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The sum of all natural numbers is not -1/12

Oh, good. There’s this claim going around that the sum of all natural numbers (1+2+3+4+5…) converges on the value -1/12. I saw that and said to myself that it’s obviously wrong, but saw the smooth patter and rapid-fire use of mathematical jargon and infinities, and no mathematician myself, couldn’t see where the error slithered in. Mathematician to the rescue: Mark Chu-Carroll explains why the story doesn’t work. Short answer: they falsely equated a summation with a converging series.

Inconsistency is death in mathematics: any time you allow inconsistencies in a mathematical system, you get garbage: any statement becomes mathematically provable. Using the equality of an infinite series with its Cesaro sum, I can prove that 0=1, that the square root of 2 is a natural number, or that the moon is made of green cheese.

What makes this worse is that it’s obvious. There is no mechanism in real numbers by which addition of positive numbers can roll over into negative. It doesn’t matter that infinity is involved: you can’t following a monotonically increasing trend, and wind up with something smaller than your starting point.

I could see the point he makes in the second paragraph, but it takes much deeper knowledge to pick out the flaw in the argument.

(Dang — I don’t even have a category for math here. Should I start one? Not that I can talk about math very often.)

Comments

  1. Markita Lynda—threadrupt says

    Perhaps you could put it under Education, Critical Thought, or Miscellania.

    Wouldn’t the sum of all natural numbers be a rather large value of infinity?

    Natural numbers do not include negative whole numbers; that set would be integers.
    .

  2. serena says

    At the youtube channel Numberphile, this is the video I’ve seen which claims this. http://www.youtube.com/watch?v=w-I6XTVZXww (sorry I don’t know how to embed/html etc, edit as needed please)
    I only passed high school algebra with my GED at age 33, so I have no understanding of how they’re claiming it works.

  3. Mark Chu-Carroll says

    I’m glad you liked my post, but I need to point out: I am not a mathematician. I’m a software engineer; I’ve got a PhD in Computer Science, not math.

    (I don’t want to take credit for something I’m not; I also don’t want to undersell what I really am. I’d be a crappy mathematician, but I’d like to think I’m a pretty good engineer.)

  4. Jason Dick says

    Well, actually, you can assign the value of -1/12 to the sum of all natural numbers. Wikipedia has a short article that shows a couple of ways of doing it:

    https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

    Yes, it’s true that the series doesn’t converge. But curiously enough, that doesn’t mean there is no way to assign a value to it. And it is incredibly bizarre that that value happens to be -1/12.

  5. says

    You’re a better mathematician than I am — so while maybe a real professional mathematician might take it for granted, I’ll just stand over in benighted ignorance, gazing in awe at your mastery.

  6. Trebuchet says

    Phil Plait has a blog post on it today, with a link to the video. I won’t claim to understand it and haven’t actually tried. He does offer a correction that it’s a sum, not a convergence.

  7. pangol says

    The thing is, this does make some sort of sense, but the recent versions kicking around are using sloppy language to make it look like something impossible is happening. Rather than talk about this one, let’s look at another similar version.

    There’s a common function in maths called factorial, written !, which is only defined for positive integers, which you get by multiplying the number by all of the positive integers up to and including itself. So, 3! = 3 x 2 x 1 = 6.

    There is another function called the gamma function, which is defined for all numbers (apart from negative integers) which agrees with factorial when you plug a positive integer in. So, Gamma(3) = 6. [Actually, this isn't quite right, it needs a bit of fine tuning, but I don't want to make this more complicated than it has to be! Apologies to mathematical pedants.]

    As an example, Gamma(1/2) is the square root of pi. Does that mean that (1/2)! is the square root of pi? No, because factorial doesn’t make sense for non-positive integers.

    This is what you’re seeing here. There is an expression for an infinite sum, which sometimes converges to a finite value – but only for a particular range of inputs. There is also a function that “extends” the infinite sum for values outside the original valid range. This gives valid answers, like -1/12 in a certain situation. But does this mean that the relates infinite sum really does converge to -1/12? Again, no.

    This is called analytic continuation, and in a pretty technical sense, there is only one way to do it “properly”. And just because it’s been used recently to claim things that aren’t stricktly true, doesn’t mean that you can’t use the result in some real situations. It’s used to prove a particularly interesting effect in quantum physics, for example.

  8. bryanfeir says

    a rather large value of infinity

    Well, no, it would be the same infinity as the usual infinity of natural numbers, or aleph-null (ℵ₀). The actual sum of numbers from 1 to n is n(n+1)/2, so for this we’d have ℵ₀(ℵ₀+1)/2, which is basically still ℵ₀ because squaring infinity gives you the same infinity.

    Infinity + n = Infinity.
    Infinity + Infinity = Infinity.
    Infinity * Infinity = Infinity.

    It isn’t until you start getting into exponentiation that this changes, and even then, only if you have the infinity in the exponent. There’s a reason why infinities tend to be counter-intuitive, even to people who have no problem with most of the rest of mathematics.

  9. says

    @Markita Lynda #1 – “Wouldn’t the sum of all natural numbers be a rather large value of infinity?”

    One might think, except that your statement relies on the same kind of sloppy reasoning that Chu-Carroll is objecting to. There is no such thing as a “larger” or “smaller” infinity: infinity is a big, amorphous domain consisting of every value larger or odder than you care to deal with at the moment. It’s like making a distinction between a larger and smaller eternity.

  10. stevem says

    re #1:

    [...]a rather large value of infinity?</blockquote?
    what are *small* infinities? Infinity is large (by def'n: "larger than any finite number". "infinity isn't even really a number but a concept) but there is only one, it can't be ranked against other infinities as there are none. Transfinites don't count. /snark

    Seriously now, I agree with MCC, how can a sum of positives EVER yield a negative? In any "system" of math, positives summed together yield a positive for the result.

  11. Mark Chu-Carroll says

    No, not really.

    There’s something called the Riemann zeta function. The Reiman Zeta can sort-of produce a value of -1/12, but it’s not really the same thing.

    At some point, I’ll write a post about it. But basically, it’s a complex-number equation, used in a particular context, where the value of the solution to the complex equation is -1/12. Solved in a different way, for a different parameterization, the Reimann zeta can be reduced to the sum of the natural numbers. They’re different solutions.

    To quote one explanation from Blake Stacey in my twitter stream: “In physics, Sometimes you’re doing a calculation and it looks like you’ll have to add up 1+2+3+4+… and so on forever. Then you look more carefully and realize that you shouldn’t–something you neglected matters. It turns out that you can swap in -1/12 for the *corrected* calculation and get a good first stab at the answer.”

  12. Irrational Rationality says

    What’s being done is a rather standard technique called analytic continuation. The Dirichlet series, when it converges, it always converges to the corresponding values of the Riemann Zeta Function. When the series doesn’t converge, a question arises as to whether or not a value can still be assigned to it. A divergent series is, in some sense, not entirely meaningful. However, there exist ways in which one can consistently assign a value to the series, treating it as a sort of symbol. Analytic continuation is one of these.

    When done properly and consistently, the only meaningful value that can be assigned to 1+2+3+… is -1/12. This is not the same as saying that the sum of all positive integers is -1/12. It isn’t, at least not quite. You cannot manipulate the individual terms in the series with ordinary operations like the assumptions of commutativity, associativity, and term-by-term integration and differentiation. These properties are not generally transfinite, and they only tend to hold in series which are uniformly convergent on an interval, such as Taylor or Fourier series.

    However, if you treat the entire series (1+2+3+…) as a single chunk, there is only one consistent finite value that you can assign to it and that value is -1/12. The methods in that YouTube video are nonsense, but there are rigorous ways of doing this. It is usually done through analytic continuation with the Rieman Zeta function. Zeta(-1) = -1/12.

    If these series are so hard to work with and make sense of, one would think you simply shouldn’t bother with them and only work with convergent series. The problem is that divergent sums and integrals show up in physics all the time. They cannot simply be thrown out. One has to invent techniques of dealing with these objects in a consistent way. For Feynman diagrams, the process is regularization and renormalization. Theorems have been proven showing that the particular choice of regularization scheme does not affect the results of the process, so a consistent answer can be found. Borel summation and Zeta function regularization are also necessary.

    Remember, in some sense, infinite series of any kind can’t be defined through simple means. This is true even if they converge. The laws of algebra can be proven, by induction, to apply to any finite string of binary operations. They don’t extend to infinite strings in any nice way. As I mentioned, commutativity and associativity don’t generally hold for all infinite series, including some which converge, like the conditionally convergent series 1 – 1/2 + 1/3 – 1/4 +….

    It’s a convention that the limit of partial sums is the value assigned to a series. This convention only works with series that converge. With non-converging series, one considers whether or not a generalization can be found. Can we find a method which still assigns the same values to convergent series as partial sum limits, but which is also able to assign values to at least some non-convergent series in a consistent and UNIQUE manner? The answer is yes, but we must remember that many of the rules of ordinary algebra do not apply to a generic series if we want to get consistent results. When we say that a series is equal to a certain value, we must be very careful what we mean by that.

  13. jackcowie says

    http://en.wikipedia.org/wiki/1+2+3+4+

    “The Ramanujan sum of 1 + 2 + 3 + 4 + · · · is also −1/12.[1]”

    This is not utter nonsense. Certainly it’s true that the series does not converge to -1/12, but that is not what numberphile actually claims at any point. All he’s doing is having some fun with a curious little area of maths which involves assigning meaningful values to divergent series.

  14. Irrational Rationality says

    Aleph null is not the same infinity as positive infinity in calculus. One is a cardinal number. The other is not actually a real number, but just a representation of the extreme limit of arbitrarily large real numbers. The closet thing to infinity as a number that you can encounter in ordinary calculus is complex infinity, and this can only be understood from the standpoint of the Riemann sphere which, strictly speaking, has a different topology than the complex plane. To equate aleph null with calculus infinity would be a category error.

  15. gussnarp says

    I’m no mathematician by any means, so this is way above my pay grade. But I do love Numberphile, so I’ve seen that video. It took far too many shortcuts for me to find it at all convincing, though I know that those guys know their stuff and they tell me that this answer is actually useful in physics. But given my ignorance, I’ve decided that this one just doesn’t matter to me. I like to understand this kind of thing, but I think I can live my life without believing that the sum is -1/12, but believing that using that approximation works for some problems. Meanwhile, I just heard an interview with a physicist named Max Tegmark who argued that we should scrap the notion of real infinities, that infinity was basically something that came up in physics and was useful in creating approximations that let us mathematically analyze problems, but that real infinities might not exist at all. I find this notion somewhat intellectually satisfying, though I also don’t know how well demonstrated it actually is.

  16. Irrational Rationality says

    I will also point out that there are other ways of getting the value of -1/12 in string theory. Adding up the positive integers only happens in the light cone gauge quantization, where it comes from adding up all of the zero modes of the string (remember, a quantum harmonic oscillator still has non-zero energy in its ground state).

    A different way to obtain it comes from considering the conformal anomaly. The only way to not have a conformal anomaly is to fix certain parameters so that a certain factor, called the central charge, disappears. The consistency of bosonic string theory requires setting the number of space time dimensions to 26, which is exactly what the -1/12 result agrees with. The conformal anomaly tends to make physicists much less uncomfortable than the summation argument in the light cone gauge.

  17. doublereed says

    Well, no, it would be the same infinity as the usual infinity of natural numbers, or aleph-null (ℵ₀). The actual sum of numbers from 1 to n is n(n+1)/2, so for this we’d have ℵ₀(ℵ₀+1)/2, which is basically still ℵ₀ because squaring infinity gives you the same infinity.

    Infinity + n = Infinity.
    Infinity + Infinity = Infinity.
    Infinity * Infinity = Infinity.

    It isn’t until you start getting into exponentiation that this changes, and even then, only if you have the infinity in the exponent. There’s a reason why infinities tend to be counter-intuitive, even to people who have no problem with most of the rest of mathematics.

    Please don’t try to explain infinity treating infinity as a real number. Infinity is not a real number. You do not add, subtract, multiple, or divide infinity. You can do this with variables and limits, but not infinity, because infinity is not a real number.

    Aleph-0 refers to countable infinity. It refers to an amount. Like the amount of the natural numbers is Aleph-0.

    That is not the same thing as the Extended Real Numbers which refers to the Real Numbers + Positive and Negative Infinity. Positive and Negative Infinity are based on convergences (going way up and going way down). They are what you put on the ends of your number lines.

    +/- Infinity and Aleph-0 are generated two different ways (one is based on counting, and the other is trying to figure out limits). There’s no need to confuse people, and there’s especially no need to further confuse people by adding/multiplying infinity when infinity is not a real number.

  18. Irrational Rationality says

    Also, if you can handle the fact that he’s kind of an asshole, Lubos Motl has a few articles on this subject:

    http://motls.blogspot.com/2011/07/why-is-sum-of-integers-equal-to-112.html
    http://motls.blogspot.com/2007/09/zeta-function-regularization.html

    He’s a dick, he’s very right-wing (but also an atheist), and he wouldn’t be popular on this site, but he is very right about physics 99% of the time. He knows physics very well and his opinions, when confined solely to physics, are very mainstream. Just don’t read anything beyond his physics and some of his mathematics articles and you should be fine.

  19. Rob Grigjanis says

    pangol @7: Γ(n) = (n-1)!, so Γ(3)=2

    Γ is used a lot in theoretical physics. For example, dimensional regularization, to render divergent integrals (e.g. from calculating Feynman diagrams) finite. You let dimension d = 4-ε, then find that the divergent pieces are terms like Γ(ε), which goes to infinity as ε goes to zero.

  20. Rob Grigjanis says

    Irrational Rationality @18: Yeah, with the caveats, Motl is brilliant. I read the physics stackexchange quite a bit, and his answers are almost always clear and insightful, displaying an impressive depth of knowledge. Stay well away from his comments on climate change.

  21. doublereed says

    More on @8

    Basically, you would never put Aleph-0 at the end of your number line. That would be silly and people would laugh at you. There is no “negative aleph-0.” That doesn’t make sense with its construction. But we do have an understanding of negative infinity on a number line.

    Neither construction, however, is based on algebra. They are not based on multiplying and dividing, but rather sequences, convergence, and orderings. This is why you should not treat infinity like a real number algebraically. That’s not how it is constructed, and it’s weird to try to apply the algebraic operations that are defined for the real numbers aren’t necessarily defined for infinity.

    For instance, “Infinity + n = Infinity” is merely a convenience that we’ve specifically defined because it doesn’t cause any problems.

  22. sc_1afdbca0f6f2896b62f4140e94e557d8 says

    I commented on MCC’s blog already. I think he was too harsh. The proof in the video is neither erroneous nor meaningless. The problem here is that the video fails to point out that the proof doesn’t work in normal algebra, the kind most mortals are used to. It was obvious to me where it diverges, namely when they said that the sum of the series 1 – 1 + 1 – 1 + 1…= 1/2. That clearly isn’t true normally, but it is also that case that in some form of abstract algebra and analysis (my terminology may be off, this past the edge of the math I studied, even being a math major in college) it is meaningful and useful.

    This isn’t like most “1=0″ proofs. If you start looking, there is a long list of famous, gifted mathematicians who used these concepts in there work. It is not fair to just say “Phil should have known better”.

  23. says

    @doublereed #17 – You reminded me of an extra credit trick question that appeared on my abstract algebra final in college. The question was:

    (infinity) / (infinity) = _______

    The first instinct is to apply multiplicative identity and give the answer as 1: after all, x/x = 1 by definition. The correct answer, of course, is “undefined” because (infinity) is not in the domain(?) of numbers, so numeric operations and rules cannot be applied.

    (I don’t recall the correct term; I think domain is correct.)

  24. machintelligence says

    The first chapter in an old (1947) book by George Gamow deals with comparing infinities in a reasonably accessible way. I first encountered it in High School in the 1960′s.

  25. Irrational Rationality says

    Infinity/Infinity and Infinity*0 are both “undefined,” because Infinity, in the calculus sense, represents a limit rather than a specific value. You cannot know what Infinity*0 is without investigating the limits that get you there.

    lim (x->inf){x^2*exp(-x)} “=” Infinity*0
    lim (x->inf){exp(x)*1/x^2} “=” Infinity*0

    Both of these limits involve Infinity times zero, but the actual value of the top one is 0 and the bottom one diverges. By manipulating the exact nature of the sequences, I can make an Infinity*0 limit converge to any real number. Thus, Infinity*0 is undefined, because the expression doesn’t contain enough information to tell us what the answer is. “Infinity” is just shorthand for some limiting process which diverges in a positive sense, and “-Infinity” is the same, mutatis mutandis. Without knowing the specific limiting process, the expression “Infinity*0″ is meaningless.

    On the other hand, Aleph Null is actually a legitimate number, but it’s not a natural number, an integer, or even a real number.

  26. inflection says

    I wish Slate didn’t make me create yet another thrice-damned social media account just to comment there. I teach math at a tech university and this makes me rage.

    The headline of the article is “When Infinity Is Actually a Small, Negative Fraction.” It’s clickbait because it’s so obviously, counterintuitively wrong. That’s the problem. It _is_ wrong. For every person who knows what Cesaro summation is and what true statement Phil might have made (but, in fact, did not), there are going to be 999 who think “there is just something wrong and nonsensical about complicated math.”

    Dammit, Phil. Do you have any idea what this is going to do among your readers? Thanks a million for creating problems for me to clean up when they get to college.

  27. ChasCPeterson says

    if you treat the entire series (1+2+3+…) as a single chunk, there is only one consistent finite value that you can assign to it and that value is -1/12.

    but…but that’s so stupid!

  28. fentex says

    I haven’t read any of this but I notice reference to addition and infinity in the same conversation.

    I suspect someone at some pint claims adding to, or up to, infinity implies something?

    Any such claim (whether here or elsewhere) is nonsense – infinity is NOT a number.

    You CANNOT add, subtract, multiply or otherwise do any operation on, to, or ‘up to’ infinity.

    Infinity is a concept that comes in many different flavours and is a basis for much philosophical pondering and reasoning but it is NOT a number that can be operated on by typical mathematical tools.

    Any time anyone introduces infinity to a function they are making an error.

  29. ChasCPeterson says

    *shrug* I am no mathematician, that’s for damn sure (I even failed calculus once, but it was my freshman year and an 8 am class and I didn;t do the homework). Zeta function regularization is far beyond my ken, and I’m not ashamed to admit it. No idea what the stuff at that link means.

    But if you tell me that you can add a whole shitload of positive integers, all of them, even, or rather even more than all of them, or whatever, and get a negative fraction, I will call bullshit every time.

    Oh, you’re not actually adding them (despite the infinity plus-signs in your series), but instead just correctly and properly “assigning a value to the whole chunk”?
    And it doesn’t concern you in the slightest that the correct and proper value you assign makes no freaking sense at all?
    ooooo-k.

  30. says

    Others have explained better than I could already about divergent series, but I will note one thing about this type of math that is relevant to an atheist site like this.

    The bit in the Numberphile video with the series 1+1-1+1… reminded me of Grandi’s Series.

    The book Calculus 4th edition by James Stewart pg 747 has a problem (#57) that states

    0=0+0+0+0+…
    =(1-1)+(1-1)+(1-1)+…
    =1+(-1+1)+(-1+1)+(-1+1)+…
    =1+(0)+(0)+(0)+…
    =1

    It then asks where this calculation went wrong (which the answer is the summation of zeros to the divergent summation of alternating 1 and -1.)

    Then the book notes that Guido Ubaldus (which I guess is Guido Grandi) claimed that this calculation that 0=1 proved the existance of god because “something has been created out of nothing.”

    Interesting that this ex nihilo argument was once used as proof of god while now it is a creationist strawman argument against evolution that you can’t get life from non-life.

    Regardless, what this all shows is when your premise is bad you can prove whatever you want.

  31. sc_1afdbca0f6f2896b62f4140e94e557d8 says

    “And it doesn’t concern you in the slightest that the correct and proper value you assign makes no freaking sense at all?”

    But it does make sense. What if I told you that 1 + 1 = 10? Would you say that it makes no freaking sense? What if I then said, “its in binary”, would you still say it makes no freaking sense? Everything in the video makes sense in the context of the mathematical system they are using. It just doesn’t happen to be the one you are used to and they don’t make it clear when they are relying on that system.

    None of this has anything to do with whether or not infinity is a number, so you should all stop talking about aleph null and different orders of infinities and the like. The only thing that matters is what operations you can and cannot do to an infinite series and under what circumstances.

    The only place they diverge from normal algebra is in the statement that 1+1-1+1-1… = 1/2. That is not an error, that is the Cesàro summation of the series. It is not some arbitrary assignment, it is well defined, unique and meaningful. But it isn’t normal algebra that produces it. And the thing that makes the video interesting is that this is counter-intuitive, yet meaningful. They don’t claim otherwise in the video, but they probably should have made it more clear.

  32. David Marjanović says

    Riemann zeta function. The Reiman Zeta [...] the Reimann

    Riemann, because it’s pronounced with “ee”.

  33. Rob Grigjanis says

    Chas @35:

    And it doesn’t concern you in the slightest that the correct and proper value you assign makes no freaking sense at all?

    In physics, there’s usually a way to see the sense, and it often involves cutoffs of some sort.

    This is a simplification: In the Casimir effect, you’re summing over harmonic modes between conducting plates to get the energy, and the sum is proportional to the sum over positive integers n (n is proportional to the frequency). Thing is, high enough frequencies will not be constrained by the plates (they’ll leak through), and will not contribute, so it makes sense to ‘regularize’ the sum by somehow discounting or playing down large frequencies. One way is to multiply each term n by an exponential exp(-an), where the number a is the ‘regularizer’ (if it goes to zero, the sum diverges).

    With a bit of calculation, and expanding in powers of a, you get, for the sum

    1/a^2 – 1/12 + (terms of order a^2)

    When you calculate the force between the plates (rather than the unobservable energy), the 1/a^2 term cancels out, and you can safely take the limit a ->0. This is how the -1/12 makes ‘sense’.

  34. says

    =(1-1)+(1-1)+(1-1)+…
    =1+(-1+1)+(-1+1)+(-1+1)+…

    As much as I like to think I fully comprehended the section on infinite series in second semester calculus, I’m confused by this.

    Why would it look like “=1+(-1+1)+(-1+1)+(-1+1)+…” instead of “(-1+1)+(-1+1)+(-1+1)…”? Aren’t they missing a -1?

  35. kevinkirkpatrick says

    My personal favorite example of how ludicrously fun conclusions can be reached by allowing “actual infinites” into formulations:

    Al and Bob are going to play a game. They have an infinite number of ping-pong balls, number 1, 2, … to infinity. In each iteration of the game, they both put 10 balls into their respective buckets in numerical order. So, round 1, they put ping pong balls # 1 – 10 in. Round 2, it’s balls 11 – 20. And so on.

    The twist: after each round, Al has to remove the smallest numbered ball from his bucket; and Bob must remove the 9 largest numbered balls. So after round 1, Al removes ball #1, and Bob removes balls #2, 3, 4, 5, 6, 7, 8, 9, and 10 (leaving just #1). Round 2, Al removes ball #2, and Bob removes 11 – 20 (leaving just #11). Thus, Al is adding 9 balls per round and Bob is adding just 1.

    The question: after an infinite number of rounds, who has more balls left?

    The answer: Al, of course. Al’s bucket will be empty. Because, just try to name the number of any ball left in Al’s bucket. #4? Removed on round 4. #1,000,000? Removed on the millionth round. However, Bob’s bucket would have an infinite number of balls left, 1, 11, 21, 31, … to infinity.

    And no. No I don’t have anything significant to contribute here.

  36. ChasCPeterson says

    Round 2, Al removes ball #2, and Bob removes 11 – 20 (leaving just #11).

    wait…what? You just had Bob both remove and leave #11. I guess you meant he removes 12-20, leaving 1 and 11.

    The answer: Al, of course.

    wait…what? You mean Bob, right?

    I guess I get the point, but it’s even more perplexing when you don’t tell it right.

  37. eigenperson says

    1 + 2 + 3 + 4 + … is not Cesaro summable. You need more than Cesaro summation and algebraic manipulation if you want to justify assigning the sum -1/12 to the series.

    So I think MCC’s gripes are very well-placed when it comes to the video and Phil Plait’s post.

    On the other hand, I don’t think it’s reasonable to say that assigning the sum -1/12 to 1 + 2 + 3 + 4 + … is “mathematical stupidity”, unless Ramanujan is mathematically stupid.

  38. ChasCPeterson says

    Everything in the video makes sense in the context of the mathematical system they are using. It just doesn’t happen to be the one you are used to and they don’t make it clear when they are relying on that system.

    The part that looked like bullshit to me was when the guy added the second series [S(2)] to itself to get 2S(2) to = S(1).
    So he writes:
    S(2)=1-2+3-4+5-6…
    + S(2)= 1-2+3-4+5…
    with the second line shifted over one numeral to magically get 1-1+1-1+1…, i.e. S(1) when added vertically.
    Isn’t that the sleight of hand?
    Because OK, I reasoned, well, he’s just implicitly adding a zero at the start of the second S(2), and it would have been clearer if he had actually written:
    S(2)= 1-2+3-4+5-6…
    + S(2)=0+1-2+3-4+5…
    But wait, if that’s kosher, then you should be able to add–or subtract–as many zeroes as you feel like.
    If you line them up like
    S(2)= 1-2+3-4+5-6…
    + S(2)=0-0+1-2+3-4+5…
    then they add vertically to 1-2+4-6+8-10…
    which isn’t S(1) at all.
    And if you don’t shift the second line at all and add vertically, you get 2-4+6-8+10… which doesn’t work either.

    But, hey, that’s in the mathematical system I use, i.e. the one that makes sense.

  39. billforsternz says

    shockna @40

    Why would it look like “=1+(-1+1)+(-1+1)+(-1+1)+…” instead of “(-1+1)+(-1+1)+(-1+1)…”? Aren’t they missing a -1?

    Bingo, that is exactly the logical flaw in all of these apparently plausible algebraic manipulations that yield nonsensical results. In each case they are shifting rows of additions/subtractions left and right, thus realigning the columns so that an infinite number of terms from adjacent rows apparently cancel each other out. The nice little “…” continuation is disguising the (mathematical) deception. The -1 you are pointing to is real and the shift should not be allowed to discard it. In other words the “…” from one row can only cancel the “…” from the adjacent row if it originates from the same, unshifted, place in the sequence.

  40. sc_1afdbca0f6f2896b62f4140e94e557d8 says

    billforsternz, no its not.

    1-1+1-1+1-1+1-1… is exactly equivilent to

    1 + -1 + 1 + -1 + 1 + -1 …

    So we can specify it as

    (1 + -1) + (1 + -1) + (1 + -1)…

    or

    1 + (-1 + 1) + (-1 + 1) + (-1 + 1)…

    Notice that the difference is only in the placement of the parenthesis. There was no shifting or leaving out a number or anything.

    And if these were convergent series you most certainly would be allowed to shift and add or shift and subtract. They teach it all the time in high school algebra for handling infinite series to help find the value they converge to. These operations may not be defined if the series is not convergent, but should we jump all over someone because they don’t know that? Didn’t recent research show that about half of scientific papers misapply statistics in a similar manner, neglecting prerequisite conditions for the applicability of statistic formulae? And that’s in a field that the authors are supposed to understand.

  41. billforsternz says

    sc_1af.. @47

    (1 + -1) + (1 + -1) + (1 + -1)…

    or

    1 + (-1 + 1) + (-1 + 1) + (-1 + 1)…

    I am tempted to yield to your apparently superior knowledge and understanding, but I concede this equivalence looks absolutely wrong to me, for exactly the same reason as I (tried to) describe in my first post. In the first expression you are starting your “…” after 6 ones/neg-ones and in the second expression after 7 ones/neg-ones. I don’t see how that’s fair or reasonable and that’s the only thing that makes it look like 0 = 1 !

    But as I say, I’m just an engineer. Just a few math papers at uni many moons ago. So I won’t bet the farm on this one !

  42. kaleberg says

    This looks like something Euler would have come up with. He was noted for playing fast and loose with limits, convergence, infinities and the like. Then again, he was usually right.

    In this case though, the confusion seems to be that most people, even pretty well educated people, interpret the word “sum” to mean the result of performing addition, but the sum is only -1/12 for a different definition of the term.

  43. moebius2778 says

    @42, no the story’s being told correctly.

    Basically, both Al and Bob have their own set of ping pong balls, each labeled 1, 2, 3, … and so forth.

    First round, Al takes balls 1 – 10 from Al’s set of ping pong balls, and Bob takes 1 – 10 from Bob’s set of ping pong balls, each places them in their respective bucket, and then they perform their operations.

    The reason that Al has no balls in his bucket after an infinite number of rounds is, try to name a ball that is in Al’s bucket. Say we name the ball labeled N. But, based on the way Al removes balls from his bucket, the Nth ball was removed on round N.

    The reason that Bob has an infinite number of balls in his bucket after an infinite number of rounds is, given any number N that is a natural number – the ball labeled ((N – 1) * 10) + 1 is in Bob’s bucket and was added on the Nth round, so the number of balls in Bob’s bucket is equal to the cardinality of the natural numbers.

    …which probably just goes to show that you need to be pretty careful when dealing with infinities.

  44. moebius2778 says

    Thinking about it further, I should note that saying “after an infinite number of rounds” may make about as much sense as saying the last number of an infinite sequence.

  45. ChasCPeterson says

    the story’s being told correctly.

    …except for the two whopping errors I identified explicitly, you mean.
    Like I said, I got the point anyway.

  46. moebius2778 says

    Your second objection doesn’t … well, may or may not work depending on how you want to calculate what happened.

    If you’re arguing that Al has more balls in his bucket than Bob, than Al must have more than 0 balls in his bucket. Therefore, there must exist an N such that the ball labeled N is in Al’s bucket (as there are no other possible balls that can be in Al’s bucket). However, it’s also provable that for any N, the balled labeled N is not in Al’s bucket.

    Again, it really depends on how you want to handle the phrase “after an infinite number of rounds”.

  47. jorjinho bolivar says

    I have the proof that this is bullshit.
    Lets’ assume this is true
    1- 1+ 1- 1+.. =1/2
    1+2+3+ 4 +.. = -1/12
    add them both

    you get the same serie with just a lil tweak
    2+1+4+3+6+5+ ..

    which is equal to 1+2+3+4+…= 1/2 – 1/12= 5/12 /=-1/12

  48. Brian O says

    One curiosity about factorials is that 0! = 1! = 1 (I’d have to dig out an old maths textbook to tell you why that is). The point being that mathematical formulae are often counter-intuitive, just like “hard” science. Also, string theory is not yet science, by the standard of verifiable experimental evidence.

  49. george gonzalez says

    This is what happens when experts go outside their field.

    What’s happening is well-meaning folks are conflating the explicit sum with the somewhat related Reinmann Zeta function.

    From the definition of the Zeta function, it LOOKS like if you plug in negative integers, particularly (-1), the function gives you the sum of the positive integers. But obviously -1/12 isn’t right. It might be “right” in some contexts, and might match up with what string theory and Casimir force equations “want” to see, and that’s fabulous and neat, but it’s obviously not “right” in a pure numerical sense.

    There are also the folks in the video, who may or may not have a clue, it looks like they’re just doing numerical card tricks and can get any answer by juggling the numbers. That’s Vegas hokum, not math.

  50. Rob Grigjanis says

    george gonzalez @56:

    …it’s obviously not “right” in a pure numerical sense.

    Right, and I’ll just add that no calculation of the Casimir effect I’ve ever seen, simply replaces the sum with ζ(-1). If any physics teacher actually did this in the classroom, they should be docked a day’s pay.

  51. kevinkirkpatrick says

    I WILL NOT POST WITHOUT PROOFREADING.
    I WILL NOT POST WITHOUT PROOFREADING.
    I WILL NOT POST WITHOUT PROOFREADING.
    I WILL NOT POST WITHOUT PROOFREADING.
    ….

    I’ll post again after I’ve finished writing that an infinite number of times. Fortunately, I’ve learned some new math that tells me I’m already about minus one twelfth of the way there.

  52. woozy says

    #54. >> which is equal to 1+2+3+4+…= 1/2 – 1/12= 5/12 /=-1/12

    Yes, but 1/2, 0, and 1 are equivalent. At least as far as “little tweaking” goes. 1 -1 + 1 – 1 + … = {1, 0, 1/2}. *or* by infinitely commuting pairs you are replacing 1- 1 + 1 – 1… = 1/2 with -1 + 1 -1 + -1 … = – 1/2. (I think replacing 1/2 with -1/2 will yield 5/12 rather than -1/12 but I haven’t tested it.)

    I’m a little disappointed by the video. Simply, an infinite series that doesn’t converge, doesn’t converge and doesn’t have a sum and that’s all there is to it. For infinite series that *do* converge the sum and the Rieman Zeta functions must be consistant so if it *did* have a sum (which it doesn’t) it *would* be -1/12 and, more to the point of the video, that “hocus pocus” would yield a consistent and correct zeta function sum (unless one added an extraneous “simple tweaking” to yields something that isn’t correct– which of course anyone can do).

    It *does* come up in physics but notice it doesn’t come up as an equality. That formula in the String Theory book did not have a equal sign (it had a bold arrow [which was somewhat confusing and misleading and non-bold arrows usually mean convergence]) because, of course, the sum doesn’t converge.

  53. chessanator says

    Another counter-intuitive infinite sum is
    1+2+4+8+16….= – 1
    This comes from p-adic metrics as follows.

    Consider a more sensible infinite sum like 1+ 1/2 + 1/4 … = 2. Each added term makes the sum closer and closer to 2 so when you do infinitely many additions you reach 2.

    All you have to do to understand the above sum is change what we mean by “close”. In the 2-adic distance, two numbers are “close” if the difference is divisible by large powers of 2. Now consider each term in the sum. 1 is 2 away from -1, 1+2 is 4 away from -1, 1+2+4 is 8 away from -1 and so on. With our new distance, the sum gets “closer and closer” to -1 so after infinitely many additions it is -1.

  54. stevem says

    I too, do not understand adding two infinite series and just shifting the second over by one value before adding each individual pair of members then totaling the infinite sum. “Hand waving” (sleight-of-hand) to the extreme.
    I.E. to add two series to get the sum of the two, one adds the total of each of the two series, IF you can rearrange the order of the values in the series, you can get any value you want.
    To be generous, I think that was the actual point of the video; to demonstrate how easy it is to misuse the “rules” of mathematics to get a completely ridiculous answer; and then to falsely associate the fact that the very same number is a result from a completely different use of math in speculative-physics, is also a demonstration of how easy it is to go totally wrong. I.E. they are using this as a “teaching moment”, to inspire the viewer to learn mathematics (theory, not just arithmetic nor simple algebra).

  55. woozy says

    Another counter-intuitive infinite sum is
    1+2+4+8+16….= – 1
    This comes from p-adic metrics as follows.

    The thing is, this *is* consistent. As (1 – x )* ( 1 + x + x^2 + x^3 + …..) = 1. It follows that 1 + x + x^2 + …. = 1/(1 -x). Thus x = -1 => 1 – 1 + 1 – 1 … = 1/2 (remember?) and, as you point out, x = 2 => 1 + 2 + 4 + 8 +…. = -1 (and 1 + 3 + 9 + …. = -2/3, etc.).

    This, of course only works, when it “makes sense”. In other words when x is within the proper radius of convergence, i.e. -1 < x < 1. (Try it. It *does* work if x = 1/2. 1 + 1/2 + 1/4 +…. *does* equal 1 and it works for x = 2/3, or -1/2 or anything between but not including -1 and 1.) However, as you point out in the p-adic metrics (the subject of my undergrad seminar, btw [although an *undergrad* seminar, in actuality hardly counts as rigor]) the 2-adic "size" of 2 is only 1/2 so it satisfies the condition.

  56. woozy says

    “I too, do not understand adding two infinite series and just shifting the second over by one value before adding each individual pair of members then totaling the infinite sum.”

    But shifting over 1 is the only one that gives a result. The others give infinite diverging series that are no closer or further away than the first toward a “solution”.

    “Hand waving” (sleight-of-hand) to the extreme.”

    Well, yes… You simply can’t assume convergence and play to see what you get. But you can manipulate symbols and come with a method that will work on convergent series.

    “I.E. to add two series to get the sum of the two, one adds the total of each of the two series, IF you can rearrange the order of the values in the series, you can get any value you want.”

    But you can’t rearrange order because that is “infinitely borrowing”. In #54 switching the order is, in essence, borrowing 1/2 (1-1+1-1+1…..) from the infinity place and never returning it. Of course, I can’t really claim that you “can’t do that” when we shouldn’t be doing *any* of the stuff that we are doing. However once we’ve broken the cardinal rule of treating a divergent series as though it is a single value, shifting it one, two, three, or seven places and adding isn’t doing any *more* harm. However rearranging the order is.

    “To be generous, I think that was the actual point of the video; to demonstrate how easy it is to misuse the “rules” of mathematics to get a completely ridiculous answer”

    I don’t think it was. I think they sincerely figured it showed the result they were claiming. And I don’t think it was bullshit nor coincidence that the final result was the “correct” Rieman Zeta function. However I think the video is disingenuous in that you simply can’t assume a divergent sum is a single value and precede, so there was really no point in starting in the first place.

    This video falls in the very uncomfortable zone of “It’s wrong but it’s not total bullshit wrong” that I find puts my teeth on edge.

  57. Jonny Mahony says

    There is an actual experiment which has to do with the casimir effect which results in that the sum of the positiv integers in -1/12 ! And THAT’S IN AN EXPERIMENT !!

  58. ChasCPeterson says

    There is an actual experiment which…

    It’s customary to document claims like this with a citation or link. Otherwise why should anybody believe you, Jonny On The Internet? Even if you yell at us?

  59. Rob Grigjanis says

    There is an actual experiment…

    No, there really isn’t. The ‘sum of positive integers’ corresponds to the energy. An actual experiment can’t measure the energy, only energy differences and energy gradients (forces). The Casimir force is a perfectly well defined, finite quantity, which is proportional to -1/12. But it is not the sum of positive integers.

  60. Rob Grigjanis says

    Chas is right, need citations.

    Simplified version (as in my #39);
    A. Zee, Quantum Field Theory in a Nutshell, Second Edition, Princeton University Press, 2010, pp70-73

    More complete treatment;
    Itzykson and Zuber, Quantum Field Theory, McGraw-Hill, 1980, pp138-141

    Zee has something to say about this nonsense.

    Appallingly, in an apparent attempt to make the subject appear even more mysterious than it is, some treatments simply assert that the sum [over positive integers] is by some mathematical sleight-of-hand equal to -1/12. Even though it would have allowed us to wormhole from (1) to (3) [divergent energy to finite force] instantly, this assertion is manifestly absurd.

  61. Rich Woods says

    Yes, I have a background in mathematics. No, I don’t much care about the detail of any of this. It was thirty years ago. One idea is likely being applied outside its context. This has been interesting reading, but I’m going to go to bed now.

    Probably.

  62. chris says

    I sometimes wonder if the trying to do arithmetic with infinities is what caused Georg Cantor to be hospitalized often for his mental health.

    I am just an engineer, though one who did lots of applied math. I saw the plot for 1-1+1-1… and thought about the Fourier Series for a triangle wave. But that is just me.

    Though this does remind me the joke that Mathworld has on Zeno’s Paradox:

    The dichotomy paradox leads to the following mathematical joke. A mathematician, a physicist and an engineer were asked to answer the following question. A group of boys are lined up on one wall of a dance hall, and an equal number of girls are lined up on the opposite wall. Both groups are then instructed to advance toward each other by one quarter the distance separating them every ten seconds (i.e., if they are distance d apart at time 0, they are d/2 at t=10, d/4 at t=20, d/8 at t=30, and so on.) When do they meet at the center of the dance hall? The mathematician said they would never actually meet because the series is infinite. The physicist said they would meet when time equals infinity. The engineer said that within one minute they would be close enough for all practical purposes.

  63. Rob Grigjanis says

    The engineer said that within one minute they would be close enough for all practical purposes.

    I’m pretty sure the engineer said “where’s the beer?”.

  64. ChasCPeterson says

    I’m pretty sure the engineer said “where’s the beer?”.

    That or “Aaaa! Girls?!!!”
    or, depending, “Aaaa! Boys?!!!”

  65. cm's changeable moniker (quaint, if not charming) says

    kaleberg @#49:

    This looks like something Euler would have come up with.

    Hmmm. From a link in Phil’s OP:

    Around 1735, Leonhard Euler gave a bizarre ‘proof’ that
    1 + 2 + 3 + 4 + · · · = −1/12

    http://math.ucr.edu/home/baez/numbers/24.pdf

    The first rule of math club is you do not argue with Euler. ;-)

  66. cm's changeable moniker (quaint, if not charming) says

    (The second rule is, of course, that you really ought to read the ((O)O)OP, and the links therein.)

  67. David Marjanović says

    Reinmann

    Rieman

    Riemann. Trust me, it makes actual sense.

    A. Zee

    That’s an awesome name.

  68. Robert Rehbock says

    1. Don’t start a math or physics category.
    2. As other posters have said, the result makes sense to physics.
    3. As other posters have it seems nonsensical to classical math types.
    My views is that math is a language. Infinity is an imaginary concept. In math and in language we can imagine purple, people eaters of infinite variety. In physics we stick to what nature allows. Others have pointed out that experiment and valuable theoretical results agree with Euler. So this is a debate between those who prefer to live in an imaginary world where infinity is and those who like the real world where infinity is less than a google. Tegmark has a nice piece recently at the Edge. For those who want to understand something check out TRF Motl. Don’t btw get put off by his other views. Some of you are like me somewhat liberal politically. He not so much so. But before calling everyone idiots and other pejoratively spend some time learning.

  69. David Radcliffe says

    I would like to make a few (obvious?) points. Addition is a binary operation — we add numbers together two at a time. We can use recursion to define the sum of a finite sequence of numbers. But we cannot define the sum of an infinite sequence of numbers within the confines of elementary arithmetic. If we say that a process such as addition is to be repeated an infinite number of times, we must be very careful to specify what this means.

    It turns out that there are many ways to define the sum of an infinite sequence. The standard approach is to compute the sum of the first N terms, and take the limit as N approaches infinity (if it exists). By this definition, the sum of all natural numbers does not exist.

    But there are nonstandard summation methods, and some of them assign the value -1/12 to the infinite sum 1+2+3+4+… . Other posts have explained why this may be useful in certain situations. But it is misleading at best to state that 1+2+3+4+… is equal to -1/12, without qualifying the statement.

  70. chris says

    cm: “http://math.ucr.edu/home/baez/numbers/24.pdf”

    Cool. It makes sense since there is a physical limit, like how many frequencies can exist in a system and in reality there is no infinite energy state.

    In my days doing structural vibration analysis there were only a certain number of eigenvalues (natural frequencies) and eigenvalues (mode shapes, one per frequency) that were of any interest. It was sufficient to stay within the boundaries of space and time where Newtonian equations were still valid. There is no reason to go into quantum levels if you are trying to make sure a tire does not vibrate off a vehicle.

    So while I understand much of the mathematics, I notice that some things were done that are outside my realm. I know my limits. In my math language I would never use infinities except in the phrase “as the limit tends towards infinity”… to get to the “close enough.” Plus imaginary numbers are extremely important as explained in Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills (I’m going to dig out my copy and review some of the bits, perhaps I won’t give it to my kid who is about to graduate with a math degree just yet).

  71. Thumper: Token Breeder says

    It doesn’t make sense on the face of it. A positive + a positive cannot = a negative, so how can a positive + a positive +a positive ad infinitum = a negative?

  72. sea turtles says

    As a math guy I have to recommend reading everything “Irrational Rationality” posted ITT. In addition to the idea of there being unique values to assign to divergent sums (even if you do not wish to accept divergent sums, you have to admit the fact that there are unique values to assign to them is pretty spooky in itself), my own opinion on how to interpret zeta-regularized sums like this one is that they implicitly carry a “memory” of something they once were, and continue to exist in a different form as a sort of ghost.

    To compare (as someone else noted), the geometric series formula tells us that 1+r+r^2+… converges to the value 1/(1-r) whenever |r|<1 in the complex plane. But the latter expression, 1/(1-r), is defined everywhere except r=1, so it continues to live past the natural death of the series (a fancy way of saying the Taylor series has radius 1 at r=0). If we think about the divergent sum value 1+2+4+…, the *form* of the left-hand side carries extra data for us, namely that it is a geometric series, and if we trace the geometric series 1+r+r^2+… from the territory where it doesn't converge *back* into the territory where it *does* converge we will get a function that uniquely extends back to the outside again to give "1+2+4+…=-1."

    Sometimes purely algebraic manipulations can achieve the same effect. The thing about many of these manipulations is that they *would* be valid if the sum converged – indeed, as another commenter noted, in the so-called "2-adic numbers" the series 1+2+4+… actually converges to -1 (these numbers have a different topology, and therefore convergence is very different). One interesting thing is that these "ghost" sums have an intrinsic graded aspect to them, and manipulations that leave the grading undisturbed can be used to find the correct regularized values where manipulations that disturb the grading will lead to inconsistencies (I believe this also has some relation to physics).

  73. Rob Grigjanis says

    @80: When someone disagrees with Motl, it is always due to naïveté and ignorance. We can now add Anthony Zee to his list of physicists who don’t understand physics.