Does 1+2+3+4+ . . . =-1/12?
(Spoiler alert: Yes it does . . . probably . . . um, maybe . . .I mean, it has to.)
An interwebs firestorm has been raging recently about a Numberphile video that makes the astounding claim that if you add up all the positive whole numbers from one to infinity, the result will be -1/12. To write it out more concisely
1+2+3+4+ . . . = -1/12 , (where the three dots indicate all the rest of the positive numbers up to infinity)
If you haven't seen the video, take a look - it's short.
Fascinating, isn't it?
Renowned science writer and astronomer Phil Plait (Bad Astronomy) blogged about the video recently, calling it "simply the most astonishing math you'll ever see." The post led to a Twitter and comment storm, fueled both by people bowled over by the calculation and a much larger number of people convinced it was nothing short of mathematical fraud.The passionate response he got to his post led Plait to write a follow up piece, partly in self defense, and partly as penance for his various mathematical sins as pointed out by his readers.
Clearly, only a fool would consider defending this absurd calculation after the reception Plait got.
So here I go . . .
First, About the Math
I'm not going to follow Plait's example of trying to explain the math that goes into the calculation. But I will point out that many of the problems that commenters and Twitterers latched onto are irrelevant if you look at the more elaborate discussion in the Numberphile's extra footage. In the much longer second video, they come to the same reviled result as in the first video, except that they use an approach first written down by Leonhard Euler.
If you dislike the initial video, you really should watch this one to see if it sways you at all.
That's much better, isn't it? I'm sure it's not perfect, but the flaws are beyond my mathematical abilities to recognize.
In any case I'm willing to believe 1+2+3+4+ . . . = -1/12 is a mathematically legitimate thing to write down for the following three reasons.
1. Euler, who was one of the greatest mathematicians of all time, proved the equation for real numbers.
2. Another great mathematician, Bernhard Riemann, generalized Euler's approach to include complex numbers, and came up with the same equation.
3. My favorite mathematician, the self-taught genius Srinivasa Ramanujan, rediscovered the equation and stood by it, even though he realized that he might be thought be mad for making the claim, writing in a letter to mathematician G.H. Hardy, "I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."
So, counting the Numberphiles' somewhat dubious derivation, there are at least four ways to prove that the sum of all the positive integers equals -1/12. And as far as I know, there's no way to prove that it doesn't equal -1/12.*
If you don't believe any of these people, then there's nothing I can do, mathematically speaking, to change your mind. I mean these guys are among the greatest. What could I add that would improve on their proofs?
But, Obviously 1+2+3+4+ . . . = -1/12 Doesn't Really Mean Anything - Right?
Of the mathematicians and physicists I've talked to about it, several of them are willing to accept that it's possible to derive the equation, but insist that it's meaningless. They tell me, if I understand them correctly, that it's some sort of numeric fluke that can't possibly have any consequences in the real world. There's just no way to add positive numbers together get a negative result in reality, especially when the numbers you're adding are getting larger and larger. In effect, it's nothing more than an artifact that results from a method that makes sense when applied to complex variables or other series, but not for the sum of positive integers. To think otherwise would be nuts, right?
The problem is, they're wrong (or so a number of physicists have told me). The equation 1+2+3+4+ . . . = -1/12 is vital for describing the real world.
As the Numberphile people point out, the dreaded equation pops up in many places in physics. They specifically note it's appearance in a string theory textbook (see page 22 in this Google book). But that's only one example and, depending on how you feel about string theory, among the least convincing ones. What's much more compelling is the fact that this sort of equation is integral to Quantum Electrodynamics (QED).
QED is the theory that explains the interaction between charged particles like electrons and protons. Along with neutrons, electrons and protons make up atoms, which in turn make up molecules and everything built of them. In other words, QED essentially describes much of the physical world we live in. And it does it extremely well. QED calculations for the spin of the electron have been confirmed to better than one part in ten trillion - making QED just about the most precise and successful theory of all time.
If QED is correct (and it appears to be the most correct theory yet developed, if experimental confirmation is a reasonable way to judge correctness), then I would argue that the things that go into QED calculations must be just as correct. Doing QED calculations requires using 1+2+3+4+ . . . = -1/12, so the equation is at least as correct as QED theory itself.
In fact, the Wikipedia page on a QED phenomenon known as the Casimir Effect shows a derivation of the effect that includes an even more audacious equation involving the sum of the cubes of the natural numbers up to infinity. Specifically, calculating the effect involves using the equation
1^3+2^3+3^3 +4^3+ . . . = 1/120, (where the notation 2^3 means 2x2x2)
(In the Wikipedia article, they have an equation that looks like this , but the stuff on the left hand side is just another way of writing 1^3+2^3+3^3 +4^3+ . . .)
The number on the right is positive this time, but it's ten times smaller than 1/12, even though each of the terms in the sum is much bigger than the corresponding terms in the equation 1 + 2 + 3 + 4 + . . . = −1/12 (except for the first term, of course, since 1^3 = 1). Both equations come from the same sort of derivation, so it's not surprising that they are both seemingly incredible and ridiculous. But if you believe in QED and the Casimir Effect, how can you not believe the pieces that go into them?
Maybe It's Just a Trick
One response I've gotten after querying my more mathematically savvy friends is that the equations are nifty tricks, and nothing more, to get rid of infinities in QED and produce the correct finite answers. I guess that's possible, but you would have to be one heck of a mathamagician to come up with a trick resulting in accuracy of a part in ten trillion.
It's even more impressive when you consider that the QED predictions came before the experiments that measured things like the electron spin to fourteen decimal places. It's one thing to design a trick to rationalize a number you already know. It's a whole other matter to come up with a trick that gives you the answers in advance of the experiment. In that case, it's not a trick, it's simply a very good theory.
Maybe It's Not Necessary, Just Handy
One final possibility that I can think of is that the equations are not really necessary for doing QED calculations, and that instead there is a correct and intelligible approach that gives answers without using nonsense like 1 + 2 + 3 + 4 + . . . = −1/12 or 1^3+2^3+3^3 +4^3+ . . . = 1/120.
I can't imagine why physicists would rather rely on trickery than doing things correctly, so I tend to dismiss the idea that some sort of mathematical conspiracy is behind it all. If it turns out that it's possible to have physical theories that describe the real world as well as QED does without relying these equations, then we might as well use those theories and forget the whole controversy.
So What's Really Wrong?
If you accept that Euler, Riemann, and Ramanujan did things properly when they found 1 + 2 + 3 + 4 + . . . = −1/12, and if you accept that it and related equations are necessary to describe the real world, then how can you not accept that the equation is true? And yet, many people still claim that there's something wrong. It doesn't make sense. It's so counter intuitive that the phrase "counter intuitive" seems far too weak a description. It's an alien, freakish, mind f----.
But that's OK. Some things are true without being conceivable. This is just the most recent example I can think of. Pythagoras and and his followers apparently committed human sacrifice because they couldn't handle the idea of irrational numbers. For centuries, ancient mathematicians struggled with unsolvable problems because they didn't know that pi is a transcendental number. And today, there are still things about quantum mechanics that defy intuitive understanding - the whole point of Schrodinger's Cat is to illustrate the absurdity of quantum superposition. But just because people didn't intuitively grasp those things, it didn't change the fact the the square root of 2 is irrational, that pi's transcendental nature means it's impossible to square the circle, and that particles can become quantum mechanically entangled just like Schrodinger's Cat.
Yes, there's a problem with 1 + 2 + 3 + 4 + . . . = −1/12. But I suspect the problem is with us and our failure to understand infinity. Why shouldn't an infinite sum of numbers going to infinity add up to a finite (and negative!) number? I don't really know what infinity means anyway, so I can't think of any way to object to a statement that includes not one but TWO infinities in it.
You might as well ask me why a bandersnatch of numbers going to bandersnatch add up to -1/12. But if you're able to mathematically sum a bandersnatch of bandersnatches, and then use that sum to describe the real world and predict the outcomes of real world experiments
So, Does 1 + 2 + 3 + 4 + . . . = −1/12 or not?
You bet your bandersnatch it does! (I think, anyway)
Thanks for bringing some clarity into this horrible mess!
Monday, May 2, 2016 at 3:37 AM
Wednesday, November 25, 2015 at 4:33 PM
Problems of summation divergent series is solved! https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/
Saturday, February 14, 2015 at 4:10 PM
Thanks, Dr. Jantzen!
Sunday, April 13, 2014 at 5:14 PM
Bernd Jantzen said...
I totally agree. And I refer to my previous comments above, where I have made similar statements.
Please note that the discussion about this topic actually went on in two new blog posts by the same author:
[titled "Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not! (I think)"]
[titled "If Not -1/12, What Does 1+2+3+4+... Equal? We Give You the Answer "]
You may see from these blog posts (and their comments) that the author changed his mind several times about this topic. Unfortunately, he did not link his follow-up posts from this first one.
Sunday, April 13, 2014 at 3:43 PM
PS: please read the 3rd paragraph at books.google.de/books?id=n8Mmbjtco78C&pg=PA73
Sunday, April 13, 2014 at 1:10 PM
I'm afraid you're greatly confused.
1. Just because Euler derived something doesn't make it true. Euler was known for "breaking the rules" all too often - that was surely part of his genial nature - but it doesn't mean that all of his results and proofs are correct, in fact his handling of infinite diverging sums is not strictly mathematically correct.
Also, n appeal to authority is a fallacy.
2. It's true that 1+2+3... results in the so-called Ramanujan sum of -1/12. But a Ramanujan sum is not at all the same as *a* sum in a traditional sense. Look it up.
3. You're engaging in trickery by juggling different definition of sums.
If we're talking about traditional sums, then no: divergent infinite series cannot be summed up and -1/12 is certainly not a sum of any such series.
Now, we can extend the definition of "summation" to cover the assignment of finite numbers to infinite divergent series. But in such a case it *must* be clear that we're talking about something entirely different than traditional summation. Because 1+2+3...=-1/12 is only mindblowing when we're thinking of traditional sums (in that case it also happens to be incorrect).
4. Now, certainly it must be admitted that the persistence with which -1/12 (and other numbers) pops up when all those different kinds of operations are performed (legally or not), as well as the fact that it is applicable to the real world does mean that it's not merely a fluke and that there is indeed a deep connection between these things on a fundamental level.
But this doesn't justify the bad math, sorry.
Sunday, April 13, 2014 at 10:19 AM
It is a trick for a simple reason - zeta function is only defined for s>1, whereas to perform the trick of "summing up" the natural numbers if set s=-1. This may be a useful trick, but it's still a trick.
Sunday, April 13, 2014 at 10:07 AM
Imre Fabian said...
For me, not a mathematician, this is a perfect example for that you can prove anything with infinity mathematics.
In my view, infinity does not exist (in the real world), nor does 0.
Infinity = anything / 0
0 = anything / infinity.
Physical argument: the smallest thing (measure) is the planck dimension (planck lenght, planck time etc.) so there cannot be an infinite number of since the beginning of time (the big bang).
Wednesday, March 5, 2014 at 4:57 AM
Definition error. The sum of n to infinity diverges, goes really big. The "regularization" converges to -1/12. The technique is proven, duh, it would be odd if they were using the regularized sum without a proof. Its like saying 2+2=10. Very misleading unless you realize that the equation has been manipulated by being turned into 2, base ten, +2, base ten, equals 10 base 2. Although the regularization is legitimate, you can't say the sum adds to -1/12. You have to say the sum can be regularized through the Riemann zeta function and through a variety of other methods to equal -1/12. Although for all intents and purposes no ones checking the addend up to infinity to make sure that it equals what we say it does so feel free to say it equals -1/12, but make sure add in that regularized part if your talking with any self respecting mathematician or physicist.
Tuesday, March 4, 2014 at 10:28 AM
Wondrful exchange guys. Thanks for the knowledge you've been sharing with us, and for the effort that takes.
Monday, February 3, 2014 at 5:22 PM
Buzz Skyline said...
Bernd, my boss happened to stop by and say basically the same thing (he's a physicist who has a lot of experience with quantum field theory) and illustrate it using the Casimir Effect. I have to admit, I'm coming around to see what you're saying. It's easier to understand with a white board handy. In any case, I'll have to think about it for a while before I have anything much to say (or ask).
Along the way, I happened to find a relatively simple way to evaluate zeta functions that I'd never seen before, for odd, negative values of n. I guess it doesn't matter, really, but it happens to make calculating some Bernoulli numbers (B_n, for n even) relatively quick and easy. I have no idea whether that's useful, interesting, or new. It entertained me though.
Tuesday, January 28, 2014 at 5:36 PM
Bernd Jantzen said...
Buzz, let me show you one way to better understand, at least in a qualitative way, how such divergent sums of positive numbers can be related to finite and sometimes even negative numbers.
Look at the physics case of the Casimir effect (http://en.wikipedia.org/wiki/Casimir_force#Derivation_of_Casimir_effect_assuming_zeta-regularization): Even when properly regularized by using the parameter s, how can a sum (and integral) of energy contributions, which are each individually positive, end up in a negative total Casimir energy at the end?
In order to explain this, let me first approximate the series representation of the zeta function by an integral which is easier to handle. Using the Euler-Maclaurin-Formula (http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula) in its 0th-order approximation, we can approximate
1^(-s) + 2^(-s) + 3^(-s) + 4^(-s) + ...
= 1/2 + integral from 1 to infinity of dx * x^(-s) + small remainder,
which is valid, as the series on the left-hand side, for s>1.
Now, as the integrand x^(-s) is positive all over the integration domain from 1 to infinity, one would expect this integral to be positive as well. And in the convergent case s>1, this is as expected. Using the primitive of the integrand, -1/(s-1) * x^(1-s), one gets:
integral = 1/(s-1) * ( 1 - infinity^(1-s) )
For s>1, the contribution from the upper boundary is infinity raised to a negative power, i.e. zero, and we and up with:
integral = 1/(s-1) > 0 for s>1.
But what happens in the divergent case, when s<1? Then the contribution from the upper boundary is divergent: infinity raised to a positive power. So the integral should be positive and infinite, as one would also expect from summing all natural numbers (the above series with s = -1) or all cubes of the natural numbers (series with s = -3). However, we are using a kind of regularization here which is based on analytic regularization. As the result 1/(s-1) for our integral is valid all over the complex half-plane where the real part of s is larger than 1, we analytically continue this result to all values of s (except s=1). Practically this means that the contribution from the upper boundary of the integral, 1/(s-1) * infinity^(1-s), is omitted although it diverges.
So our analytically regularized integral is still given by minus the primitive of the integrand at the lower boundery, which is now a negative contribution:
integral = 1/(s-1) = -1/(1-s) < 0 for s<1.
Summarizing, the negative and finite result is obtained by dropping the positive and infinite contribution from the upper boundary of the integral. This prescription to drop such an infinite contribution arises from the analytic continuation in the parameter s.
In physics (example of the Casimir effect) such a prescription makes sense because constant energy shifts (even if they are infinite) are irrelevant for the Casimir force.
Tuesday, January 28, 2014 at 4:17 PM
Bernd Jantzen said...
No conspiracy. But some people tend to oversimplify things when they present them to non-experts. Then they are presenting them in a way which is actually not correct, but behind the wrong picture is a true meaning which though would have been more complicated to explain in a correct way. That's what the Numberphile people have done in my opinion. I always try to avoid this, but sometimes it is hard.
Tuesday, January 28, 2014 at 3:10 PM
Buzz Skyline said...
Well, finite, and *correct* I assume, is what your after. I'm sure there are many ways to deal with infinities. I imagine the ways that give you precise answers are fewer.
So your proposal is that the answer is more along the lines of a conspiracy of physicists who are telling us this stuff to impress. It's possible, I suppose, but hard to believe.
Tuesday, January 28, 2014 at 2:46 PM
Bernd Jantzen said...
I have also done a lot of QED calculations, and I have never needed to sum divergent series without a proper regularization. Probably the physicists you are speaking about just wrote down such "identities" with divergent series as a very simple picture of what they did, omitting all notion of regularization for sake of simplicity? Maybe they even wanted to impress you by showing you what kind of "crazy things" they are doing? (Things which aren't crazy at all, once you properly define your regularization procedure.)
As I told you before, physicists tend to be pragmatic. So they might even work with "identities" like 1+2+3+4+... = -1/12 because they know that, in principle, it can also be done properly and mathematically. Anyway, theories in physics do not depend on such "identities" being true, they just depend on being regularizable and renormalizable such that their predictions are finite in the end.
Tuesday, January 28, 2014 at 2:32 PM
Buzz Skyline said...
By saying that the theories rely on the equation, I'm saying that, as I understand it, the theories would make different predictions if the "sum of the infinite series" (as Ramanujan says) equaled something other than -1/12. If you can do without them, they why to physicists who do QED calculations tell me they need them?
Tuesday, January 28, 2014 at 2:04 PM
This is proof of my Theory of Dark Numbers. This theory postulates that Dark Numbers exist (but yet unobserved) to tame the large expansion of currently observable whole numbers 1+2+3+4+... so that the infinite series is equal to -1/12 instead of infinity. Tesla alluded to using Dark Numbers when he built his transport ray machine. According to my theory, the number line contains 4.9% ordinary numbers, 26.8% dark Robinson hyper-reals and 68.3% dark numbers.
Tuesday, January 28, 2014 at 1:47 PM
Bernd Jantzen said...
Thanks, Davide, for your comments. I really like Evelyn Lamb's statement which you cited: You may assign -1/12 to the divergent series, but it is misleading to call this the sum of the series.
And, Buzz (are you the same as "Buzz Skyline" or a different person?), it is sad when the only remaining argument for a mathematical equation like 1+2+3+4+… = -1/12, is the correctness of theories in physics (like QED). As a physicist I tell you, there is no meaningful theory in physics which relies on "identities" like 1+2+3+4+… = -1/12. Whenever infinities and divergences are encountered in physics, they are properly regularized (and eventually renormalized, especially in QED). And that is also what Euler, Ramanujan & co. did: They attached a meaning to divergent series by choosing a certain regularization.
Tuesday, January 28, 2014 at 1:42 PM
Keep in mind, I'm saying
If QED is true
and it relies on 1+2+3+4…= -1/12
then 1+2+3+4…= -1/12 must be true.
I have no reason to want 1+2+3+4…= -1/12 to be true. I am happy with it being infinite. But then what's going on with QED? I don't accept that luck can lead to such precise predictions.
Tuesday, January 28, 2014 at 1:37 PM
Ramanujan (and Euler) wrote
Ramanujan's exact words are ". . . the sum of an infinite number of terms of the series
1+2+3+4…= -1/12 . . ."
I am only repeating what he said.
See for yourself here http://books.google.com/books?id=Of5G0r6DQiEC&pg=PA53&dq=gratified#v=onepage&q&f=false
Are you saying he is wrong or that "=" means something other than equals?
Tuesday, January 28, 2014 at 1:29 PM
Davide Castelvecchi said...
I am sorry Physics Buzz, but you are talking nonsense. The statement "include all the terms to infinity" is devoid of meaning. And as Evelyn Lamb explained, you are misrepresenting what Ramanujan & co. did. I don't think there is anything more I can say at this point other than suggesting that you read the sentence "There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series" again and again. Perhaps it would have been advisable to do so before blogging about this on a website that's supposed to be educational.
Tuesday, January 28, 2014 at 1:09 PM
I think the point is that no one is saying they converge. It's the same as for the series 1-1+1-1+1-1+...
It doesn't converge, but if you include all the terms to infinity (not the limit as you go to infinity, but the value when infinite terms are included), it equals 1/2.
If you approach it by thinking in terms of convergence, as you point out, there's nothing you can do with zeta(-1), zeta (-3), or 1-1+1-1+1 . . . And yet, using the values that Euler, Reimann, and Ramanujan calculated for those things give you meaningful answers to real, testable theories.
I can't see a way to rectify those two things - either the equations and the theories are true, in some sense anyway, or they're each wrong in ways that precisely (to 12 or more decimal places) correct the errors the other one produces. The second option seems much harder to believe than the first.
Tuesday, January 28, 2014 at 12:49 PM
Davide Castelvecchi said...
Analytical continuation is not a way to make divergent series converge. It just isn't.
Analytical functions are by definition those that admit a power
expansion at every point. That means that *locally* you can write them
as the sum of their Taylor expansion. (And no, not all functions are
analytic, even if they admit infinitely many derivatives: sometimes
you can write the Taylor expansion but that does not equal the
function you started with. But I digress.)
Being analytic does not mean that the function is the same as its
Taylor expansion at a point. The Taylor expansion, in particular, may
start diverging at points where the function is perfectly well
defined. Taylor expansions have a well-defined convergence radius in
the complex plane, and that radius may be infinite (as it is for the
case of e^z, say) but it is often finite.
So, if you start with a given Taylor series, there can be a function
that the Taylor series was the expansion of (at a given point), and
that function may be well-defined on a larger domain than the Taylor
series was. But that doesn't mean that you are making the original
Taylor series converge. It still diverges: it's just that it only
represents the function _within_ its radius of convergence.
If you want to play with an example, take f(z)=1/(z-1). f(z) is
perfectly well defined at any complex number z, as long as it's not 1.
Now take its Taylor expansion at z=0, meaning
f(0) + f'(0)z + 1/2f"(0)z^2 + 1/6 f'''(0)z^3 + ...
Its radius of convergence is 1, which means that the Taylor series
*diverges* for any |z|>1. The fact that f(z) is well defined for *all*
|z|>1 doesn't magically make the series converge there. It's the
*function* that is well defined there, not the Taylor series. If you
had started with the Taylor series, and then realized that it was the
Taylor series of 1/(z-1), then you could say that f(z) was an
analytical continuation of the Taylor series. But again, doing that
doesn't make the series any more convergent than it was before. This
is stuff that has been figured out once and for all 200 years ago.
About 1+2+3... Evelyn Lamb has already said everything that there is to say: "There is a meaningful way to associate the number -1/12 to the series 1+2+3+4…, but in my opinion, it is misleading to call it the sum of the series."
End of story.
Tuesday, January 28, 2014 at 12:10 PM
The twelfth root of 2 is how we divide the musical scale, each note is 1/12th of the octave, wonder if there is a connection. Its often assumed that all music strictly cultural but the evidence does not support it, for example tone/tempo of voice carries emotional information you can understand even if you don't know the language and have no other clues, play a film in a foreign language while blindfolded and you can still tell what the characters are feeling by tone/tempo of voice alone. The variances in tempo and pitch that convey this information are indistinguishable from musical variances, we speak with music only some of which carries abstract associative meaning but most of which carries universally understood emotional state information. This suggests that the emotional content of music is conveyed by adherence to universal rules that are just as solid as the rules of physics with the 12th root of two the simplest generator of the frequencies for both tempo and tone.
Sunday, January 26, 2014 at 4:04 PM