1-1+1-1...) + (1-1+1-1...)=1-1+1-1... It follows that 2Z=Z. If Z=1/2, then we arrive at 1=1/2, which is clearly a contradiction.

[@AnAverageRecon]

There is no logical way you can get a NEGATIVE fraction from only adding positive numbers.

[@hypersonicmonkeybrains3418]

Damn, i was hoping the answer would be 42.

[@harishkumaar9085]

Since all the basics of maths are ignored here. You could possibly get that too , by careful manipulation.

[@codywang1801]

hitchhiker lol

[@hmmm3210]

42 likes rn

[@sniper1326]

@@harishkumaar9085 the proof shown here isn't the real one...but this is the simplest one...

[@tommyvasec5216]

@Dr Deuteron Negative Infinity! Do I get an A?

[@sordidknifeparty]

This is definitely mathematical hocus-pocus, as one of the primary postulates of mathematics is that the sum of two positive numbers is a positive number. That being true the sum of any number of positive numbers is also a positive number. That being true the sum of all positive numbers is also a positive number. I'm not sure what happened here that allows you to get this obviously incorrect answer oh, but it is obviously incorrect

[@dp2404]

The mistake starts with the divergent series 01010101. He "makes" it converge to 1/2 and then goes on to use rules for convergent series and gets these absurd results.

[@cygil1]

This is an excellent proof of the fact that if you attempt to sum a divergent series, you get a garbage result.

[@erroid]

+cygil1 which is not so garbage for physicists if they say those number occur evrywhere

[@douggwyn9656]

+erroid The problem is that since it is garbage logic, you can't trust it in applications. In another thread I gave the example of a bridge designer who uses an infinite series to approximate local stresses on a long bridge; if the approximation mathematically shows that the stress on the bridge exceeds any supportable value, but he recalls this video and substitutes -1/12 which would be more than safe (if it were only correct), would you want to drive over his bridge?

[@alexkfridges]

+erroid "string theorists" not physicists ;)

[@Trias805]

Well, according to the video, the theory is matched with experiment resulsts, so we cannot completely disregard this, however mindblowing it is.

[@douggwyn9656]

String theory doesn't have any experimental results yet, and anyway mathematical verification is to be found in rigorous, logical proof, not in physical experiments. In this particular case, the actual sum of all positive integers is provably divergent (to +infinity), not -1/12, and the errors in the reasoning have been pointed out several times in the commentary: The -1/12 comes from something else (Riemann zeta) that is not equal to the original series but is a substitute for it. No justification has been given for making the substitution, but even if there were some form of justification, it could not be on the grounds of numerical equality, since obviously -1/12 is not equal to +infinity.

[@absenceofgravitas]

1 + 1 - 1 + 1 -1 + 1.. doesn't "sum" to 1/2, it's a DIVERGENT SERIES. This is 1st year Calc stuff.

[@prim16]

It really shocks me that they blatantly ignore every Calculus law, and they present it as if it's fact, without any method of disproving Calculus theorems.

[@autitToGo]

as a matter of fact they give 1/2 as given as if it was a sort of definition of the sum and, ok, I do not know Strings theory and it may be that there it can be useful to be defined that way, but 1) they should say clear, that they are not using regular math, sure not the regular definition of sum. 2) They keep on using regular properties of sums and their definitions indeed, they are therefore mixing up different definitions of numbers and of operations, that is working within different groups or fields, what cannot be done.

[@w1darr]

@@autitToGo You hit the nail, unfortunately, your answer is covered in the third row and the majority of the guys watching would never read and let the video mislead them.

[@farhanbadrkiani7259]

S=1-1+1-1+1... S=1-(1-1+1-1...) S=1-S 2S=1 S=1/2

[@bash555]

Farhan Badr Kiani Your 2nd line is false. That operation can only be applied on a convergent series, and S is not convergent.

[@janreiter3950]

I don't agree that it's possible to "shift" the terms for an infinite series. I've learned that when calculating with infinite series you must assign the sum to be N long and let N go towards infinity. If we said that we could shift the terms we could do this: S = 1 + 2 + 3 + 4 + ... (Like in the video) WIth subtratction: S - S = 0 = 1 + 2 + 3 + 4 + 5 + ... - (1 + 2 + 3 + 4 + 5 + ...) = 0 But if we shift it: S - S = 0 = 1 + 2 + 3 + 4 + 5 + ... - (1 + 2 + 3 + 4 + 5 + ... ) = 1 + 1 +1 +1 + 1 + 1 + ... And: S - S = 0 = 1 + 2 + 3 + 4 + ... - (1 + 2 + 3 + 4 + ...) = -1 - 1 - 1 - 1 - 1 - ... If shifting is allowed then: -1 - 1 - 1 - 1 - 1 - ... = 1 + 1 + 1 + 1 + 1 + 1 + = 0 This makes no sense.

[@MrAkashvj96]

Now the problem here is that the sums 1+1+1+1+1+1....... and the sums -1-1-1-1-1-1-...... are both indeterminate and can take the form of any real number (yes I'm absolutely serious). This is because to evaluate the 2 sums you have to consider a 3rd sum and the answer to both these sums varies with your pick for the 3rd sum. But 0 is not indeterminate, it's a fixed value and that's why that's the answer here.

[@joejackson3090]

I don't know a ton about math, but I agree. I kind of thought of the equation as an infinitely long zipper; it doesn't matter where or if it ends, you can't start the zipper with only one starter-block-thing, it won't zip if you don't match them up.

[@caperUnderscore26]

What if the hotel was in front of the paving machine for the Infinite road ? WOuld it exist then?

[@angweeshuen]

***** But the infinite hotel is a paradox, no? A contradiction when the hotel proclaims that it is fully booked, yet it still can accommodate more guests.

[@sivad1025]

Infinite sums create weird contradictions. But there's still infinite space. You're trying to limit infinity but infinity is a concept of Neverending beyond comprehension.

[@64-Kare]

You can't multiply infinity by 2! It would result in nonsense like "2 infinities are greater than 1 infinity".

[@coreymagin]

How can you just average the stopping point and call that a solution? You should call that an average?

[@oralboytoy]

You made two blatant mathematical fallacies in your video. 1. The sum of the first series you showed is absolutely not 1/2. It will never be 1/2. This is a divergent and discrete, oscillating series. You calculated the arithmetic mean of the series for every finite truncation, which will never be equal to the sum, because the sum doesn't exist. 2. When you add two series, you can't simply shift all of the terms to the right or to the left for the convenience of whatever result you're trying to attain. I can easily disprove that. Consider two series. The first is (1+2+3+4+5+6+7+...). The second is (-1 -2 -3 -4 -5 -6 -7 -...). The second series is simply the negation of the first. Obviously their sum is the convergent constant series, (0+0+0+0+0+0+...). However, if we inexplicably decide to shift all of the negative number in the second series to the RIGHT as you did in the video, and then add the terms of the two series vertically, we'll now get the series (1+1+1+1+1+1+1+...) which is a divergent series, and not a remotely accurate result. If string theory is based on this illogic, then theoretical physicists should refine their arithmetic abilities.

[@Prochillah90]

thats exactly what i thought! why even "shifting" in the first place? for what reason? in your example you could shifte one more time without a reason and you would get (1+2+2+2+2+2+2...) for me thats the same as saying "ok now once we know we have this result, we could add a banana to it! and for that we got banana(1+2+2+2) So with that we can prove that mathematics are really made for monkeys".

[@TheSurpremeLogician]

I don't understand why they shift it in the first place.

[@qantj]

Infinity does weird things to mathematics.

[@oralboytoy]

TheMentallord I'm a calculus teacher with a math degree. UT, class of '14.

[@TheMentallord]

+jg bubba then you should be fired immeadiatly. no calculus teacher should ever say that the sum of 2 divergent series is 'obviously' 0, because they are DIVERGENT. you cant say that infinity - infinity = 0, thats just plain wrong.

[@BrianBell4073]

This is bollox. For S1, you can't just stop on odd or even. Infinity is infinity. It is a concept. It isn't a number. You have to keep going.

[@numberphile]

Yet sometimes these ideas/series appear in nature and physics, where saying things like "forever", "infinity", or "it just blows up" can't be accepted so easily. (I also imagine you have already seen our Grandi's Series video kzbin.info/www/bejne/hnTYkHWEg65orpY which covers the multiple ways in which S1 can be argued to equal 1/2....) www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help

[@RetroAdvance]

If it can't be accepted so easily, then the series 1+2+3... itself isn't an appropriate model, as simple as that. Rather there are several infinities at play, and this is just a trick to cancel out the infinite "junk" out of them, given the right conditions/context. Otherwise I could also claim that S = 1+2+3+4... = 1, I only forgot to tell you that my condition/context is that I divide it again by itself, S/S :P

[@BrianBell4073]

+RetroAdvance.....when you get down to the hard sums then treating infinity as a number allows you to prove anything. I think the flaw is that treating infinity as a number for a 'well behaved' series gives a common sense result.....which is then extrapolated to the series which are not 'well behaved. I suspect the 'well behaved' series results are nothing more than a fluke and should not be extrapolated

[@RetroAdvance]

I think it is actually another concept, an analytical continuation, there can be a function that also assigns a value to a divergent sum. But this value has a different meaning, it's the "imaginary part" so to speak. The problem is only that it is not introduced as such in the video. All that is said or hinted at is "but if you go to infinity you will get -1/12 as a conventional limes", which simply is not the case as infinity is bigger than every finite sum of the series.

[@tomkrausz2166]

BrianBell4073 yes I'm sure you're better than a proof in a published textbook

[@keithhigh7773]

Shame you did not mention the great Indian (largely self taught) mathematician Srinivasa Ramanujan who first postulated this idea back in the early 20th Century. He died in 1920 aged 32. Even today, the work he left behind is still proving both challenging and useful.

[@sreenavenugopalan936]

Absolutely 💯

[@ritwikbasak4960]

True

[@zebt7477]

He didnt because he knew that this infinite summation result is false

[@tibschris]

All this rests on Grandi's series summing to a finite value. Well for that to work, we have to assume 1+1-1+1-1+... is even a finite number at all. Why should we, when it isn't even convergent?

[@filipgagacev6697]

This video is like that episode of fairly odd parents where timmy got a mathematician to prove to his teacher that 2+2=5

[@trevinbeattie4888]

That equation is obviously true … for large values of 2 and small values of 5. ;)

[@Bruno_Noobador]

@@trevinbeattie4888 (QED)

[@alan137]

That mathematician was Stephen Hawking, a physicist 😄

[@fangzhouli6363]

Totally wrong. The value -1/12 comes out due to analytical continuation. Assigning a value to the sum of an infinite divergent sequence is neither mathematical nor physical.

[@iPlayDotaReligiously]

But it is used in string theory! U think its not physic?

[@fangzhouli6363]

@@iPlayDotaReligiously You are changing my argument. I was referring the way they proved was totally wrong. In string theory book it simply says "by regulating the theory, you can evaluate the summation as -1/12", so technically the book just avoids introducing more concepts to the readers.

[@mehul050]

You should learn about the person who came up with this sum. Srinivasan Ramanujan. ♥️

[@ahmadtaufikchoo2270]

Your self confidence is mind-blowing. Congratulations.

[@ericzgrey]

You can get any arbitrary value by assigning any arbitrary amount to the series which has no definition.

[@jarednicholas3180]

Seriously guys, just stop... There's an error in this methodology ftom the start

[@adryanus9]

i'm just astonished how a infinite sum of positive numbers is a negative number, and people act like it is physicaly possible... i bet it happens in "theoretical physics", that is still unproven and higly theorectical, it might explain everything, or it might just be wrong as everything... i have a really low education in maths, but an infinite sum of a alternating series, doesnt converge to a number, is diverging, and certainly not 1/2

[@endoscopisis]

how the fuck S1 converges to infinity? its values are 0,1,0,1,0,1,.... it does not converge!

[@15october91]

Please see Mathologer's video on this.

[@christopherstark887]

Please see "Why -1/12 is a gold nugget" with Dr. Edward Frenzel watch?v=0Oazb7IWzbA

[@bikashthapa7316]

i think mathologer have corrected this video

[@ruszki]

... in a very asshole way. They could make a "What's really going on here?" video, but instead, they chose a really wrong style to present their obviously true argument.

[@culwin]

Which one? Mathologer's first one? Or the latest one? Maybe he'll make a third one that really pounds the point home for the zillionth time.

[@15october91]

culwin the second.

[@A_Turner]

Why are these guys so poorly funded that they’re writing their maths on old envelopes instead of fresh paper?

[@sxnchou]

beats me 😂😂

[@michaelmiller2210]

the numberphile brown paper is iconic

[@FragMentEditing]

I don't think everyone in these comments realizes that they're using averages, since those sequences don't have an actual sum.

[@comsumptionfordistraction4538]

Mathologer just dropped a mathematical diss track against this video.

[@culwin]

The equivalent of releasing a diss track to a 4 year old song that everyone already dissed.

[@OHYS]

Bikram and Bishal LOLOLOL SAMMMEEE

[@kcccc5682]

The diss track fell flat though as all of mathologers "points" had already been debunked in the follow up post and video linked in the video description.

[@philosophpascal]

what do you mean with "debunked"? this is obvious garbage,Mathologers points hold and he is right. also, what link do you mean?

[@VYScuti]

P-Zombie how is this wrong?

[@brumels1570]

that's why arithmetic with divergent infinite series requires care else u end up with nonsense.

[@LyraLyraPantsOnFyra]

The issue I see immediately is the idea of taking the average of the two possible results depending on if infinity is an odd or even number... infinity is not a number and so the function 1-1+1-1+1... is undefined. Taking the average is ridiculous.

[@numberphile]

Here's an extra article from Tony (in the video) --- bit.ly/TonyResponse

[@Gunno77]

Basically a longer explanation of the mistakes he makes in the video - if we use non-conventional methods to force values onto things that have no value, we can use those forced values in conventional ways. It doesn't make it any more correct to do so in terms of those conventional mathematical operations.

[@majorkatzmann2240]

What I don't really understand is why you are allowed to assume that 1 - 1 + 1 ... is 1/2. In standard mathematics this series is divergent so it doesn't sum to anything. So how exactly should the sum of an infinite series be defined that 1 + 2 + 3 + ... would be -1/12?

[@Theelepeltjel]

It's called Cesaro summation, an alternative summation method also used in Fourier theory: if a series has a normal limit, it equals the Cesaro sum, but some divergent series (such as S1 here) do converge in Cesaro. It's explained clearly in the video linked in the description, and the wikipedia page actually gives this series as an example.

[@qqleq]

They prove that in another vid. There are BTW different ways to prove it, if x = 1-1+1-1+1-1+... Then 1-x = 1-1+1-1+1-1+1... = x, so 1-x = x that leaves nothing but x = ½ is the simplest prove. If it is valid, I do not know. But it does seem to make very good sense and it holds in numerous examples, too...

[@JohannVII]

It's a Cesáro summation - the Wikipedia article on it has a good explanation for anyone who's taken calculus (people who haven't likely won't be familiar with the concepts involved). It's not a standard mathematical summation, and this video is more than a bit disingenuous.

[@enriquemiguelarroyochavela3897]

You are right. It is not standard mathematics, I mean it's no an analytical result is more like a statistical result. Check the other video

[@nihil1]

Cesaro Summation (but there are other ways). To calculate an infinite sum, you have to define how to do the summation as you cannot actually sum an infinite number of times. The standard way is to calculate if the sum of the n first terms with n increasing to infinity converges to something (defining what 'convergence' is is an entire topic on its own). Then you attach that limit value to the sum. But there are other ways to define infinite sumations that will give you answers even for divergent series.

[@floyo]

This shows why physicists are not mathematicians

[@reizinhodojogo3956]

ye theres many things u could do to unprove this

[@1RajatS]

true🥰🥰🥰😍😍😂😂😂

[@jackbrennan3709]

Although in Moriartys words as physicists we are willing to stomach a little less rigor…

[@jackbrennan3709]

I should really say looseness …

[@iamthegreatest3914]

Bro most of the greatest mathematicians are also physicist like Newton , Gauss , Euler etc

[@sadas3190]

The trick is S2 is divergent so you cannot just shift it along to evaluate a rational number out of it.

[@atifrafique3764]

yes , my point too , i want to shift 10 places or 20 places or even infinte places to right or left and call it 2S , f+ck me

[@mcumadbroi2478]

I'm confused why you can just shift the numbers along when you add them to themselves

[@michaelbauers8800]

As long as it's clear how the operations will repeat, even though it's an infinite process, they will just show enough math to convince you it can be done forever, and they don't actually have to do it forever. For example if you say x = 1 + 2 + 3 + ...; and then say 4x = 4 + 8 + 12 + ..., it's assumed the reader trusts that the pattern makes sense. And then they do the trick of taking two equations involving infinite series, and add or subtract them, and once again, it's assumed by the mathematician that you can see the pattern, and agree it's valid. I would say every step they did was 100% valid except for saying 1 - 1 + 1 -1 +1 ... = 1/2 (the Grandi series). That's the magic step that let's this happen. The reason they can come up with such a crazy answer for the Grandi series is because infinite series can be weird, and by doing the math in different orders, they can come up with different answers.

[@brendanharris9197]

me as well, I can't understand why you can do that in step 2

[@michaelbauers8800]

I can see how it's confusing. I presume you mean the way they did 2 times s sub 2? They didn't change the arithemetic IMO. All they did was shift the numbers. If you take 1 + 2 + 3 and add to that 3 + 4 + 5, and line that up shifted, you get 1 + ( 2 + 3) + (3 + 4) + 5 = 18. So same answer. And that's analgous to what they did. The only difference is that there were an infinite number of terms

[@brendanharris9197]

+Michael Bauers that sort of makes sense, except the part that I am confused about is how you can add them off sequence like that, then end up with a resultant string that to me is kind of gibberish because you've modified it. If they were just going for a singular numerical answer it get it wouldn't matter. I know I'm not going to disprove what hundreds of physicist and mathematicians and that isn't my point at all, I just don't see it.

[@michaelbauers8800]

Infinite sums are not that intuitive. But I can see no problems with adding s sub 2 to s sub 2 and getting 1 - 1 + 1 - 1 + 1 ... The tricky part was saying 1 - 1 +1 -1 + 1 = 1/2 because that's what's called the Cesaro sum. They took a diverging series, rearranged terms and got an answer when you thought there was no answer :)

[@shadowatom]

I think the biggest problem with how this was presented is that it was never made clear that these are not what they equal in the traditional sense. What you're getting are values assigned to the series, like ID numbers in a sense (maybe not the best way to think about it, but it at least gets you off the idea that it actually "equals" that value). All of them diverge. They've just found ways to assign values to each one.

[@FunkyDexter]

But these "values" as inconsistent. You can literally assign any value to the series if you manipulate them in the right way.

[@AmericanRoads]

The whole results are hinged upon an ASSUMPTION that 1+1-1+1-1+1-1+... = 1/2, which is pretty shaky to say the least, don't you think so?

[@gogl0l386]

If you had no problem accepting the way they used algebra in this video, then 1-1+1...=1/2 is not so controversial. S=1-1+1... 1-S=1-(1-1+1...)=1-1+1-1...=S 1-S=S 1=2S 1/2=S 1-1+1...=1/2

[@cristian-bull]

My brain only complained about that part.

[@deljohnson3264]

It is Cesaro summable (average of the partial sums converges) and Abel summable (a_n * r^n converges for 0

[@atticusbeachy3707]

+GogL0L If you use algebra the way they do in this video, you can prove that 1+1+1+1+1+ ... = 0 = -1 Proof: Let S = 1+2+3+4+5+6... Let T = 1+1+1+1+1+1... Now, let us calculate S+T: S+T = 2+3+4+5+6+7... which is actually S-1 (this is S, without the 1 in it.) Hence, T = -1 !!! Now, let us calculate S-T S-T = 0+1+2+3+4+5... which is actually S shifted with one zero Hence, T = 0 !!!!!!!!! Wow, a sum of infinite number of 1s is equal to -1 and 0 simultaneously ! That's what happens when Numberphille defines finite sums of divergent series... (proof copied from Milen Cenov's comment)

[@vinko8237]

Thank you for the honesty (credit to Cenov) and to getting to the real point here.

[@BrijeshAhlawat]

For those of you who are unfamiliar with this series, which has come to be known as the Ramanujan Summation after a famous Indian mathematician named Srinivasa Ramanujan, it states that if you add all the natural numbers, that is 1, 2, 3, 4, and so on, all the way to infinity, you will find that it is equal to -1/12.

[@aditube8781]

How did he think of this ??????

[@RapeNetanyahu]

@@aditube8781 You can say that about most of Ramanujan's mathematical contributions, to be fair

[@aryan_verma_1729]

I hav a doubt If 1-1+1-1.... is a diveegent series and has two ans i.e 0 and 1 then how can we take average of both ans and say that is the correct one....i guess this breaks the law of infinite sequence and series chapter ...this rule is not given in it

[@onionman8160]

It's not a conventional sum though, nor did Ramanujan pretend that it is. It is specifically used to sum divergent series, which is useful but it's not the same as a typical sum.

[@keithhigh7773]

A great tragedy that he died so young, but we must be grateful that he left so much ground breaking maths.

[@McDaniel77]

Nonsense is still nonsense, no matter how you explain it.

[@robertweixelbaum6467]

if you assume that S is infinite, than the equation S - 1/4 = 4S is correct. But you can't subtract S on both sides. With infinite sums you cannot calculate as with normal numbers.

[@eoinlanier5508]

Manipulations of infinite sums are perfectly allowed, with certain restrictions (you can, for example, reorder a finite number of terms within an infinite sum but not an infinite number of terms). There are more restrictions the less absolutely convergent the series is, for instance some sums can have an arbitrary number of zeroes added, and some cannot.

[@akk92278]

Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined. ...

[@seanshirley5938]

Thank you for helping me make sense of this. Some of the assumptions in this video made zero sense to me until you put it in this context.

[@newmanhiding2314]

Finally, someone who actually says what it is.

[@sauravraj5271]

@@newmanhiding2314 finally someone who just copy paste the words from google.

[@Chazulu2]

Using the methods used in this video I was able to show that the summation from n=2 to inf of {S(Pn)*[Pn+G(Pn)]+n} = 1 where S(Pn) is the summation of all of the numbers with non trivial factors strictly greater than Pn. Pn is the nth prime if you include 1 as being prime P1. Thank you all for the inspiring video.

[@Chazulu2]

Update. Showed that G(Pn)= 1+ 1/(Pn+1) -Pn.??? 🤷‍♂️

[@Bollibompa]

​@@Chazulu2 Use the methods presented in this video with heavy caution. They are in no sense of the word robust or deterministic since the underlying framework requires another approach, i.e. analytical continuation of the Riemann Zeta function, regularization and renormalization.

[@Chazulu2]

@Bollibompa I agree, tho I think the result I got is indeed interesting. Like, it's obviously "nonsensical" in the same way that -1/12 is, but the magnitude of the result is just less than the prime number. Note, the result I got was G(Pn)=-(Pn+[1/n^n-1*S(Pn)]-1/Pn-1) Where S(Pn) is the sum of all positive natural numbers with no divisors less than or equal to Pn other than the trivial divisor of 1. I posted the handwritten work on mathoverflow, but they blocked it and sent me to mathstackexchange. I had little desire to post to their presumed sister site after having already been jerked around significantly. If you or others are interested in how I got the result, I'd be happy to post a picture of my work somewhere. I took the sum of all primes to the even powers, multiplied the summation by the prime to get them all to all of the odd powers, added those sums together to get the prime to all of the even and odd powers then took a difference of two squares and canceled a common facror in the numerator and denominator of the general expression. I then essentially shifted the S(Pn) function described earlier and multiplied it by the subsequent prime (or the prime + the prime gap). I also had to get the bounds of the summation to all start at n=2, so I rewrote a +1 as the geometric series. Then, since they were all integer summations from n=2 to inf, I asked the question if canceling the summations is logically consistent. It was a lot of fun, and I would be happy to talk to someone capable and willing to read over my work... let me know if you want me to post the picture somewhere specific (it's like a 1.5 pages)

[@Chazulu2]

Lol, I forgot to circle back to why I think that it's interesting. It could be related to the intuition that in the limit, the gap between prime numbers should be bound by the size of the most recent prime number (even if composite numbers are maximally dense) I have no clue if or how it could relate to the twin prime Conjecture, as I'm not a professional. If analytical continuation relies heavily on the first derivative of a function and the 0th derivative, then the 1/2 vertical line could be a reflection of the 0, 1, oscillation leading to 1/2 used in the 1-1+1-1+1-... portion of the discussion in this and related videos.

[@shardulnikam9322]

Oh, Ramanujan, you left us too early. Would have been interesting to see what else he had in his locker.

[@Wyvern07_]

So by only adding positive numbers you get a negative number?

[@kevinwilbert3966]

unintuitively yes

[@Zenmuron3]

--Wyvern07-- of course not. There are thousands of ways to disprove this silly video

[@Zenmuron3]

Avana Sure. The series created by n, where n is a natural number, is a divergent series because the sequence of partial sums, Sn = n = {1, 2, 3, 4, ...} is a divergent sequence. Proof: Let Sn be the sequence of partial sums of the series of n. Let M be any natural number. Let N be M + 1 (The smallest natural number larger than M). Then for any n > N, we have Sn = n > N > M. Thus this sequence is unbounded and increasing (increasing is easily proven using induction). Thus this sequence is divergent to infinity. By definition, the sequence of partial sums Sn is divergent if and only if the series created by n is divergent. I don't want everything you own, I would rather have you learn real mathematics. Do not believe what you watch "smart" people say and do, PROVE IT your self!

[@Zenmuron3]

When you read mathematical proofs, as a reader you look to break the logic. When I say, for example, "let M be any natural number" you are supposed to try to find a "M" (natural number) that will be a counter example to my argument. However, when I say "let Sn be the sequence of partial sums" you are not able to change this because this is from the definition for what a series (or sum of all) is. Your question is slightly confusing me, but I think I understand. My argument is quite the opposite of "random" and is in-fact very precise. There's a reason why I chose everything the way it is, and it's because it works logically. The reason why he can't use "a random string" to solve the equation "with that" is because this equation has no solution because it is a divergent series (diverges to infinity, obviously). If numbers and mathematics really fascinates you, then it's better to read actual mathematical documents and literature.

[@sandraviknander7898]

The thing is that, this isn't a mathematical result much less a mathematical proof it's more of a physical definition made becaus it works to describe certain physical phenominon. This video realy should have ben uploaded to sixtysymbols instead of numberfiles.

[@lamb4078]

Honestly, if you don't do all the crazy extra stuff, as long as you keep adding positive integers there is no reason logically to expect it to equal anything that isn't ultimately positive.

[@finneganmcbride6224]

But you don’t “keep adding” because infinity isn’t a number, so you can’t add an infinite number of times. You don’t stop the set at the end because there is no end. So you don’t start at the beginning and keep adding. You have to somehow do it all at once.

[@lamb4078]

@@finneganmcbride6224 Like I said, logically, adding all *positive* integers should never result in a negative fraction. A statement like that is too abstract for such a pragmatic premise.

[@itzvihaan6734]

This is absolute bs Look up Mathologer's explanation to why this video is complete wrong

[@pianoplaynight]

Unless you're a programmer 😂

[@MikehMike01]

Everything in this video is a fabrication. They might as well say it summed to a pumpkin

[@adisingh4422]

I thinks it flawed in averaging 0 and 1 for that first sum.

[@siddharthraturi3914]

There's actual proof for it. The -1/12 result is also correct proven in the Riemann Zeta function.

[@Shuizid]

@@siddharthraturi3914 There might be actual proof - but this videos is mathematically incorrect on several levels.

[@doomshroom7682]

Yeah that's a divergent series with no sum. It is either 1 or 0 but it never converges.

[@gflow8357]

Siddharth Raturi But the analytic continuation of the Riemann Zeta function is not this sum.

[@felipelanger8851]

You're right. In calculus, we study convergent and divergent series, and 1-1+1-1+1 is a case of divergent series. Every single statement they do is based on the mistake that 1-1+1-1... Converges to 1/2

[@blakecarlson1057]

I love how he says it's not a bunch of mathematical hocus pocus one second then says you have to do the mathematical hocus pocus in order to reach such a result.

[@mrbdzz]

Hocus pocus indeed. The fundamental problem here is that (1 - 1 + 1 - 1 + ... ) does not actually converge to 1/2 nor any other number. This is a classical case of applying a false statement, which allows one basically to get whatever as the end result. In this case, 1+2+3... = -1/12. BTW, 1+2+3+... does not converge either. However, this does not mean that the related physics are readily wrong. The Cesaro sum, which is a transfomation of the series, actually gives you 1/2. The cesaro sum gives correct limits for converging series, and limits for some non-converging series, too. But there are other transformations, which also yield correct limits for convergible series, but other limits than 1/2 for the 1-1+1-... series. If these things work in physics, it tells you that the physics are actually more related to the Cesaro sums (or other transformations) of these series instead of the series themselves. It wouldn't be the first time the physicists take little shortcuts in their math, but I think we can forgive them doing that if the end results match with experiments.

[@douggwyn9656]

Note that the Cesaro sum for 1+2+3+4+... is infinite, so that series behaves even worse than 1-1+1-1+... Your point about the physics is valid: the original model that has been set up makes a bad prediction, and "regularization" amounts to a distributed smoothing procedure that "tames" the divergence. Why the particular procedure is adopted has to my knowledge never been clearly explained, other than by saying that it seems to "work". At the very least, better explanation is needed, or better yet, a model should be developed that doesn't yield infinity for what are physically finite quantities.

[@Hasan...]

I thought the same too, but I tried with the other possible answers of: S1 = 1 - 1 + 1 - 1 + 1 .. and the answer has to be either 0 or 1, correct? But even if you take 0 or 1, and continue with the rest (with no more assumptions), you end up getting S = 1+2+3+4+5.. = either 0, or, -1/3 !! Try it out. Can you explain this?

[@douggwyn9656]

The explanation could be that you can get a wide variety of different finite results by manipulating terms of a divergent series. None of them is "correct".

[@nialv22]

+Doug Gwyn is the real hero in this comment section.

[@fortawesomm]

why did he shift the numbers when multiply S2?

[@awlabrador]

I saw this for this first time today when my son drew my attention to it, and I immediately knew that S1 wasn’t convergent and that the entire argument fell apart from that point. It gave me a headache to sit through. I felt like the person who yells at the television telling the teenagers not to go into the barn alone and unarmed in the middle of the night while that masked serial killer is still on the loose. Except I don’t watch those kinds of shows. One huge benefit, though, was in being able to have a nice discussion with my son about math and physics, mathematicians, theorists, and experimental physicists. And yes, I’m a physicist, too.

[@RSLT]

Note that you can say X^2+1 =0 has no solution, and that could be correct. There are different levels of math. In level zero, 1+2+3-... doesn't have a limit (like x^2+1=0 has no solution). However, at a higher level, the roots of X^2+1=0 are well defined. Your laptop or cellphone car,... works based on complex analysis principal. At that level, there is no question that 1+2+3...=-1/12

[@Gunno77]

@@RSLT 1+2+3... *is associated with* -1/12. It's incorrect to use "=" here without that clarification.