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
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.
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.
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.
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 RUclips.
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.
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
@@Synthesis_Softwaretech 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.
@@Synthesis_Softwaretech 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 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.
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.
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.
@@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.
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
@@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.
@@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?
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
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
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.
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.
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
@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.
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.
***** 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. ... ?
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 ruclips.net/video/PCu_BNNI5x4/видео.html 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
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
+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
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.
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
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.
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".
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.
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
@@quantspazar6731The 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
@@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
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.
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
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
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..
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
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
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 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.
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 RUclips video. Don't get me wrong, a section at the start of the video defining operators sounds fun and all, but... :)
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 )
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.
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
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.
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
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
S = 1 - 1 + -1.... Ok so the claim is made that since we don't know if the answer is 0 or 1, the answer is 0.5. That doesn't follow. If we can't say what the answer is, then it's undefined, not an average.
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.
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....
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.
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".
+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.
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?
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
@@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.
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.
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
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
There are multiple ways to show it is, such as with Ramanujan's summing, Cesàro sum and Abel sum etc., see en.wikipedia.org/wiki/Summation_of_Grandi%27s_series.
I'll copy my working from my original comment. You can shift the sum over two spaces so you never have to deal with 1-1+1-1+... : 2*S2=1 - 2 + (3-4+5-6+...) + (1-2+3-4+...) 2*S2=1 - 2 + (3+1) - (4+2) + (5+3) - ... 2*S2=1 - 2 + 4 - 6 + 8 - 10 + ... 2*S2=1 - 2(1 - 2 + 3 - 4 + ...) 2*S2=1-2*S2 4*S2=1 S2=1/4
I never said it was equal. These are ways to interpret a value for the sum, which doesn't imply strict equality via convergence, which the sum obviously doesn't do. Admittedly this is one of the aspects the video should've been more clear about.
+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?
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.
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..
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.
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.
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.
yeah these comments are hilarious. there's uni proffesors and physics text books proving the sum of all number = -1/12 but these people think they know better.
I defy you to cite one textbook that purports to prove that the sum of all natural numbers equals -1/12. Certainly the one cited in this video not only didn't claim to prove that, it didn't even claim that.
+SGDTV well People in Physics may be happy with the First sum, but in the first Five weeks of studying math, your Prof will actually take this as an example of wrong convergence
Yeah, there's a rule that you can't take the limit of a number that doesn't converge, because it's meaningless and undefined. It would be more consistent with the accepted rules of math to say S1 = 1/2 +/- 1/2 . Saying it yet another way, instead of just writing " . . . ", if you write out the series to an arbitrary last number "n" , then you should see where that reasoning breaks the accepted maths. But I'm sure others have shown this because it's not difficult.
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.
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
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.
@@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
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
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.
I want to know what law allowed the addition of S2 (producing 2S2) to shift the lower term one place to the right before allowing addition to take place. That honestly is what I have a problem with.
That does seem to imply that the last term of the bottom S2 , let's call it n, doesn't have a corresponding -(n-1) term on the top S2, implying that 2S2 tends towards infinity.
+Vecheslav Novikov No, it doesn't. Both are infinite series, so there is no end to them. It's impossible for them to not have a corresponding term to them, even if common sense implies otherwise (because of the shift). Common sense has no place in logic and maths, since it can easily be illogical.
+Brian Streufert All the shifting is is taking advantage of the Commutativity while adding numbers (and subtractions are nothing different than adding a negative number).
+Brian Streufert It has nothing to do with any mathematical properties, it's simply a bit of foresight for visual purposes. 1+1 = 1 + 1 All he did was line up numbers so that you can easily see how the addition will take place.
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.
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?
IronicTB -1/12=sum(n) from 1 to inf =1 + sum(n) from 2 to inf =1 + sum(n+1) from 1 to inf => sum(n+1) from 1 to inf = -13/12 => sum(n+1) from 1 to inf - sum(n) from 1 to inf = -13/12 - (-1/12) => -1 = sum(n+1) from 1 to inf - sum(n) from 1 to inf = -1 = sum(1) from 1 to inf = -1 + sum(1) from 0 to inf => 0 = sum(1) from 0 to inf = sum(1) from 1 to inf = 1 but 0 = 1 cant be true no matter what.
First, It is NOWHERE mentioned in the book that the sum of all natural numbers is equal to -1/12. Go to the part where he shows the book and look closely. It's an ARROW, not an equal sign. And yes, it does make a difference! Second, most mathematicians would probably feel uncomfortable with the proof presented in the video. Notations are not well defined, false assumptions/contexts are made at the beginning (for example just assuming 1-1+1-1+...=1/2 even though it's a divergent series) etc. Third, I don't know much about string theory, but I do know enough about the mathematics involved in the video. I know about the zeta function, about analytic continuation and Ramanujan summation. And by that I mean I didn't just read the Wikipedia page, I actually studied these topics in depth. I don't know about your mathematical background (maybe you can tell me), but If you want I can point out what's wrong with the methods used in the video and explain some of the background to shed some light on it.
Hold on... He is telling us that SOMETHING LESS then S = 3xS. For example S=5 (in my imaginary world) "something less then 5"=4. Then S = 3xS that means 3x5=15. does that mean 5=15?
It's wrong from a physics perspective too, as was appreciated by quantum field theorists in the 1960s, if not before. But they use the trick anyway, because they haven't figured out how to formulate their models so that singularities don't arise in the first place. If you could show how to avoid the divergences it is likely that you would win the Nobel prize in physics.
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.
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
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.
@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
Louis Victor There are a lot of manipulations these two make which can only work for convergent series. For example, if one lets S equal an infinite series and then performs algebraic operations to find a value for S, then the value for S is only valid if the sum converges. For example, let S = 1 + 2 + 4 + 8 + 16 + ... + 2^n + ... Multiply both sides by 2 to get 2S = 2 + 4 + 8 + 16 + ... Notice the right hand side is S - 1. 2S = S - 1 So subtracting S from both sides, S = -1. This result is correct in the 2-adic numbers, but not in standard summation. Using standard summation, you have S = ∞. So, when you subtracting S from both sides, you are subtracting ∞ from ∞, an operation which is strictly disallowed since it can lead to erroneous conclusion. Another manipulation which is made in the video is adding two series together, term by term. Suppose you have two infinite sums: ∑an and ∑bn If ∑an and ∑bn both converge, then ∑an + ∑bn = ∑(an + bn). But, you cannot combine the two series if either diverges. Yet, in the video, they combine divergent series in this way. I think there are a few more illegal manipulations they do in the video, but I don't want to rewatch it right now. Mathologer explains these ideas quite well.
I did it the other way round. The 34 min video from Mythologer is way better and more complete. This video just tells you 'half the truth'. The 'normal' answer ofcourse being n(n+1)/2 It has all to do with 'cheating' by taking averages, and not the actual sum. Or even taking averages of averages.
@@nihlify Oh no, infinite sums absolutely do exist, and sometimes they converge (like, for example the sum of all inverse powers of 2, which converges towards 1). But when they diverge, you can't just take an average of the values they oscillate between and expect it to work.
@@irokosalei5133 It's not "used", in that there are only a couple of applications in QED and the one shown in that book is wrong, leading to the multiverse instead of the observed data.
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.
Nice try but that doesn’t work. Since B-A=A-A=0, so dividing right hand side by zero results in infinity. Thus, the only solution is the trivial solution which is when both A and B = 0
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.
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!
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.
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.
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.
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.
This is awesome, but I personally feel that this method is too assumptious in many ways, even from the beginning, where you had 1 - 1 + 1 - 1 + ..., and chose to take the average, as 0.5. I feel that this is a mathematical flaw, as opposed to the actual sum of all natural numbers. Because if you think about it, it kind of obviously is not the sum of all natural numbers.
There's actually two parts that don't work in regular calculus. Firstly, yes, the first sum is a divergent series (has no obvious answer) and saying "it's 1/2" is like saying anything divided by 0 is 0. Might make sense using certain theorems (rules), but in regular calculus it's nonsense. Secondly, you can't add/subtract/multiply an infinite sequence from another infinite sequence. If you could, you could effectively produce any number you wanted. The sum "1+1+1+1..." subtracted from itself but shifted one number over would end up with just 1.
It's not really an assumption, use the infinite geometric sequence sum formula (a/(1-r)) and the answer comes out as 1/2. This is the limit of the sequence. The infinite sequence approaches 1/2. And after doing a lot of maths you'll realise approaching is the same as equalling.
But it is quite a leap of faith to say +1-1+1-1+1... = 1/2. I mean, it is either 0 or 1. It isn't 1/2. If we didn't have that result, then you can't get this weird sum, right?
@@Magst3r1 It is very clearly defined as ½, in any mathematical context which is capable of rigorously assigning it a value. Borel summation, Holder and Cesaro summation, Abel summation, amd Ramanujan summation, among many other methods, all define it as ½.
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.
***** 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."
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.
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?
I think we should quit using the equals sign for these kinds of sums, because it's more accurate to say that -1/12 represents 1+2+3+4+5+6... rather than saying it is equal to it.
A reputable definition wouldn't really be necessary. By "nature" I mean "inherent features". The sum of the series is infinite, but when you take a closer look at the series itself rather than the sum of its parts, you can derive that it's different from other sums that lead to infinity. That difference we represent with -1/12.
believe or not Anthony but it is equal to negative one twelfth. just like the value of 1+1+1+1+1+1+.... is nothing but zero. the fact that the sum of all natural nos is -1/12 was given by Indian mathematician Ramanujan
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
@@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
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 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.
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.
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".
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.
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.
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.
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?
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".
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.
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...
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
I get a little annoyed about these 1+2+3+4+...=-1/12 videos since they don't explain the errors in the arguments presented. Normal people might get confused into believing that their intuitive belief about 1+2+3+4+... = infinity is wrong which it is not. You should introduce the concept of generating functions.
Can't help but feel the random shifting to the left kind of makes the logic somewhat random. But I guess you gotta do something, and stick with it; then call it math and hope it doesn't cause things to explode.
All the people saying this is bullshit, first off, are you a mathematician? And secondly, if it weren't true, why would it be used in String Theory textbooks?
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.
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
+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.
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 :)
I'm not supporting this video, but to answer your question, the so-called Cesàro sum of a series can be thought of as a generalization of the standard sum, in that it converges to the same value when the series is convergent, it diverges to infinity when the series diverges to infinity, and in cases where the original series does neither, *sometimes* the Cesàro process (limit of average of partial sums) converges to a finite value. Whether or not the Cesàro value is usable depends on the problem you were trying to solve when the non-convergence popped up.
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.
+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.
+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.
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.
+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.
+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.
This broke my brain so hard. I just wanted to watch the video about 0.577, but now I'm down the rabbit hole. If I fully understand this, do I break out of the Matrix?
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.
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.
@@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
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.
Guys, reality check here. When you do something in maths and you end up with a clearly incorrect answer, as you did here, it's time to recheck your maths. Clearly, one of the manipulations you did was incorrect, in this case I believe that the "intuitive" sum of the grandi series is simply wrong. And "shifting" sums to add them is also a bit, well, shifty. It's beyond me how you happily accept this erronous result. It's pretty much akin to dividing by zero somewhere and coming up with 1=2. Even if you didn't know that dividing by zero is not admissible, you would not accept the result and look for the flaw in your reasoning. Same here.
A blog with more links and info - www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help
wtf a new comment from the channel after 10 years
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
Luckily our finite lifespans tell us that his plan is doomed to failure and you will die a rich person.
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.
TrackpadProductions this video just proves that math is broken at some points not that the sum of all natural numbers is negative
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.
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.
i always multiply both sides by zero. Seems to fix things up pretty well.
Just like my future! :D
I try differentiating both sides always....funnily I get the same result as multiplying with zero
Zero is not really a value.
@@youme1414 it is but u don't get the point
I mean, at this point, it seems to be a more logical way to go about it than whatever that was.
A simple stack overflow bug. God will patch it in the next update.
Wow! You're the forefather of Albert Einstein.
Quality.
Still no updates, support is clearly messing
@@waitweightwhite793 Just hope they don't wipe the drive and do a fresh OS install...
But you won't get negative of irrational numbers.
SO error in java would be some integers
My IQ increased by -1/12 after watching this
Infinite intelligence!!!!
@@YT7mcquite the opposite. This is why most people here belive on this video despite ot having numerous errors.
@@bartekordektend to agree here.... I reckon this is a series that politicians are forced to believe in before takeing office.
@@bartekordekWhat errors?
@@N269Could either of you point out a single error? Considering there are numerous, a single one shouldn’t be too hard.
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.
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 RUclips.
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.
Means decreased 😓
I really hope you didn't watch this 12 times
Huh, shoulda only happened once but never.
Top 10 pranks that went too far.
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
@@Synthesis_Softwaretech 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.
@@Synthesis_Softwaretech 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.
@@harishkumaar9085 these western people are just assholes don't waste energy to argue with them they copy our Indian culture and nothing much
@@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.
Me before watching this video: liar
Me after watching this video: cheater
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.
why not?
Jax Infinite series are often defined by their order. When there is no set end of the sequence, you can’t just reorder things.
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.
Its ramanujan infinity sum
Mathematics when RUclips removes the dislike button:
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"
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.
@@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.
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
@@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.
@@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?
This video represents negative knowledge; if you watch it, you will know less about mathematics than when you started.
Jerry Steffens video represents more knowledge than you can comprehend
@@saratoga123321 you're negatively missing the jokes.
It does, since it is false.
@@saratoga123321 It has error in the very first line...
@@dropdatabase2569 r/woooosh
To quote a math teacher from my uni: "It's extremely unpleasant to approximate solutions that don't exist."
paul zapodeanu unpleasant. But not always useless
Aeroscience in this case very useless
Wang Dave not at all. Eventually this stuff lead into the Riemann zeta function. Which is very useful.
Such a boring maths teacher you got
@@huhun23 There are other ways to prove it using basic arithmetic such that a 5th class student can understand. No need of zeta functions
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
Thank you! Guys are easily deceived by their ignorance.
Can we say it's a negative number ?
You can do three proofs by contradiction that adding integers will always give something positive, integer and rational.
Please explain in 100 iq terms. I do not understand.
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
After watching this I have some idea why string theory went off the rails.
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.
Hahaha!! Only physicists...😓
This is a result that explains the Casimir Effect...physically. The analytical continuation of the Reimann Zeta function.
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.
@Brandon Neifert dont get excited that 44 is out of 200
“So now do you believe me?”
Me: *No*
numberphile is disseminating wrong maths and false claims. This vid should have been an april fool. But its still up after 7 years.
@@jamesgrist1101 I agree
@@jamesgrist1101 You can also prove it with Rieman Zeta function.
@@jamesgrist1101 maybe numberphile did not explain the topic so well but THAT DOES NOT mean that the equation is wrong, kid.
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
I think this video perfectly illustrates Proof by Contradiction:
Start with nonsense, end with nonsense.
it all started when they used infinity as a number
How did they start with non sense?
@@x_theandrey9614 Where?
Exactly.
@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.
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…
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.
@@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.
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.
+Enzo Molinari *claps*
Gregery Barton what a buzzkill
+Enzo Molinari Im pretty sure you can prove anything with this premise. Event that Kim Kardashin is smart
+Enzo Molinari Im pretty sure you can prove anything with this premise. Event that Kim Kardashin is smart
***** 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. ... ?
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.
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 ruclips.net/video/PCu_BNNI5x4/видео.html 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
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
+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
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.
BrianBell4073 yes I'm sure you're better than a proof in a published textbook
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
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.
Exactly, this seems very contrived
this comment should be pinned
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".
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.
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
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
Exactly
@@quantspazar6731The 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
@@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
@@quantspazar6731 but have you considered that the sum of all natural numbers is -1/12
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.
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
What if one of the person that goes in was pregnant
no, they would declare the house has as many people as it had before
That's just bad practice as a burglar, assuming the house is empty.
What of that one person is a serial killer with suicide mentality
One of the angriest RUclips comment sections since the incident with the forest in Japan
You'll have to explain that reference to me.
@@RWBHere Logan Paul incident
That analogy is inaccurate because this was here before then
A little different, because nobody liked Logan Paul in the first place.
Because this video is spreading lies and making people stupider and less interested in math. It’s immoral.
"So we take the average of the two" - [raises eyebrow]
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
This should be a pinned commercial, well said 👏
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..
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
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.
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
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
Dear God,
I'd like to file a bug report (see attached video)
Amen.
My thoughts exactly XD
I don't get it
+Declan Peters It means even infinite sum number could result in -1/12. That odd especially infinite is larger than -1/12.
+Vecheslav Novikov looooooooooooool
The most correct comment I've ever seen (this one is a little broken).
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.
mikosoft One minus one plus one minus one - Numberphile
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.
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 RUclips video. Don't get me wrong, a section at the start of the video defining operators sounds fun and all, but... :)
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 )
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.
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
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.
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
Ikr, this video feels like a scam
I mean, there is another way to prove it
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
@@nycolasfelix8828 Come on man, 2S1=2 -2 2 -2 ..., that doesn't converge to any value just like S1
@@giacomoverardo6446 I absolutely agree , you just put a 1 there
S = 1 - 1 + -1.... Ok so the claim is made that since we don't know if the answer is 0 or 1, the answer is 0.5. That doesn't follow. If we can't say what the answer is, then it's undefined, not an average.
The US government is trying this with the national debt.
the best comment ever!
@@gangulic you guys still alive? Just curious 😁
@@ayushdhakal333 allo allo zis iz night hawk can you ear mi?
@@gangulic wow he's alive
Dr. Srinivash Ramanujan✅
Damn, i was hoping the answer would be 42.
Since all the basics of maths are ignored here. You could possibly get that too , by careful manipulation.
hitchhiker lol
42 likes rn
@@harishkumaar9085 the proof shown here isn't the real one...but this is the simplest one...
@Dr Deuteron Negative Infinity! Do I get an A?
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.
I think they don't read the comments
@@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
Username checks out
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....
@@gamester2495 I think that formula only works when -1
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.
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".
I don't understand why they shift it in the first place.
Infinity does weird things to mathematics.
TheMentallord
I'm a calculus teacher with a math degree. UT, class of '14.
+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.
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?
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
@@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.
Now this is brilliant
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
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.
Here you go: www.bradyharanblog.com/blog/2015/1/11/this-blog-probably-wont-help
Brance Finger Thanks for clearing it up, imo the video should get removed from RUclips.
Brance Finger "Guys, I've never studied infinite series or any math, really, but this is totally not true."
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.
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
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
You haven't even convinced me that 1-1+1-1... = 1/2
There are multiple ways to show it is, such as with Ramanujan's summing, Cesàro sum and Abel sum etc., see en.wikipedia.org/wiki/Summation_of_Grandi%27s_series.
Watch the linked video at the beginning. It basically says S=1-1+1-1... and 1-S=1-(1-1+1-1...)=1-1+1-1+1...=S » 1-S=S » S=1/2
Cesaro sum doesn't prove equality. Equality is defined as convergence.
I'll copy my working from my original comment. You can shift the sum over two spaces so you never have to deal with 1-1+1-1+... :
2*S2=1 - 2 + (3-4+5-6+...) + (1-2+3-4+...)
2*S2=1 - 2 + (3+1) - (4+2) + (5+3) - ...
2*S2=1 - 2 + 4 - 6 + 8 - 10 + ...
2*S2=1 - 2(1 - 2 + 3 - 4 + ...)
2*S2=1-2*S2
4*S2=1
S2=1/4
I never said it was equal. These are ways to interpret a value for the sum, which doesn't imply strict equality via convergence, which the sum obviously doesn't do. Admittedly this is one of the aspects the video should've been more clear about.
obvious error, the sum of +1-1+1-1+1-1 ..is NOT 1/2 .. this is a DISCONTINUED function that alternate between 1 and 0 ... YOU DO NOT AVERAGE IT !
4+8+16….. Shouldn’t be = 4S.
This is an excellent proof of the fact that if you attempt to sum a divergent series, you get a garbage result.
+cygil1 which is not so garbage for physicists if they say those number occur evrywhere
+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?
+erroid "string theorists" not physicists ;)
Well, according to the video, the theory is matched with experiment resulsts, so we cannot completely disregard this, however mindblowing it is.
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.
There is no logical way you can get a NEGATIVE fraction from only adding positive numbers.
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..
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.
4-4/4-4=1/2 prouf this question solved
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.
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.
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.
Mathematician: **calculates something, result doesn't make any sense.**
Mathematician: "I define this as correct."
it does make sense?
@@Sadnessiuseless stay in school kids
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.
You have to realize the reason they assign the value -1/12 to this sum is because it is useful in some way.
@@SuperRaidriar I feel like there's a logical physical explanation for that which doesn't include abusing analysis
I love how everyone's a mathematician all of a sudden.
yeah these comments are hilarious. there's uni proffesors and physics text books proving the sum of all number = -1/12 but these people think they know better.
I defy you to cite one textbook that purports to prove that the sum of all natural numbers equals -1/12. Certainly the one cited in this video not only didn't claim to prove that, it didn't even claim that.
Is it so improbable that mathematically educated people would watch video's about mathematics? Am I missing something here?
+SGDTV well People in Physics may be happy with the First sum, but in the first Five weeks of studying math, your Prof will actually take this as an example of wrong convergence
Yeah, there's a rule that you can't take the limit of a number that
doesn't converge, because it's meaningless and undefined. It would be
more consistent with the accepted rules of math to say S1 = 1/2 +/-
1/2 . Saying it yet another way, instead of just writing " . . . ", if
you write out the series to an arbitrary last number "n" , then you
should see where that reasoning breaks the accepted maths. But I'm sure
others have shown this because it's not difficult.
how the fuck S1 converges to infinity? its values are 0,1,0,1,0,1,.... it does not converge!
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.
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
@@mutt8553 It seems like a bit of a gimmick to me so not all that serious
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.
@@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
You can also claim that 2+2=-1/12 if you want, but without evidence it doesn't really matter.
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
Wow you’re so smart 😮 😱
@@xenqor5438 That's not nice at all, he's trying his best, let him arrive to the conclusion O_O
The problem/flaw of all this begins at the assumption that the "average" of 1-1+1-1+1.... equals 1/2
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
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.
I want to know what law allowed the addition of S2 (producing 2S2) to shift the lower term one place to the right before allowing addition to take place. That honestly is what I have a problem with.
That does seem to imply that the last term of the bottom S2 , let's call it n, doesn't have a corresponding -(n-1) term on the top S2, implying that 2S2 tends towards infinity.
+Vecheslav Novikov No, it doesn't. Both are infinite series, so there is no end to them. It's impossible for them to not have a corresponding term to them, even if common sense implies otherwise (because of the shift). Common sense has no place in logic and maths, since it can easily be illogical.
+Brian Streufert All the shifting is is taking advantage of the Commutativity while adding numbers (and subtractions are nothing different than adding a negative number).
+Brian Streufert It has nothing to do with any mathematical properties, it's simply a bit of foresight for visual purposes.
1+1 = 1 + 1
All he did was line up numbers so that you can easily see how the addition will take place.
+Brian Streufert Is a basic property of sums: (a+b) + c = a + (b+c).
In other words you can add the terms in the order you prefer.
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.
No, they were making more money each day and ended up with negative money xd
It's funny because they actually love hiring physics majors
The limit of the sequence of partial sums of the sequence is not 1/2. It does not exist. Stick to physics.
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?
This shows why physicists are not mathematicians
ye theres many things u could do to unprove this
true🥰🥰🥰😍😍😂😂😂
Although in Moriartys words as physicists we are willing to stomach a little less rigor…
I should really say looseness …
Bro most of the greatest mathematicians are also physicist like Newton , Gauss , Euler etc
2,818 people think they're smarter than the majority of mathematicians and scientists.
IronicTB
-1/12=sum(n) from 1 to inf
=1 + sum(n) from 2 to inf
=1 + sum(n+1) from 1 to inf
=> sum(n+1) from 1 to inf = -13/12
=> sum(n+1) from 1 to inf - sum(n) from 1 to inf = -13/12 - (-1/12)
=> -1 = sum(n+1) from 1 to inf - sum(n) from 1 to inf
= -1 = sum(1) from 1 to inf
= -1 + sum(1) from 0 to inf
=> 0 = sum(1) from 0 to inf
= sum(1) from 1 to inf
= 1
but 0 = 1 cant be true no matter what.
IronicTB The "majority" of scientists is 2 people?
DHMdM I can assure you that quite a lot of important people agree with the two people in this video.
IronicTB
Such as...? (by the way, ever heard of 'appeal to authority'?)
First, It is NOWHERE mentioned in the book that the sum of all natural numbers is equal to -1/12. Go to the part where he shows the book and look closely. It's an ARROW, not an equal sign. And yes, it does make a difference!
Second, most mathematicians would probably feel uncomfortable with the proof presented in the video. Notations are not well defined, false assumptions/contexts are made at the beginning (for example just assuming 1-1+1-1+...=1/2 even though it's a divergent series) etc.
Third, I don't know much about string theory, but I do know enough about the mathematics involved in the video. I know about the zeta function, about analytic continuation and Ramanujan summation. And by that I mean I didn't just read the Wikipedia page, I actually studied these topics in depth.
I don't know about your mathematical background (maybe you can tell me), but If you want I can point out what's wrong with the methods used in the video and explain some of the background to shed some light on it.
Dude shifted a whole infinite series to the right and said the answer is 1/2 😂 What is this a Kangaroo 🦘 Class?
How can you just average the stopping point and call that a solution? You should call that an average?
This video is like that episode of fairly odd parents where timmy got a mathematician to prove to his teacher that 2+2=5
That equation is obviously true … for large values of 2 and small values of 5. ;)
@@trevinbeattie4888 (QED)
That mathematician was Stephen Hawking, a physicist 😄
If 1+2+3+4+5...=-1/12, does that mean that 1>1+2+3...?
don't open pandora's box
Hold on... He is telling us that SOMETHING LESS then S = 3xS. For example S=5 (in my imaginary world) "something less then 5"=4. Then S = 3xS that means 3x5=15. does that mean 5=15?
Josh Yord yes
It's wrong from a physics perspective too, as was appreciated by quantum field theorists in the 1960s, if not before. But they use the trick anyway, because they haven't figured out how to formulate their models so that singularities don't arise in the first place. If you could show how to avoid the divergences it is likely that you would win the Nobel prize in physics.
Even zero is more than this sequence
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.
Yeah teaching false assumptions and statements without specifying in what branch we are actually talking about....
I don't think so
@@canyoupoop😅
No, but @@canyoupoop
Yeah my math teacher taught me that when a sequence approaches infinity as its limit, the series will be divergent
@@Grgrqryeah unless |r|
I prefer to see this as a demonstration that 1-1+1-1+1-1... does NOT equal 1/2
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
@@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
@Sari Çizmeli Mehmet Ağa infinity is equal to two times infinity plus 1. Infinity is odd.
You are so right, showing that the limit does not exist is quite simple
S=1-1+1-1+1-...
S=1-(1-1+1-1+1...)
S=1-S
2S=1
S=1/2
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.
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
You can't compare two infinity if you want to compare two Infinity you need limit to compare infonity
Physicians*
Mathematicians care about this and don't trick people with false calculations
@Tom Petitdidier it's just hard coded subjectivity induced by scientists to make things less complicated and more useful.
@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
Who else came here from the Mathologer video because it was 34 minutes long?
The Mathologer video is _very, very_ different in content from this one. I suggest watching that one too.
This here is no proof, lots of forbidden manipulations. The Mathologer knows what he's talking about.
What forbidden manipulations?
Louis Victor There are a lot of manipulations these two make which can only work for convergent series.
For example, if one lets S equal an infinite series and then performs algebraic operations to find a value for S, then the value for S is only valid if the sum converges.
For example, let S = 1 + 2 + 4 + 8 + 16 + ... + 2^n + ...
Multiply both sides by 2 to get
2S = 2 + 4 + 8 + 16 + ...
Notice the right hand side is S - 1.
2S = S - 1
So subtracting S from both sides, S = -1.
This result is correct in the 2-adic numbers, but not in standard summation. Using standard summation, you have S = ∞. So, when you subtracting S from both sides, you are subtracting ∞ from ∞, an operation which is strictly disallowed since it can lead to erroneous conclusion.
Another manipulation which is made in the video is adding two series together, term by term.
Suppose you have two infinite sums: ∑an and ∑bn
If ∑an and ∑bn both converge, then ∑an + ∑bn = ∑(an + bn). But, you cannot combine the two series if either diverges. Yet, in the video, they combine divergent series in this way.
I think there are a few more illegal manipulations they do in the video, but I don't want to rewatch it right now.
Mathologer explains these ideas quite well.
I did it the other way round. The 34 min video from Mythologer is way better and more complete.
This video just tells you 'half the truth'. The 'normal' answer ofcourse being n(n+1)/2
It has all to do with 'cheating' by taking averages, and not the actual sum. Or even taking averages of averages.
taking an average has absolutely no place here. This is a summation, not statistics.
You can't sum up infinity
@@nihlify Oh no, infinite sums absolutely do exist, and sometimes they converge (like, for example the sum of all inverse powers of 2, which converges towards 1). But when they diverge, you can't just take an average of the values they oscillate between and expect it to work.
Exactly! And then just go on to use rules for convergent series 🤦♀️
right???
>This sum is actually used in physics so we know it's true!
>*shows a book on string theory*
bruh
I don't know what drives someone to use greentext on RUclips
It's used in Quantum Electrodynamics so yeah it is.
@@irokosalei5133 It's not "used", in that there are only a couple of applications in QED and the one shown in that book is wrong, leading to the multiverse instead of the observed data.
wow i never knew youtube greentext was used in quantum electrodynamics
It’s used in physics in the sense that you have to adhere to the same mathematical limits for it to work. It isn’t a number like planck's constant
Mathologer just dropped a mathematical diss track against this video.
The equivalent of releasing a diss track to a 4 year old song that everyone already dissed.
Bikram and Bishal LOLOLOL SAMMMEEE
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.
what do you mean with "debunked"? this is obvious garbage,Mathologers points hold and he is right. also, what link do you mean?
P-Zombie how is this wrong?
If A=B, then
AB=B^2
AB-A^2=B^2-A^2
A(B-A)=(B-A)(B+A)
A=B+A
A=2A
1=2
Math makes so much sense now!
Nice try but that doesn’t work. Since B-A=A-A=0, so dividing right hand side by zero results in infinity.
Thus, the only solution is the trivial solution which is when both A and B = 0
Seriously guys, just stop... There's an error in this methodology ftom the start
Well, this didn’t come out on April 1st, so I’m confused?
You're asking us if you're confused?
????? are you??????
@@GabrielTravelerVideos They're asking if this is a joke because of how it uses bad maths :^)
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.
This series actually exists ! It's Ramanujan's Infinite sum .
Nonsense is still nonsense, no matter how you explain it.
So by only adding positive numbers you get a negative number?
unintuitively yes
--Wyvern07-- of course not. There are thousands of ways to disprove this silly video
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!
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.
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.
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.
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.
This is awesome, but I personally feel that this method is too assumptious in many ways, even from the beginning, where you had 1 - 1 + 1 - 1 + ..., and chose to take the average, as 0.5.
I feel that this is a mathematical flaw, as opposed to the actual sum of all natural numbers. Because if you think about it, it kind of obviously is not the sum of all natural numbers.
The whole confusion I think, stems from assuming the first sum is 1/2.
There's actually two parts that don't work in regular calculus.
Firstly, yes, the first sum is a divergent series (has no obvious answer) and saying "it's 1/2" is like saying anything divided by 0 is 0. Might make sense using certain theorems (rules), but in regular calculus it's nonsense.
Secondly, you can't add/subtract/multiply an infinite sequence from another infinite sequence. If you could, you could effectively produce any number you wanted. The sum "1+1+1+1..." subtracted from itself but shifted one number over would end up with just 1.
Barry Monahan
Precisely. However, it is also remarkable that such seemingly nonsensical results have actual use in other disciplines.
+Barry Monahan you just blew my mind
0 actually, because shifted or not it is the same series, unless you mean the first ininite series subtracted by itself, except with n-1 elements.
It's not really an assumption, use the infinite geometric sequence sum formula (a/(1-r)) and the answer comes out as 1/2. This is the limit of the sequence. The infinite sequence approaches 1/2. And after doing a lot of maths you'll realise approaching is the same as equalling.
Ok I’ve never been this frustrated watching a RUclips video
But it is quite a leap of faith to say +1-1+1-1+1... = 1/2. I mean, it is either 0 or 1. It isn't 1/2. If we didn't have that result, then you can't get this weird sum, right?
if we take either 1 or 0 then answer would be either -1/6 or 0
Grandi's series can only be 0 or 1 if it stops. A method that give answers of 0 or 1 is an Eilenberg-Mazur Swindle.
It's undefined, like his second premise. The only one with a solution is 1+2+3+4+... which is positive infinity
@@Magst3r1 It is very clearly defined as ½, in any mathematical context which is capable of rigorously assigning it a value. Borel summation, Holder and Cesaro summation, Abel summation, amd Ramanujan summation, among many other methods, all define it as ½.
@@eoinlanier5508if you think this in any way is correct then is -12(1+2+3+4+n)=1 ?
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.
***** 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."
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.
AMGwtfBBQsauce
I'd already made that point. In the comment immediately before his.
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
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?
I think we should quit using the equals sign for these kinds of sums, because it's more accurate to say that -1/12 represents 1+2+3+4+5+6... rather than saying it is equal to it.
But -1/12 does not represent the sum of that series.
Doug Gwyn It represents the nature of the infinite series.
Where can I find a reputable definition of the "nature" of a series?
A reputable definition wouldn't really be necessary. By "nature" I mean "inherent features". The sum of the series is infinite, but when you take a closer look at the series itself rather than the sum of its parts, you can derive that it's different from other sums that lead to infinity. That difference we represent with -1/12.
believe or not Anthony but it is equal to negative one twelfth. just like the value of 1+1+1+1+1+1+.... is nothing but zero. the fact that the sum of all natural nos is -1/12 was given by Indian mathematician Ramanujan
So is the *real* answer to a math test question simply the average of all of the test takers' answers?
You'll actually get a pretty accurate answer
@@rohitgejje3717 I was about to say that
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
@@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
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
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
There is no last number tho.
@@-entr0pY That’s why I said technically. At any point in time there is one less number in the second set than the first
@@godbroccoli11 But that logic doesnt work because it implies the series end at some point where one other number can be left out.
@@-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.
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.
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".
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.
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.
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.
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?
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".
+Doug Gwyn is the real hero in this comment section.
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.
There are more shenanigans later - with divergent sums, you can't shuffle terms around willy nilly, etc.
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...
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
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.
@@TacticusPrime We call them Imaginary Numbers for a reason. Not 1/2.
I get a little annoyed about these 1+2+3+4+...=-1/12 videos since they don't explain the errors in the arguments presented. Normal people might get confused into believing that their intuitive belief about 1+2+3+4+... = infinity is wrong which it is not. You should introduce the concept of generating functions.
Can't help but feel the random shifting to the left kind of makes the logic somewhat random. But I guess you gotta do something, and stick with it; then call it math and hope it doesn't cause things to explode.
So... the universe is susceptible to integer overflow?
Dilandau3000 Omg Dilandau.... I didn't know you were interested in math ! ! ! Your 'Let's Play's' are awesome!
You have nailed it.
The bit bucket sprung a leak.
This joke is so specific and wonderful. I know I'm late commenting on this, but well done.
Go home math, you're drunk.
Bro added an infinite amount of inches to his jawline but it came out to be -1/12 inches
All the people saying this is bullshit, first off, are you a mathematician? And secondly, if it weren't true, why would it be used in String Theory textbooks?
I usually like numberfile videos... except this one.
I'm confused why you can just shift the numbers along when you add them to themselves
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.
me as well, I can't understand why you can do that in step 2
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
+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.
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 :)
The sum of 1 to infinity is given by Indian Mathematician 'Srinivasa Ramanujan'
@@akshit5363
His summation is not the same as the normal summation function that we regularly use.
It's different
What gives you the RIGHT to average to get the answer? Seems arbitrary.
I'm not supporting this video, but to answer your question, the so-called Cesàro sum of a series can be thought of as a generalization of the standard sum, in that it converges to the same value when the series is convergent, it diverges to infinity when the series diverges to infinity, and in cases where the original series does neither, *sometimes* the Cesàro process (limit of average of partial sums) converges to a finite value. Whether or not the Cesàro value is usable depends on the problem you were trying to solve when the non-convergence popped up.
The inclusion of a reference to another video which gives a derivation granted the right.
Two wrongs don't make a right.
you can obtain the same answer by making 1-S1=1-1+1-1...=S1
So S1=1-S1
and S1=1/2
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.
+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.
+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.
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.
+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.
+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.
This broke my brain so hard. I just wanted to watch the video about 0.577, but now I'm down the rabbit hole. If I fully understand this, do I break out of the Matrix?
Well, this is wrong
nobody:
mathematicians:
when physicist do math: ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12
You can get whatever number you want if you get to pick your own rules and varibles.
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.
@@daddymuggle this result is used in quantum physics
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.
@@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
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.
I love numberphile, but this is the biggest crock of crap I have seen so far.
Guys, reality check here. When you do something in maths and you end up with a clearly incorrect answer, as you did here, it's time to recheck your maths. Clearly, one of the manipulations you did was incorrect, in this case I believe that the "intuitive" sum of the grandi series is simply wrong. And "shifting" sums to add them is also a bit, well, shifty. It's beyond me how you happily accept this erronous result. It's pretty much akin to dividing by zero somewhere and coming up with 1=2. Even if you didn't know that dividing by zero is not admissible, you would not accept the result and look for the flaw in your reasoning. Same here.