15 ms·

Does 1+2+3+... Really Equal -1/12?

197 points by CarolineW 10y ago

sorennielsen 10y ago
Well... Guess that if I watch an infinity number of videoes like this I will end up not getting more stupid like I did with this one. Waste of time.
lifthrasiir 10y ago
Somehow the title is missing an ellipsis, which made me confused for a second.
- CarolineW 10y ago
  Now added - apologies for the confusion.
dozzie 10y ago
> Does 1+2+3 Really Equal -1/12? Of course not. It's a misconception propagated by ignorant people who stumbled upon Ramanujan summation.
- CarolineW 10y ago
  Out of interest - and I'd really like you to answer honestly although obviously I have no way of knowing if you don't - have you read the article, or are you simply responding to the title?
  - dozzie 10y ago
    Only responding to the title. Such titles annoy me, and I happen to have this bad habit of complaining in such cases.
    - CarolineW 10y ago
      Thank you for your honest answer. The article has a discussion that is good, careful, nuanced, and quite complete. It also has lots of links to related discussions, including talking about why - although this specific claim is "obvious nonsense" - there is actually some sense underneath it, although deeper than most people ever go. If you're going to complain you should probably have an opinion on how to improve it. What title would you give such an article?
      - dozzie 10y ago
        I would have an opinion if I read the article, which I do not intent at the moment (I find my own code more interesting than mathematical concepts I have no use for).
        qubex 10y ago
        Then mightn't your time be more productively spent paying attention to your code you mention rather than commenting on articles you have not read about concepts you reckon you have no use for?
        mcguire 10y ago
        Compiling!
amelius 10y ago
This is what you get when mathematicians start "hacking". Instead, they should have invented a proper notation conveying what is really meant by the sum and the "..." ellipsis. This notation apparently does not cut it.
- nothrabannosir 10y ago
  > This is what you get when mathematicians start "hacking". I.e. physics.
- MichaelBurge 10y ago
  ζ(s) has been in use since 1859, when Riemann introduced it. Riemann's paper isn't as thorough as it could've been, but it seems clear he knew exactly what he was doing, being one of the founders of Complex Analysis: http://www.claymath.org/sites/default/files/ezeta.pdf http://www.claymath.org/sites/default/files/ezeta.pdf This series is ζ(-1). I don't know about the Youtubers or Scientific American, but mathematicians studying Complex Analysis know exactly what it means.
MichaelBurge 10y ago
No need to invoke the Riemann Zeta function and analytic continuation. In high school they summed this series(valid for x < 1): 1 + x + x^2 ... = 1/(1-x) Plug in x = 2 to get: 1 + 2 + 4 + 8 ... = -1 There's a million of these series in your dusty old Calculus textbook. Or you could look in 'generatingfunctionology' to find others. Complex analysis makes it more interesting, because the additional Cauchy-Riemann constraints make the solution unique. And so people are more willing to say that the unique solution is the "true" answer. Really, I say this is just another example of the complex numbers being weird. I took a couple different courses in it(one in college, 2 online) because it was clear something interesting was happening with them, but there was never a unifying theme. I can kind of spot a rule like "analytic functions preserve 90-degree angles" in the Cauchy-Riemann equations, but it hardly explains all the crazy theorems. That series is just another example of someone taking a well-behaved series, analytically continuing it into the complex numbers, and now it's clear something interesting is happening but it's not clear what.
- thaumasiotes 10y ago
  Who says it's only valid for |x| < 1? That series isn't just a cute bit of trivia, it is the basis for signed integer mathematics in your CPU. "True" enough for you?
  - messe 10y ago
    > Who says it's only valid for |x| < 1? Anybody who understands the definition of convergent series. > That series isn't just a cute bit of trivia, it is the basis for signed integer mathematics in your CPU. "True" enough for you? Eh. That's only half true. It's related, certainly, by means of a similarity between 2-adic numbers and two's complement representation, but I'd hardly call it the basis for signed integer mathematics.
  - MichaelBurge 10y ago
    Are you doing something like this: * Define the series over the 2-adic numbers rather than the reals * Associate every 2-adic integer with its corresponding sequence in the inverse limit Z/2^nZ for n = 1..infinity * Truncate the sequence to a certain precision(an index i), and make an equivalence class a ~ b if a and b agree on the first i elements. You get a field Z_(2^n), and identities in 2-adic analysis should carry down to the equivalence classes. Here 1 + 2 + 4 ... = -1 in the 2-adics. That isn't the way I would've thought about signed integers, but it seems like it would work.
mcbits 10y ago
I think a good analogy to make this seem less mysterious would be the various methods of determining an average. E.g. we have no problem saying the "average family" has 2.4 children despite the impossibility of any family having 2.4 children. The number is just useful in other calculations for income, expenses, etc. From there it's not hard to imagine that the -1/12 result could be useful and justifiable as an intermediate step in other calculations despite being an "impossible" destination on its own.
hhjkjhkjhhlkj 10y ago
Animated gifs made me ill. Couldn't read article :(
tinco 10y ago
Tl;dr: No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising. Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to. edit: For fun, a short story: If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.
- oneloop 10y ago
  The arrogance here is breathtaking. Have you ever considered that maybe people who know more than you have come to the conclusion that what you consider the bad definition is actually the good one, by the criteria that matter, and vice versa?
  - flipp3r 10y ago
    Have you?
  - avian 10y ago
    He didn't say "bad definition". He said "unintuitive" and I agree. The video is obviously meant for an audience that is not intimately familiar with infinite series or their applications in string theory. I'm sure most people found it most unintuitive as it is very much in conflict with how sums behave in everyday life. The fact that the video failed to discuss the important details is well worth criticizing, as the article explains.
  - raverbashing 10y ago
    > The arrogance here is breathtaking. Oh we can definitely agree to that > that what you consider the bad definition is actually the good one Sum is pretty much well defined for different sets used in mathematics. In none of them the sum of infinite positive numbers is a negative number. Here's a definition for a bad definition: it is ambiguous
vaidhy 10y ago
I had found this YouTube video before that explained it in a way I can understand - http://m.youtube.com/watch%3Fv%3DjcKRGpMiVTw http://m.youtube.com/watch%3Fv%3DjcKRGpMiVTw
- CarolineW 10y ago
  I think there's something wrong with the link you've provided: > Our systems have detected unusual > traffic from your computer network. > Please try your request again later. Perhaps this is the video you intended: https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw
  - ryanschneider 10y ago
    I came here to post this video as well, Mathologer is a great channel, I often prefer his explanations to Numberphile's. Highly recommended.
beefsack 10y ago
As soon as I see people dropping GIFs and image macros into intelligent discussions like this, I can't help but immediately become sceptical of what they're saying. It happens a lot in fairly serious technical computing blog posts and I've been trying to wrap my head around why people do it.
- ColinWright 10y ago
  Many times now when people have looked at my carefully written, carefully reasoned, well-laid out writings, have gone: Aaarrrggghhhh !!! WALL OF TEXT !!! Aaarrrggghhhh !!! It seems that many people need humorous (for some definition of "humorous") images and animations to make them think tat what they are reading is entertainment. I hate it, but it is an increasing trend, and I'm not surprised. Disappointed, but not surprised.
  - 91823791 10y ago
    I am surprised, because this presentation style used to be restricted to children's books. I was unable to finish this article because I refuse to have Riemann mixed up with memegenerator.net.
    - nommm-nommm 10y ago
      Unable is not the same as unwilling.
      - ars 10y ago
        I had a hard time as well - the moving image was very distracting, I had to cover it to try to concentrate on what the text was saying.
        nommm-nommm 10y ago
        That is not what the comment I was replying to said. There was a "refusal" to mix academic content with Internet memes.
TheCondor 10y ago
It doesn't satisfy the Cauchy criteria, it can't converge. The trick step of assuming that 1+0+1+0.... converges is also not Cauchy. It's interesting and string theory and some other physics get results by using -1/12 but it's not strictly correct. No more than the omni proof
pesenti 10y ago
For the mathematically inclined, Terence Tao's explanation provides an easier intuition of the relationship using real-variable methods: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...
Tergmap 10y ago
This was in one of Rumanajan's textbooks that he sent to Hardy. It changes the semantics of established notation instead of invnting new.
olsgaard 10y ago
In the numberfile video he doesn't explain the way he calculates 2・S_2. He says he shift it, but doesn't explain why that is valid. Shifted version: 1-2+3-4+5-6 ... 1-2+3-4+5 ... sum: 1-1+1-1+1-1 ... Multiplied version: 2-4+6-8+10-12 ... The multiplied version shifts between +(2n) and -(2n). Following the logic that S_1 = 0.5, because that is the average between 0 and 1, I would argue that the multiplied version of S2 should equal 0, as that is the average between a postitive constant and its negative (but the variance is going to be infinite. Doesn't that have a say?). What if we triple shift? Triple shifted version: 1-2+3-4+5-6+7-8+9 ... 1-2+3-4+5-6 ... sum: 2-3+3-3+3-3+3 ... Look! now 2・S2 is equal to 2!
- diogofranco 10y ago
  I don't think your math adds up. Your triple shifted version would be between -1 or 2 depending on the cut right? So, still 1/2.
  - olsgaard 10y ago
    Yeah, that was a brainfart on my end. What about the multiplied version? That is how I would intuitively understand 2S_2, and I still don't accept that shifting is the same.
whack 10y ago
Just watched the entire video. Using the same "proof" that they used, I can also prove that 1+1+1+1.... = 0 Proof: S1 = 1 + 2 + 3 + 4..... S2 = 0 + 1 + 2 + 3 + 4... S1 - S2 = (1 + 2 + 3 + 4 + ...) - (0 + 1 + 2 + 3 + ...) = 1 + 1 + 1 + 1.... S2 == S1 (by definition, since all you're doing is adding a 0) => S1 - S2 == 0 Therefore 0 = 1 + 1 + 1 + ..... Obviously this is pure nonsense. You can't just "shift things around" and use elementary mathematics when dealing with infinite series that don't converge. Maybe there's a more convincing proof out there, but the one they presented in the video is bogus.
- Bahamut 10y ago
  Didn't watch the video, but what is mentioned in the article is correct - analytic continuation is used to extend this series to an analytic function on the complex plane, and it happens to be the Riemann zeta function. By uniqueness of analytic continuation (due to a theorem or lemma that when an analytic function is constant at an infinite number of points, it is constant everywhere), we know this is the only valid extension that makes sense to work with (consider being analytic similar to what it means for a function to be differentiable in the real numbers).
- deleted 10y ago
  [deleted]
- zeroer 10y ago
  > Obviously this is pure nonsense. This is begging the question. Why can't 1 + 1 + ... = 0? Also, I wouldn't be so sure that S2 == S1. You can't re-arrange infinitely many terms in an infinite series and still be guaranteed the sum is the same.
  - whack 10y ago
    S2 is identical to (0 + S1). Are you suggesting that (0 + S1) != S1 ?
    - ebola1717 10y ago
      I don't know for certain if (0 + S1) ?= S1, but infinite series require care. Consider this: Let S1 = 0 + 0 + 0 + ... = 0 Then surely, S1 = (1 - 1) + (1 - 1) ... = 1 - 1 + 1 - 1 + ... -1 + S1 = -1 + 1 - 1 + 1 - 1 ... = (-1 + 1) + (-1 + 1) ... = 0 But then -1 + 0 = 0
Ruud-v-A 10y ago
Here’s what’s really going on: You can define a function f from a subset of the complex numbers to the complex numbers where f(z) = \sum_{n=1}^\infty 1/n^z. Be careful with the domain of this function: the series does not converge for all z. You can plug in -1 and see that symbolically, f(-1) = 1 + 2 + 3 + .... But the series does not converge for z = -1, and it is simply not true that \sum_{n=1}^\infty n = -1/12; the series does not converge; equating it to something is a nonsensical thing to do. What is going then? Even though f is not defined for all complex numbers, there exist functions from the complex numbers to the complex numbers that -- restricted to the domain of f -- are equal to f. They "continue" f to all of the complex numbers. And if one imposes a restriction on these continuations (namely that they are analytic), then it turns out that there is a unique analytic continuation of f: the Riemann zeta function. And zeta(-1) = -1/12. Don’t confuse the definitions here. The Riemann zeta function can be defined as the analytic continuation of the series, the series is not defined in terms of the Riemann zeta function!
- nwjtkjn 10y ago
  What bothers me is that it seems there could be many ways of writing 1 + 2 + 3 + ... as the "specialization" of a formal series depending on a variable z, and for which the formal series converges to an analytic function on some domain away from the specialization. I can imagine there's a way of doing this in such a way that the function's analytic continuation to the specialization evaluates to any number you want, not just -1/12. However, I'm having trouble cooking up such an example.
  - noobermin 10y ago
    It indeed makes you wonder whether it is true or not and whether it justifies the usage. This is one of the many reasons I left particle physics, so I have a biased perspective.
  - Chinjut 10y ago
    This is a good concern. FWIW, though, there ARE natural summability methods which turn mere indexed sequences of terms into sums, without any particular choice as to how to interpret those sequences as specializations of formal series, which happen to assign the sequence whose n-th term is n a sum of -1/12. For an example of one such method, see the end of my post elsewhere in this thread.
psyc 10y ago
The other YouTube popularization that regularly irks me is when people beam about how "there are lots of different sized infinities", without explaining that when mathematicians use the word "size" and "infinity" in that context, it means cardinality (technical term) of an infinite set, within an explicitly defined set theory that includes the axiom of infinity (all technical concepts).
egjerlow 10y ago
I'm gonna be contrarian and say, yes it does - it's been physically proven! The casimir effect, as other people here have mentioned, depends on this being true, and it has been experimentally tested. To me it's one of those things where you just go 'damn' because of the perplexing relations that exist between math and physics. If anything hints at what the hell goes on in this universe, to me it's stuff like this.
- quantumhobbit 10y ago
  The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out of these imaginary numbers. Once we do that we notice that these numbers are really useful for calculating real physical things. So maybe imaginary numbers aren't so imaginary. Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.
  - chowells 10y ago
    I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.
quantumhobbit 10y ago
I forget who said, "In mathematics, you don't learn things, you just get used to them." But it applies here.
- CarolineW 10y ago
  According to this link[0] it was John von Neumann: Young man, in mathematics you don't understand things. You just get used to them. [0] https://en.wikiquote.org/wiki/John_von_Neumann https://en.wikiquote.org/wiki/John_von_Neumann
xg15 10y ago
I want familiar before with the concept of analytic continuations and so this part really confused me: ("The" is the appropriate article to use because the analytic continuation of a function is unique.) Let f: N -> N, f(x) = x be a function on the natural numbers. Then I could define two functions g(x) and h(x) on N u {foo} that behave just like f for natural numbers. However, g(foo) is 42 while h(foo) is 666. Wouldn't g and h both be valid analytic continuations of f in N u {foo}, according to the definition of analytic continuations explained in the article? I'm wondering about that as the uniqueness seems to be an important property for the rest of the explanation, yet it is simply assumed here without any further explanation.
- Bahamut 10y ago
  Uniqueness requires proof - it relies on a theorem/lemma having to do with if an analytic function is constant in a region, then it is constant everywhere. Uniqueness then arises as if there are two analytic continuations that are the same in that region, then the difference is 0 there, and thus 0 everywhere, resulting in the two being the same. That is a non-trivial theorem/lemma though, and the proof is typically gone through at a graduate complex analysis course.
- derkha 10y ago
  To clarify Bahamut's point: The functions must be analytic - https://en.wikipedia.org/wiki/Analytic_continuation#Initial_discussion https://en.wikipedia.org/wiki/Analytic_continuation#Initial_.... Regardless of the actual value of foo, at least one of g or h won't be differentiable at foo, so they can't both be analytic (assuming you choose some open superset of N u {foo}, otherwise the theorem breaks down even sooner).
throw94 10y ago
From Zen and the Art of Motorcycle Maintenance <i> “The law of gravity and gravity itself did not exist before Isaac Newton." ...and what that means is that that law of gravity exists nowhere except in people's heads! It 's a ghost!" Mind has no matter or energy but they can't escape its predominance over everything they do. Logic exists in the mind. numbers exist only in the mind. I don't get upset when scientists say that ghosts exist in the mind. it's that only that gets me. science is only in your mind too, it's just that that doesn't make it bad. or ghosts either." Laws of nature are human inventions, like ghosts. Law of logic, of mathematics are also human inventions, like ghosts." ...we see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past.” </i>
- j1vms 10y ago
  To go a bit further, everything as one perceives it is necessarily in one's mind (or nervous system, to be precise). One consciously or unconsciously determines that something is happening outside of their mind, which makes that determination simply a philosophical one. This is a reminder to all, as to why the highest degree granted at any conventional institution is labelled a Doctorate of Philosophy in a given subject. I would go as far as to say that just as the Uncertainty Principle prescribes a limit on what is knowable in quantum physics, so does philosophy suggest limits on what will ever be rationally proven through human perception, ideas, and knowledge.
jiiam 10y ago
No. Source: https://en.wikipedia.org/wiki/Betteridge%27s_Law_of_Headlines https://en.wikipedia.org/wiki/Betteridge%27s_Law_of_Headline... EDIT: Some further elaboration: I'm sick of the question. The answer is: not in any sense that would be meaningful to the people to whom this stuff is being told. You're just being misleading by implying that the sum of positive integers can converge. I don't want to hear any "But if you take this analytic continuation..." or "But in a certain sense...", they're just misleading as the thousands of proofs that 1=0.
CorvusCrypto 10y ago
From my point of view, what Numberphile did was wildly successful. I think the video's "wow" factor, as the author put it, is doing its job of getting people interested in maths. Yes they could have explained some things a little better. However as noted by the author and some of her other field-mates she referred to at the bottom of the article, I think really the only sore people are the mathematicians. I also think that this is because they already know the rest of the story behind this interesting result. Once we know something in detail we as humans tend to scoff at incomplete explanations because in our eyes it does injustice to the topic. However, to the normal viewer this video probably made maths look incredibly interesting and more than likely even caused them to research it a bit more. I would hazard to say an article like the one Dr. Lamb wrote would not have that same effect, though it is technically more correct. Numberphile to me is more about reinstating the interest in maths in a society where you are usually introduced to the topic by doing repetitive, seemingly impractical calculations and this video of theirs as referenced in the article has definitely done that.
- foofoo55 10y ago
  The 6 or so people I know who have seen the original Numberphile video responded with statements like "I'll never understand math", "that's just stupid", and "I don't get it". None expressed a greater interest in maths as a result; quite the opposite. Just like a good host at a party, a good educational video should leave the viewer felling positive and better about themselves. So clearly, in this case, it depends on the audience.
  - knownothing 10y ago
    > Just like a good host at a party, a good educational video should leave the viewer felling positive and better about themselves. Is the purpose of education to make the other person feel positive and good about themselves?
    - StavrosK 10y ago
      No, the purpose of education is to teach you stuff. A good way to educate is to make people feel good about themselves.
empath75 10y ago
As a computer programmer, this actually feels fairly intuitive. If you had a function that computed the sums of convergent series and returned errors for inputs for which it was undefined, and replaced it with one that computed the reimann zeta function, you'd still get the same results for convergent series, but would get results for the formerly undefined inputs. Thought of this way, it's pretty clear that you're talking about performing two different calculations, and they aren't equal.
JoeCamel 10y ago
I loved this video https://www.youtube.com/watch?v=XFDM1ip5HdU https://www.youtube.com/watch?v=XFDM1ip5HdU "An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math."
Chinjut 10y ago
I didn't think the Numberphile video on this was well done, in its "Math is a bunch of inscrutable magic you laypeople will never understand, and every attempt you (as represented by our camera man) make to voice your dumb intuitions is dumb, you big dumb dumbs!" way, but I even more was annoyed by the sneering "You're not allowed to do that! There are clear fixed rules and single, permanent, all-purpose definitions!" dismissals of the 1 + 2 + 3 + 4 + ... = -1/12 result in the backlash. I'm going to copy and paste the explanation I originally wrote at Quora (https://www.quora.com/Whats-the-intuition-behind-the-equation-1+2+3+-cdots-tfrac-1-12/answers/3900029 https://www.quora.com/Whats-the-intuition-behind-the-equatio...), because I think it captures well everything I'd like to say about this at every level of the discussion: The sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this: First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and so, in some sense, X = 1/2. Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and so, in some sense, Y = X/2 = 1/4. Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and so, in some sense, Z = -Y/3 = -1/12. In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't. That's it. It's that simple. Everything else I'm going to say is just to comfort those who are uncomfortable with the game we've just played. Note that I've said "in some sense" several times in the above argument. That's because, while we all know how to add and subtract a finite collection of numbers in the ordinary way, when it comes to adding and subtracting an infinite series of numbers, there are many different ways of interpreting what this should mean. Just knowing how to add finitely many numbers doesn't automatically tell us what it means to add a whole infinite series of them. And when it comes to summation of infinite series, it turns out there's not just one nice notion of "summation"; there are many different ones, which are nice for different purposes. One such notion is "Keep adding things up, one by one, starting from the front, and see if the results get closer and closer to some particular value; if so, that value is the sum". On that account of what summation means, you clearly won't get any finite answer for 1 + 2 + 3 + 4 + ...; since the terms never get any smaller, the partial sums will never settle down to a finite value (and certainly not a negative one like -1/12!). They instead, in a natural sense, should be understood as summing to positive infinity. And there's nothing wrong with this! You are not wrong to feel that 1 + 2 + 3 + 4 + ... is positive and infinite, and math does not deny this; there absolutely is an account of summation corresponding to this intuition. It's just not the only account of summation worth thinking about. We could instead consider other notions of "summation", including ones designed precisely so that arguments like the one we made at the beginning (which are very natural arguments to make!) counted as legitimate ways to reason about such "summation". And then, by definition, we will have that 1 + 2 + 3 + 4 + ... = -1/12, on such accounts of "summation". (In doing so, we will lose certain familiar properties such as "A sum of positive terms is always positive". But this is how generalizations work; generalizations very often lose familiar properties. Even the textbook, limit-based account of infinite summation loses familiar properties like "The order of summation doesn't matter". Even finitary summation of integers loses the familiar property "If a sum is zero, so are all the summands" from basic counting. But there is a web of resemblances to more familiar kinds of summation which can justify, in certain moods, thinking of each of these generalizations as a form of summation itself.) If you insist that "Keep adding things up and see if the results get closer and closer to some particular value" is the only account of summation you're interested in, you'll object to the argument we gave at the beginning, saying "You're not allowed to do that kind of shifting over and adding to itself reasoning all willy-nilly; look at what nonsense it produces!". But it can be made sense of, and is even fruitful to make sense of, in certain contexts in mathematics, and there is no need to blind ourselves to this insight. Again, that's it. It's that simple. Everything else I'm going to say is just to comfort those who are still uncomfortable. For those who want a more systematic, formal account of series summation of a sort which validates the above manipulation, read on: [Comment too long, will be continued in reply]
lisper 10y ago
http://blog.rongarret.info/2014/01/no-sum-of-all-positive-integers-is-not.html http://blog.rongarret.info/2014/01/no-sum-of-all-positive-in...
- nkurz 10y ago
  Encountering it now, I think your followup article "Does it matter if the sum of all integers is -1/12?" makes an excellent case for why it does in fact matter a lot: The fact that the unsound reasoning in this particular case led to a conclusion that superficially resembles a conclusion that can also be arrived at by sound reasoning just makes it that much worse. It encourages people to think: because this mode of reasoning led to a "correct" conclusion in that case, then it will probably lead to correct conclusions in other cases. If the problem were confined to mathematics I might not make such a big deal out of it, but it's not. The problem of people uncritically accepting conclusions drawn by unsound methods of reasoning pervades our society and causes real damage. http://blog.rongarret.info/2014/01/does-it-matter-if-sum-of-all-integers.html http://blog.rongarret.info/2014/01/does-it-matter-if-sum-of-...
TheRealPomax 10y ago
tl;dr: there is no "real" or "one" kind of maths, you'll have to first pick which field of mathematics you're working with. Arithmetics? Then no: 1+2+3+... is a divergent sum. Using calculus, specifically analytic continuation on the an Euler series? Then yeah, you can apply the rules in such a way that you get this answer and it's a mighty useful identity that can be exploited in complicated proofs.
powertower 10y ago
Is it saying that the complex version of the series (1+0i)+(2+0i)+(3+0i)... = -1/12?
blueprint 10y ago
No, it doesn't.
numlocked 10y ago
Reading a couple of the linked posts, I thought this was the best take on it from mathematician Jordan Ellenberg (it certainly made sense to my comp-sci brain): It's not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + .... = -1/12. When we ask “what is the value of the infinite sum,” we've made a mistake before we even answer! Infinite sums don't have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video. A different protocol assigns it the value -1/12. Neither answer is more correct than the other.
rrmm 10y ago
Terry Tao shows how to get from side of equality to the other: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri... It goes into detail and shows the derivation and internal consistency of the method.
pfortuny 10y ago
It boils down to what you mean by "equal."
ojknkjnsdf 10y ago
I don't understand why people keep playing with infinite series like this... it's not sound math and doesn't lead to anything useful. Just a cheap way for worthless mathematicians to feel clever.
owenversteeg 10y ago
Glad SciAm is helping to spread correct info. I think that basically, the problem with many math videos in this style is that they tend to make uncommon, if interesting, assumptions without explaining what they are or why they make the assumption in order to get clicks. This is my comment from the last time this video was mentioned: > "The sum of the series 1+2+3+4+5+6... = -1/12" is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series. Even mathematicians working with Cesàro sums surround such statements with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." [0] Precisely none of the times I've heard the "1+2+3+4...=-1/12" bullshit has the person stating it prefaced their statement with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." > If you say that "1+2+3+4...=-1/12" without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that "this 95 cent pack of gum is free" because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum. > The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers. > [0] Madras, Neal. "A Note on Diffusion State Distance." arXiv preprint arXiv:1502.07315 (2015). My favorite post on the subject still has to be this: http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-bad-astronomer/ http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-...
serge2k 10y ago
Knew it was that numberphile video before I clicked.
chris_wot 10y ago
So when you have an infinite series of the form 1-1+1-1+1-1+... then because there is no single point of convergence and so you average the two values 0 and 1 to give 0.5. Why does an average of the two values that you can converge on help? Also, what about something like: sin(π/2) + sin(3π/2) - sin(5π/2) + sin(7π/2) - ... what would this be?
- tamana 10y ago
  The average is simply a summary description of the behaviorat infinity. Averages are a standard way to summarize complex phenomena. More precisely, the sume is 0.5+/-0.5, which is greater than 0 and less than 1 and ambiguous in between.
dahart 10y ago
Good time to remember 0^0=1. Or some people (like Knuth) like to assign that definition because it's convenient, and it makes everything work out in certain situations. I don't think it's bad to say that 0^0=1 when you hear the whole story, which is that there are multiple right answers given how we define exponents and operations on 0. But it's misleading to say 0^0=1 and stop there. Similarly, after reading about this today, it's making more sense to me that we can, if we choose, define 1+2+3+...=-1/12, and it makes sense in some contexts. It's just that this isn't the only answer and it isn't the whole story. I liked the article, and ended up reading a bunch of Evelyn's column earlier today. I see some people complaining about the pictures... I thought they were funny and relevant, my only nit pick is I can barely read the light blue text on the wink gif, the colors are so hard for me to look at.
mcv 10y ago
As it happens, I have a proof that 1 == 2. Of course my proof involves a step that involves division by zero, but if it's legal to define undefined results, then why can't I do that if it helps my proof? Because as far as I can tell (with my admittedly limited understanding of mathematics), that's basically what's going on here: they define the result of 1-1+1-1+1-1... to be 1/2, which it can of course never be. The result is never 1/2; it's either 1 or 0. Taken to infinity, the only reasonable definition for that sum is undefined. If I can say that 1/2 is fine too, then I should also be able to attach my own definition to 1/0. Also, if string theory really relies on such questionable mathematical steps, then that would make me question string theory even more. As far as I understand, string theory makes no testable predictions, which suggests to me that no results based on this questionable mathematical trick have been experimentally verified. If there is some real, experimentally verified physics that relies on the sum of all natural numbers to be -1/12, then I'd love to be corrected (though I doubt I'll understand it).

197 points by CarolineW 10y ago

Well... Guess that if I watch an infinity number of videoes like this I will end up not getting more stupid like I did with this one. Waste of time.

lifthrasiir 10y ago

Somehow the title is missing an ellipsis, which made me confused for a second.

CarolineW 10y ago
Now added - apologies for the confusion.

dozzie 10y ago

> Does 1+2+3 Really Equal -1/12? Of course not. It's a misconception propagated by ignorant people who stumbled upon Ramanujan summation.

CarolineW 10y ago
Out of interest - and I'd really like you to answer honestly although obviously I have no way of knowing if you don't - have you read the article, or are you simply responding to the title?
- dozzie 10y ago
  Only responding to the title. Such titles annoy me, and I happen to have this bad habit of complaining in such cases.
  - CarolineW 10y ago
    Thank you for your honest answer. The article has a discussion that is good, careful, nuanced, and quite complete. It also has lots of links to related discussions, including talking about why - although this specific claim is "obvious nonsense" - there is actually some sense underneath it, although deeper than most people ever go. If you're going to complain you should probably have an opinion on how to improve it. What title would you give such an article?
    - dozzie 10y ago
      I would have an opinion if I read the article, which I do not intent at the moment (I find my own code more interesting than mathematical concepts I have no use for).
      - qubex 10y ago
        Then mightn't your time be more productively spent paying attention to your code you mention rather than commenting on articles you have not read about concepts you reckon you have no use for?
        mcguire 10y ago
        Compiling!

amelius 10y ago

This is what you get when mathematicians start "hacking". Instead, they should have invented a proper notation conveying what is really meant by the sum and the "..." ellipsis. This notation apparently does not cut it.

nothrabannosir 10y ago
> This is what you get when mathematicians start "hacking". I.e. physics.
MichaelBurge 10y ago
ζ(s) has been in use since 1859, when Riemann introduced it. Riemann's paper isn't as thorough as it could've been, but it seems clear he knew exactly what he was doing, being one of the founders of Complex Analysis: http://www.claymath.org/sites/default/files/ezeta.pdf http://www.claymath.org/sites/default/files/ezeta.pdf This series is ζ(-1). I don't know about the Youtubers or Scientific American, but mathematicians studying Complex Analysis know exactly what it means.

MichaelBurge 10y ago

No need to invoke the Riemann Zeta function and analytic continuation. In high school they summed this series(valid for x < 1): 1 + x + x^2 ... = 1/(1-x) Plug in x = 2 to get: 1 + 2 + 4 + 8 ... = -1 There's a million of these series in your dusty old Calculus textbook. Or you could look in 'generatingfunctionology' to find others. Complex analysis makes it more interesting, because the additional Cauchy-Riemann constraints make the solution unique. And so people are more willing to say that the unique solution is the "true" answer. Really, I say this is just another example of the complex numbers being weird. I took a couple different courses in it(one in college, 2 online) because it was clear something interesting was happening with them, but there was never a unifying theme. I can kind of spot a rule like "analytic functions preserve 90-degree angles" in the Cauchy-Riemann equations, but it hardly explains all the crazy theorems. That series is just another example of someone taking a well-behaved series, analytically continuing it into the complex numbers, and now it's clear something interesting is happening but it's not clear what.

thaumasiotes 10y ago
Who says it's only valid for |x| < 1? That series isn't just a cute bit of trivia, it is the basis for signed integer mathematics in your CPU. "True" enough for you?
- messe 10y ago
  > Who says it's only valid for |x| < 1? Anybody who understands the definition of convergent series. > That series isn't just a cute bit of trivia, it is the basis for signed integer mathematics in your CPU. "True" enough for you? Eh. That's only half true. It's related, certainly, by means of a similarity between 2-adic numbers and two's complement representation, but I'd hardly call it the basis for signed integer mathematics.
- MichaelBurge 10y ago
  Are you doing something like this: * Define the series over the 2-adic numbers rather than the reals * Associate every 2-adic integer with its corresponding sequence in the inverse limit Z/2^nZ for n = 1..infinity * Truncate the sequence to a certain precision(an index i), and make an equivalence class a ~ b if a and b agree on the first i elements. You get a field Z_(2^n), and identities in 2-adic analysis should carry down to the equivalence classes. Here 1 + 2 + 4 ... = -1 in the 2-adics. That isn't the way I would've thought about signed integers, but it seems like it would work.

mcbits 10y ago

I think a good analogy to make this seem less mysterious would be the various methods of determining an average. E.g. we have no problem saying the "average family" has 2.4 children despite the impossibility of any family having 2.4 children. The number is just useful in other calculations for income, expenses, etc. From there it's not hard to imagine that the -1/12 result could be useful and justifiable as an intermediate step in other calculations despite being an "impossible" destination on its own.

hhjkjhkjhhlkj 10y ago

Animated gifs made me ill. Couldn't read article :(

tinco 10y ago

Tl;dr: No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising. Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to. edit: For fun, a short story: If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.

oneloop 10y ago
The arrogance here is breathtaking. Have you ever considered that maybe people who know more than you have come to the conclusion that what you consider the bad definition is actually the good one, by the criteria that matter, and vice versa?
- flipp3r 10y ago
  Have you?
- avian 10y ago
  He didn't say "bad definition". He said "unintuitive" and I agree. The video is obviously meant for an audience that is not intimately familiar with infinite series or their applications in string theory. I'm sure most people found it most unintuitive as it is very much in conflict with how sums behave in everyday life. The fact that the video failed to discuss the important details is well worth criticizing, as the article explains.
- raverbashing 10y ago
  > The arrogance here is breathtaking. Oh we can definitely agree to that > that what you consider the bad definition is actually the good one Sum is pretty much well defined for different sets used in mathematics. In none of them the sum of infinite positive numbers is a negative number. Here's a definition for a bad definition: it is ambiguous

vaidhy 10y ago

I had found this YouTube video before that explained it in a way I can understand - http://m.youtube.com/watch%3Fv%3DjcKRGpMiVTw http://m.youtube.com/watch%3Fv%3DjcKRGpMiVTw

CarolineW 10y ago
I think there's something wrong with the link you've provided: > Our systems have detected unusual > traffic from your computer network. > Please try your request again later. Perhaps this is the video you intended: https://www.youtube.com/watch?v=jcKRGpMiVTw https://www.youtube.com/watch?v=jcKRGpMiVTw
- ryanschneider 10y ago
  I came here to post this video as well, Mathologer is a great channel, I often prefer his explanations to Numberphile's. Highly recommended.

beefsack 10y ago

As soon as I see people dropping GIFs and image macros into intelligent discussions like this, I can't help but immediately become sceptical of what they're saying. It happens a lot in fairly serious technical computing blog posts and I've been trying to wrap my head around why people do it.

ColinWright 10y ago
Many times now when people have looked at my carefully written, carefully reasoned, well-laid out writings, have gone: Aaarrrggghhhh !!! WALL OF TEXT !!! Aaarrrggghhhh !!! It seems that many people need humorous (for some definition of "humorous") images and animations to make them think tat what they are reading is entertainment. I hate it, but it is an increasing trend, and I'm not surprised. Disappointed, but not surprised.
- 91823791 10y ago
  I am surprised, because this presentation style used to be restricted to children's books. I was unable to finish this article because I refuse to have Riemann mixed up with memegenerator.net.
  - nommm-nommm 10y ago
    Unable is not the same as unwilling.
    - ars 10y ago
      I had a hard time as well - the moving image was very distracting, I had to cover it to try to concentrate on what the text was saying.
      - nommm-nommm 10y ago
        That is not what the comment I was replying to said. There was a "refusal" to mix academic content with Internet memes.

TheCondor 10y ago

It doesn't satisfy the Cauchy criteria, it can't converge. The trick step of assuming that 1+0+1+0.... converges is also not Cauchy. It's interesting and string theory and some other physics get results by using -1/12 but it's not strictly correct. No more than the omni proof

pesenti 10y ago

For the mathematically inclined, Terence Tao's explanation provides an easier intuition of the relationship using real-variable methods: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...

Tergmap 10y ago

This was in one of Rumanajan's textbooks that he sent to Hardy. It changes the semantics of established notation instead of invnting new.

olsgaard 10y ago

In the numberfile video he doesn't explain the way he calculates 2・S_2. He says he shift it, but doesn't explain why that is valid. Shifted version: 1-2+3-4+5-6 ... 1-2+3-4+5 ... sum: 1-1+1-1+1-1 ... Multiplied version: 2-4+6-8+10-12 ... The multiplied version shifts between +(2n) and -(2n). Following the logic that S_1 = 0.5, because that is the average between 0 and 1, I would argue that the multiplied version of S2 should equal 0, as that is the average between a postitive constant and its negative (but the variance is going to be infinite. Doesn't that have a say?). What if we triple shift? Triple shifted version: 1-2+3-4+5-6+7-8+9 ... 1-2+3-4+5-6 ... sum: 2-3+3-3+3-3+3 ... Look! now 2・S2 is equal to 2!

diogofranco 10y ago
I don't think your math adds up. Your triple shifted version would be between -1 or 2 depending on the cut right? So, still 1/2.
- olsgaard 10y ago
  Yeah, that was a brainfart on my end. What about the multiplied version? That is how I would intuitively understand 2S_2, and I still don't accept that shifting is the same.

whack 10y ago

Just watched the entire video. Using the same "proof" that they used, I can also prove that 1+1+1+1.... = 0 Proof: S1 = 1 + 2 + 3 + 4..... S2 = 0 + 1 + 2 + 3 + 4... S1 - S2 = (1 + 2 + 3 + 4 + ...) - (0 + 1 + 2 + 3 + ...) = 1 + 1 + 1 + 1.... S2 == S1 (by definition, since all you're doing is adding a 0) => S1 - S2 == 0 Therefore 0 = 1 + 1 + 1 + ..... Obviously this is pure nonsense. You can't just "shift things around" and use elementary mathematics when dealing with infinite series that don't converge. Maybe there's a more convincing proof out there, but the one they presented in the video is bogus.

Bahamut 10y ago
Didn't watch the video, but what is mentioned in the article is correct - analytic continuation is used to extend this series to an analytic function on the complex plane, and it happens to be the Riemann zeta function. By uniqueness of analytic continuation (due to a theorem or lemma that when an analytic function is constant at an infinite number of points, it is constant everywhere), we know this is the only valid extension that makes sense to work with (consider being analytic similar to what it means for a function to be differentiable in the real numbers).
deleted 10y ago
[deleted]
zeroer 10y ago
> Obviously this is pure nonsense. This is begging the question. Why can't 1 + 1 + ... = 0? Also, I wouldn't be so sure that S2 == S1. You can't re-arrange infinitely many terms in an infinite series and still be guaranteed the sum is the same.
- whack 10y ago
  S2 is identical to (0 + S1). Are you suggesting that (0 + S1) != S1 ?
  - ebola1717 10y ago
    I don't know for certain if (0 + S1) ?= S1, but infinite series require care. Consider this: Let S1 = 0 + 0 + 0 + ... = 0 Then surely, S1 = (1 - 1) + (1 - 1) ... = 1 - 1 + 1 - 1 + ... -1 + S1 = -1 + 1 - 1 + 1 - 1 ... = (-1 + 1) + (-1 + 1) ... = 0 But then -1 + 0 = 0

Ruud-v-A 10y ago

Here’s what’s really going on: You can define a function f from a subset of the complex numbers to the complex numbers where f(z) = \sum_{n=1}^\infty 1/n^z. Be careful with the domain of this function: the series does not converge for all z. You can plug in -1 and see that symbolically, f(-1) = 1 + 2 + 3 + .... But the series does not converge for z = -1, and it is simply not true that \sum_{n=1}^\infty n = -1/12; the series does not converge; equating it to something is a nonsensical thing to do. What is going then? Even though f is not defined for all complex numbers, there exist functions from the complex numbers to the complex numbers that -- restricted to the domain of f -- are equal to f. They "continue" f to all of the complex numbers. And if one imposes a restriction on these continuations (namely that they are analytic), then it turns out that there is a unique analytic continuation of f: the Riemann zeta function. And zeta(-1) = -1/12. Don’t confuse the definitions here. The Riemann zeta function can be defined as the analytic continuation of the series, the series is not defined in terms of the Riemann zeta function!

nwjtkjn 10y ago
What bothers me is that it seems there could be many ways of writing 1 + 2 + 3 + ... as the "specialization" of a formal series depending on a variable z, and for which the formal series converges to an analytic function on some domain away from the specialization. I can imagine there's a way of doing this in such a way that the function's analytic continuation to the specialization evaluates to any number you want, not just -1/12. However, I'm having trouble cooking up such an example.
- noobermin 10y ago
  It indeed makes you wonder whether it is true or not and whether it justifies the usage. This is one of the many reasons I left particle physics, so I have a biased perspective.
- Chinjut 10y ago
  This is a good concern. FWIW, though, there ARE natural summability methods which turn mere indexed sequences of terms into sums, without any particular choice as to how to interpret those sequences as specializations of formal series, which happen to assign the sequence whose n-th term is n a sum of -1/12. For an example of one such method, see the end of my post elsewhere in this thread.

psyc 10y ago

The other YouTube popularization that regularly irks me is when people beam about how "there are lots of different sized infinities", without explaining that when mathematicians use the word "size" and "infinity" in that context, it means cardinality (technical term) of an infinite set, within an explicitly defined set theory that includes the axiom of infinity (all technical concepts).

egjerlow 10y ago

I'm gonna be contrarian and say, yes it does - it's been physically proven! The casimir effect, as other people here have mentioned, depends on this being true, and it has been experimentally tested. To me it's one of those things where you just go 'damn' because of the perplexing relations that exist between math and physics. If anything hints at what the hell goes on in this universe, to me it's stuff like this.

quantumhobbit 10y ago
The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out of these imaginary numbers. Once we do that we notice that these numbers are really useful for calculating real physical things. So maybe imaginary numbers aren't so imaginary. Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.
- chowells 10y ago
  I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.

quantumhobbit 10y ago

I forget who said, "In mathematics, you don't learn things, you just get used to them." But it applies here.

CarolineW 10y ago
According to this link[0] it was John von Neumann: Young man, in mathematics you don't understand things. You just get used to them. [0] https://en.wikiquote.org/wiki/John_von_Neumann https://en.wikiquote.org/wiki/John_von_Neumann

xg15 10y ago

I want familiar before with the concept of analytic continuations and so this part really confused me: ("The" is the appropriate article to use because the analytic continuation of a function is unique.) Let f: N -> N, f(x) = x be a function on the natural numbers. Then I could define two functions g(x) and h(x) on N u {foo} that behave just like f for natural numbers. However, g(foo) is 42 while h(foo) is 666. Wouldn't g and h both be valid analytic continuations of f in N u {foo}, according to the definition of analytic continuations explained in the article? I'm wondering about that as the uniqueness seems to be an important property for the rest of the explanation, yet it is simply assumed here without any further explanation.

Bahamut 10y ago
Uniqueness requires proof - it relies on a theorem/lemma having to do with if an analytic function is constant in a region, then it is constant everywhere. Uniqueness then arises as if there are two analytic continuations that are the same in that region, then the difference is 0 there, and thus 0 everywhere, resulting in the two being the same. That is a non-trivial theorem/lemma though, and the proof is typically gone through at a graduate complex analysis course.
derkha 10y ago
To clarify Bahamut's point: The functions must be analytic - https://en.wikipedia.org/wiki/Analytic_continuation#Initial_discussion https://en.wikipedia.org/wiki/Analytic_continuation#Initial_.... Regardless of the actual value of foo, at least one of g or h won't be differentiable at foo, so they can't both be analytic (assuming you choose some open superset of N u {foo}, otherwise the theorem breaks down even sooner).

throw94 10y ago

From Zen and the Art of Motorcycle Maintenance <i> “The law of gravity and gravity itself did not exist before Isaac Newton." ...and what that means is that that law of gravity exists nowhere except in people's heads! It 's a ghost!" Mind has no matter or energy but they can't escape its predominance over everything they do. Logic exists in the mind. numbers exist only in the mind. I don't get upset when scientists say that ghosts exist in the mind. it's that only that gets me. science is only in your mind too, it's just that that doesn't make it bad. or ghosts either." Laws of nature are human inventions, like ghosts. Law of logic, of mathematics are also human inventions, like ghosts." ...we see what we see because these ghosts show it to us, ghosts of Moses and Christ and the Buddha, and Plato, and Descartes, and Rousseau and Jefferson and Lincoln, on and on and on. Isaac Newton is a very good ghost. One of the best. Your common sense is nothing more than the voices of thousands and thousands of these ghosts from the past.” </i>

j1vms 10y ago
To go a bit further, everything as one perceives it is necessarily in one's mind (or nervous system, to be precise). One consciously or unconsciously determines that something is happening outside of their mind, which makes that determination simply a philosophical one. This is a reminder to all, as to why the highest degree granted at any conventional institution is labelled a Doctorate of Philosophy in a given subject. I would go as far as to say that just as the Uncertainty Principle prescribes a limit on what is knowable in quantum physics, so does philosophy suggest limits on what will ever be rationally proven through human perception, ideas, and knowledge.

jiiam 10y ago

No. Source: https://en.wikipedia.org/wiki/Betteridge%27s_Law_of_Headlines https://en.wikipedia.org/wiki/Betteridge%27s_Law_of_Headline... EDIT: Some further elaboration: I'm sick of the question. The answer is: not in any sense that would be meaningful to the people to whom this stuff is being told. You're just being misleading by implying that the sum of positive integers can converge. I don't want to hear any "But if you take this analytic continuation..." or "But in a certain sense...", they're just misleading as the thousands of proofs that 1=0.

CorvusCrypto 10y ago

From my point of view, what Numberphile did was wildly successful. I think the video's "wow" factor, as the author put it, is doing its job of getting people interested in maths. Yes they could have explained some things a little better. However as noted by the author and some of her other field-mates she referred to at the bottom of the article, I think really the only sore people are the mathematicians. I also think that this is because they already know the rest of the story behind this interesting result. Once we know something in detail we as humans tend to scoff at incomplete explanations because in our eyes it does injustice to the topic. However, to the normal viewer this video probably made maths look incredibly interesting and more than likely even caused them to research it a bit more. I would hazard to say an article like the one Dr. Lamb wrote would not have that same effect, though it is technically more correct. Numberphile to me is more about reinstating the interest in maths in a society where you are usually introduced to the topic by doing repetitive, seemingly impractical calculations and this video of theirs as referenced in the article has definitely done that.

foofoo55 10y ago
The 6 or so people I know who have seen the original Numberphile video responded with statements like "I'll never understand math", "that's just stupid", and "I don't get it". None expressed a greater interest in maths as a result; quite the opposite. Just like a good host at a party, a good educational video should leave the viewer felling positive and better about themselves. So clearly, in this case, it depends on the audience.
- knownothing 10y ago
  > Just like a good host at a party, a good educational video should leave the viewer felling positive and better about themselves. Is the purpose of education to make the other person feel positive and good about themselves?
  - StavrosK 10y ago
    No, the purpose of education is to teach you stuff. A good way to educate is to make people feel good about themselves.

empath75 10y ago

As a computer programmer, this actually feels fairly intuitive. If you had a function that computed the sums of convergent series and returned errors for inputs for which it was undefined, and replaced it with one that computed the reimann zeta function, you'd still get the same results for convergent series, but would get results for the formerly undefined inputs. Thought of this way, it's pretty clear that you're talking about performing two different calculations, and they aren't equal.

JoeCamel 10y ago

I loved this video https://www.youtube.com/watch?v=XFDM1ip5HdU https://www.youtube.com/watch?v=XFDM1ip5HdU "An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math."

Chinjut 10y ago

I didn't think the Numberphile video on this was well done, in its "Math is a bunch of inscrutable magic you laypeople will never understand, and every attempt you (as represented by our camera man) make to voice your dumb intuitions is dumb, you big dumb dumbs!" way, but I even more was annoyed by the sneering "You're not allowed to do that! There are clear fixed rules and single, permanent, all-purpose definitions!" dismissals of the 1 + 2 + 3 + 4 + ... = -1/12 result in the backlash. I'm going to copy and paste the explanation I originally wrote at Quora (https://www.quora.com/Whats-the-intuition-behind-the-equation-1+2+3+-cdots-tfrac-1-12/answers/3900029 https://www.quora.com/Whats-the-intuition-behind-the-equatio...), because I think it captures well everything I'd like to say about this at every level of the discussion: The sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this: First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and so, in some sense, X = 1/2. Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and so, in some sense, Y = X/2 = 1/4. Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and so, in some sense, Z = -Y/3 = -1/12. In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't. That's it. It's that simple. Everything else I'm going to say is just to comfort those who are uncomfortable with the game we've just played. Note that I've said "in some sense" several times in the above argument. That's because, while we all know how to add and subtract a finite collection of numbers in the ordinary way, when it comes to adding and subtracting an infinite series of numbers, there are many different ways of interpreting what this should mean. Just knowing how to add finitely many numbers doesn't automatically tell us what it means to add a whole infinite series of them. And when it comes to summation of infinite series, it turns out there's not just one nice notion of "summation"; there are many different ones, which are nice for different purposes. One such notion is "Keep adding things up, one by one, starting from the front, and see if the results get closer and closer to some particular value; if so, that value is the sum". On that account of what summation means, you clearly won't get any finite answer for 1 + 2 + 3 + 4 + ...; since the terms never get any smaller, the partial sums will never settle down to a finite value (and certainly not a negative one like -1/12!). They instead, in a natural sense, should be understood as summing to positive infinity. And there's nothing wrong with this! You are not wrong to feel that 1 + 2 + 3 + 4 + ... is positive and infinite, and math does not deny this; there absolutely is an account of summation corresponding to this intuition. It's just not the only account of summation worth thinking about. We could instead consider other notions of "summation", including ones designed precisely so that arguments like the one we made at the beginning (which are very natural arguments to make!) counted as legitimate ways to reason about such "summation". And then, by definition, we will have that 1 + 2 + 3 + 4 + ... = -1/12, on such accounts of "summation". (In doing so, we will lose certain familiar properties such as "A sum of positive terms is always positive". But this is how generalizations work; generalizations very often lose familiar properties. Even the textbook, limit-based account of infinite summation loses familiar properties like "The order of summation doesn't matter". Even finitary summation of integers loses the familiar property "If a sum is zero, so are all the summands" from basic counting. But there is a web of resemblances to more familiar kinds of summation which can justify, in certain moods, thinking of each of these generalizations as a form of summation itself.) If you insist that "Keep adding things up and see if the results get closer and closer to some particular value" is the only account of summation you're interested in, you'll object to the argument we gave at the beginning, saying "You're not allowed to do that kind of shifting over and adding to itself reasoning all willy-nilly; look at what nonsense it produces!". But it can be made sense of, and is even fruitful to make sense of, in certain contexts in mathematics, and there is no need to blind ourselves to this insight. Again, that's it. It's that simple. Everything else I'm going to say is just to comfort those who are still uncomfortable. For those who want a more systematic, formal account of series summation of a sort which validates the above manipulation, read on: [Comment too long, will be continued in reply]

lisper 10y ago

http://blog.rongarret.info/2014/01/no-sum-of-all-positive-integers-is-not.html http://blog.rongarret.info/2014/01/no-sum-of-all-positive-in...

nkurz 10y ago
Encountering it now, I think your followup article "Does it matter if the sum of all integers is -1/12?" makes an excellent case for why it does in fact matter a lot: The fact that the unsound reasoning in this particular case led to a conclusion that superficially resembles a conclusion that can also be arrived at by sound reasoning just makes it that much worse. It encourages people to think: because this mode of reasoning led to a "correct" conclusion in that case, then it will probably lead to correct conclusions in other cases. If the problem were confined to mathematics I might not make such a big deal out of it, but it's not. The problem of people uncritically accepting conclusions drawn by unsound methods of reasoning pervades our society and causes real damage. http://blog.rongarret.info/2014/01/does-it-matter-if-sum-of-all-integers.html http://blog.rongarret.info/2014/01/does-it-matter-if-sum-of-...

TheRealPomax 10y ago

tl;dr: there is no "real" or "one" kind of maths, you'll have to first pick which field of mathematics you're working with. Arithmetics? Then no: 1+2+3+... is a divergent sum. Using calculus, specifically analytic continuation on the an Euler series? Then yeah, you can apply the rules in such a way that you get this answer and it's a mighty useful identity that can be exploited in complicated proofs.

powertower 10y ago

Is it saying that the complex version of the series (1+0i)+(2+0i)+(3+0i)... = -1/12?

blueprint 10y ago

No, it doesn't.

numlocked 10y ago

Reading a couple of the linked posts, I thought this was the best take on it from mathematician Jordan Ellenberg (it certainly made sense to my comp-sci brain): It's not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + .... = -1/12. When we ask “what is the value of the infinite sum,” we've made a mistake before we even answer! Infinite sums don't have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video. A different protocol assigns it the value -1/12. Neither answer is more correct than the other.

rrmm 10y ago

Terry Tao shows how to get from side of equality to the other: https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/ https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri... It goes into detail and shows the derivation and internal consistency of the method.

pfortuny 10y ago

It boils down to what you mean by "equal."

ojknkjnsdf 10y ago

I don't understand why people keep playing with infinite series like this... it's not sound math and doesn't lead to anything useful. Just a cheap way for worthless mathematicians to feel clever.

owenversteeg 10y ago

Glad SciAm is helping to spread correct info. I think that basically, the problem with many math videos in this style is that they tend to make uncommon, if interesting, assumptions without explaining what they are or why they make the assumption in order to get clicks. This is my comment from the last time this video was mentioned: > "The sum of the series 1+2+3+4+5+6... = -1/12" is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series. Even mathematicians working with Cesàro sums surround such statements with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." [0] Precisely none of the times I've heard the "1+2+3+4...=-1/12" bullshit has the person stating it prefaced their statement with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." > If you say that "1+2+3+4...=-1/12" without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that "this 95 cent pack of gum is free" because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum. > The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers. > [0] Madras, Neal. "A Note on Diffusion State Distance." arXiv preprint arXiv:1502.07315 (2015). My favorite post on the subject still has to be this: http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-bad-astronomer/ http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-...

serge2k 10y ago

Knew it was that numberphile video before I clicked.

chris_wot 10y ago

So when you have an infinite series of the form 1-1+1-1+1-1+... then because there is no single point of convergence and so you average the two values 0 and 1 to give 0.5. Why does an average of the two values that you can converge on help? Also, what about something like: sin(π/2) + sin(3π/2) - sin(5π/2) + sin(7π/2) - ... what would this be?

tamana 10y ago
The average is simply a summary description of the behaviorat infinity. Averages are a standard way to summarize complex phenomena. More precisely, the sume is 0.5+/-0.5, which is greater than 0 and less than 1 and ambiguous in between.

dahart 10y ago

Good time to remember 0^0=1. Or some people (like Knuth) like to assign that definition because it's convenient, and it makes everything work out in certain situations. I don't think it's bad to say that 0^0=1 when you hear the whole story, which is that there are multiple right answers given how we define exponents and operations on 0. But it's misleading to say 0^0=1 and stop there. Similarly, after reading about this today, it's making more sense to me that we can, if we choose, define 1+2+3+...=-1/12, and it makes sense in some contexts. It's just that this isn't the only answer and it isn't the whole story. I liked the article, and ended up reading a bunch of Evelyn's column earlier today. I see some people complaining about the pictures... I thought they were funny and relevant, my only nit pick is I can barely read the light blue text on the wink gif, the colors are so hard for me to look at.

mcv 10y ago

As it happens, I have a proof that 1 == 2. Of course my proof involves a step that involves division by zero, but if it's legal to define undefined results, then why can't I do that if it helps my proof? Because as far as I can tell (with my admittedly limited understanding of mathematics), that's basically what's going on here: they define the result of 1-1+1-1+1-1... to be 1/2, which it can of course never be. The result is never 1/2; it's either 1 or 0. Taken to infinity, the only reasonable definition for that sum is undefined. If I can say that 1/2 is fine too, then I should also be able to attach my own definition to 1/0. Also, if string theory really relies on such questionable mathematical steps, then that would make me question string theory even more. As far as I understand, string theory makes no testable predictions, which suggests to me that no results based on this questionable mathematical trick have been experimentally verified. If there is some real, experimentally verified physics that relies on the sum of all natural numbers to be -1/12, then I'd love to be corrected (though I doubt I'll understand it).