Does 1+2+3+… Really Equal -1/12?

Artistic interpretation of infinity.

Infinity is not a number, it is a place. It lies just to the left of the Undiscovered Country. Nobody knows what it is like there, because nobody has seen it. All we can ever do is approach it and say what it isn’t.

Funny thing about Infinity is that they way you get there matters. If you were to head out toward it on a straight line taking one step at a time, and I were to follow taking two steps forward and one step back, we would not necessarily arrive at the same neighborhood. That would depend on how fast we were walking.

None of us would ever get there anyway, not while we’re stuck in time. Infinity isn’t in time. It sure isn’t in our intuitions. Which explains the latest flap.

There’s a video going round (below) which shows that we can assign the value of -1/12 to the infinite sum 1 + 2 + 3 + 4 + … Those dots don’t end until Infinity, always a key something weird is going on. I’ll assume you’ve watched the video.

The path the gentlemen in the video took was not a straight line, which is how they arrived at -1/12 (See also these videos, particularly the third, for more on the path.). But if you were to go straight—the simple sum, i.e. “the limit”—you’d end up right where intuition suggests, at a whopping big number, unimaginably big. Why the difference? Mathematical truths, like all truths, are conditional on the premises assumed and those premises include the paths.

Well, bizarre, right? Yet why shouldn’t Infinity be bizarre? Why a “flap”? Turns out P.Z. Myers, self-proclaimed “rationalist”, saw the video but could not understand it, and since he could not understand it he concluded it therefore could not be true (a line of argument which he frequently employs). So he put up the post “The sum of all natural numbers is not -1/12.” “I saw [the video] and said to myself that it’s obviously wrong”. All the proof he needed.

Switch on the Wayback Machine and slide back to 1990 when Marilyn Vos Savant explained it’s better to switch doors in the Monty Hall Probability Problem (see this for an explanation), a highly non-intuitive result. Thousands of genuine PhD mathematicians reacted like Myers and said “No way this result can be true because I don’t understand it!” Which proves probability is notoriously difficult—and that academic certification is far from a guarantee of infallibility.

Myers also enlisted the support of his own PhD mathematician, Mark Chu-Carroll, who explained carefully but failed to appreciate the difference between limits (a technique which the proofs in the videos do not use) and Cesaro sums (which they do). Chu-Carroll also forgot the -1/12 result was first given to us by Leonhard Euler, perhaps the fattest mathematical brain ever.

The gentleman in video number three (above) was also careful about explaining how Grandi’s series—1 -1 + 1 – 1 + 1 – 1 + …—could, out at Infinity, be +1 or 0, on or off, spin up or down, and that if we consider the series in a sort of probabilistic sense, it can be given the value 1/2. Sounds a bit like quantum mechanics, no? It is this assignment that makes the magic happen and is why the infinite sum can be -1/12. Anyway, Myers didn’t bother to investigate any of this before going off. Which is what makes him a rationalist.

Enter our friend Lubos Motl, arch defender and knight-errant of string theory (he will not see her virtue impugned), who has the habit of writing even simple numbers in Latex, who took Myers to task in the post “Sum of integers and oversold common sense.” Motl also takes pains at showing there is more than one way to sum a series.

Phil “Bad Astronomer” Plait joined in the fray at Slate, which might not have been the wisest move. If there is any place on the Internet where the people already know all they need to know, this is it. And they already knew the infinite sum could not be -1/12. Poor Phil had to issue multiple corrections for being too glib with his language.

Most civilians and rationalists don’t know there is a (let us call it) tension between the kind of math physicists do and the types mathematicians themselves use. Physicists are a little bolder, even playful. Mathematicians are more staid. Full disclosure: I learned my math from physicists. And you may be surprised to learn that there are even warring camps inside each field about the very fundamentals of mathematics. Too much for us here today, except to note that this latest incident is part of the never-ending war of ideas.

Oh, read the history of the Heaviside function for a fun example (I don’t have a link).

Update: now with even more Infinity!

Mathematics isn’t the only place where we meet Infinity. Take the idea of Omniscience, which is knowing everything, and everything includes Infinity. Suppose one knew 10100 facts, a googol of facts. That’s a lot of knowledge, but still far from infinite knowledge. How about knowing a 10^{10^{100}}, a googolplex, of facts? Some estimate there are only about 1082 particles in the universe. If you knew a googolplex of facts you’d be able to name each particle. You’d know where everything is located, including my Kindle which went missing a week ago, so I’d appreciate an email.

A googolplex of facts is already unimaginable, but 10^{10^{100}} is just as far from Infinity as 10100 is. We still have a long way to go before reaching Omniscience. And what’s it like when we get there? Boggles the mind to think about, just like, in a much smaller way, the sum above does. The lesson is intuition, particularly knee-jerk reaction, only takes you so far. And usually to the wrong place.

Update The HTML superscripts weren’t rendering properly, so I switched to Latex for the googolplex.


Thanks to our friend Luis Dias for alerting us to this topic.

Categories: Philosophy

50 replies »

  1. This is the Obi-Wan Kenobi move, right? “What I told you was true … from a certain point of view.”

    “On a clear day you can see a point just behind your head” doesn’t have the same ring to it somehow.

  2. Not being a mathematician (or even a really smart guy)I could not really understand how the series sum would be -1/12. I also have a problem visualizing a graphical proof. A lot of infinite sum series can be shown graphically to converge at a certain point, even if you don’t know math well enough to prove it. In other words, the graphical trend supports the proof. I do not see any graphical support for the infinite sum of integers being -1/12. To me this would imply that there is some abrupt change that occurs somewhere between N(really, really big number) and N(infinity). Like some sort of inflection point. Which would imply that we could find said inflection point algebraically.

  3. But consider this when speaking of large numbers, it is estimated that there are 400 billion stars in the Milky Way Galaxy and our national debt is ~$17 trillion therefore, the number of dollars of national debt is 42,500 times the number of stars in our galaxy. Thank you, I’m having a drink now.

  4. This is just another example where the manmade rules of a natural system do not adequately explain the system.

  5. The Grandi Series going to 1/2 is a place where I throw my magic bucket. Saying that the answer is a 1/2 makes my “intuition” hiccup. For me, 0, 1, and 1/2 are all partial answers. The crooked path taken is when they play the 1+S game. I admire such games. I repeat them to my kids hoping it will help them when they hit these course and have to “prove” stuff. You can always add 0. You can always multiply by 1. These are games that are fundamentally good. They can also lead you astray wandering down paths imaginary. Imaginary paths have been found to be joyous though. The imaginary somehow exactly matches the reality of electronic circuits. Except I always saw the imaginary there as just being conveniently 90 degrees from natural. In the video one of the mathematicians says “-1/12 is used to find the 26 dimensions of string theory”. String theory has helped us understand stuff, I think. Not sure how much. Lots of folks are getting paid to look into that. I like that they are getting paid to look into that. I am not smart enough to join their club. I had to learn to understand 32.2 and when to use it and when not to use it. Practically though I almost never have to use it. Right now -1/12 is magic to me. It might help explain the mysterious nature of the electron. Maybe the physicist who suggested that there is really only 1 electron in the universe is right and -1/12 is the link to show it does work that way.

    I will not be throwing -1/12 haphazardly into my analysis though. I never face infinite series equations practically. There are always inflows, outflows and accumulation. I am happy to have -1/12 in my head to answer a trivia question, but 0, 1/2, 1 are the states ensorceled in its creation and 1/2 is only a pseudo state. If eventually -1/12 makes the matter energy transporter work or possibly the hyperspace drive, I will embrace it blindly. Until then, I admire the brilliance of the folks who derived the equations. I was never able to add 0 without prompting. People who can do it intuitively have my awe.

  6. Strange. Why did one video embed while the other remained as a link?
    Must be a result of that 1-1+1-1… thing.

  7. OK, my mathematics was also learned from physics (and from card games in an ill-spent youth), but I have always understood that for a limiting sum, there must be convergence and for convergence, you need that awful stuff, delta epsilon proofs. I don’t know from Grandi series (whatever those might be), but would appreciate enlightment, but I am convinced that string theory is not physics/science but rather mathematical metaphysics, so the fact that the result sum of -1/12 is used in string theory, is not a convincing argument as far as I’m concerned.

  8. Nobody knows what it is like there, because nobody has seen it. All we can ever do is approach it and say what it isnâ€™t.

    Maybe. I was in an establishment in Bangkok that offered infinite delights. I admit I was rather distracted so they did, in fact, remain countless but if memory serves me right they added up to more than -1/12. (Like maybe that’s where I got my strange brain disease).

  9. DAV,

    Embedding fixed. Thanks.

    Bob K,

    The Grandi series isn’t a limit in the standard delta-epsilon sense. The Grandi series has no limit. But, as the video shows, it can have a probability-like interpretation.

    Also watch the extended video in the link above; it’s about 22 minutes. Show’s Euler’s proof, of course given long before the advent of string theory. It’s the string physicists who are seeking ontological interpretations of the epistemological result. Whether they are right is, as you hint, another question.

    P.S. Don’t let Motl know you’re a doubter.

  10. Bob Kurland ,

    I have to go along with your assessment of string theory. Another case of confusing math with reality. Talk about reification!


    On a side note, Bud Abbott made a rather damning remark about the school system in the first link (and this was way back when):
    Lou asks: Didn’t you go to school, stupid?
    Bud says: Yeah! And I graduated that way, too.

  11. I love how PJ calls in an expert who isnt an expert.

    we need discipline called expertology

  12. Learning maths from a physicist can get you into all sorts of bad habits. Like renormalisation. That said, I learned mine from mathists and think that video’s a delightful explanation.

  13. Saying it’s equal to -1/2 is fine, I’d say, if you view it as a notational convenience for comparing to other divergent series, instead of saying the series is actually equal to that number.

    I would rant about it, but a lot of the maths I do uses generating functions, which do the same sort of thing, and almost no one complains about those not converging.

  14. Bob Kurkland,

    String Theory reminds me of Game Theory. According to game theorists, Game Theory has many real world applications. Push further and ask for an example and the best you get is, “String theory was used by the US during the Cold War.” Push again … and silence.

    So the best I can gather is US policymakers convinced themselves mutual assured destruction was not a rational policy, all based on a two-by-two grid.

    Yet Game Theory holds a special place in academia, as does String Theory.

  15. I’ve watched the second video and looked at Wikipedia (and other websites) on the Riemann zeta function, analytic continuation, the 1+2+3+… sum, and am now in the throes of cognitive dissonance–right and left frontal lobes are in active opposition. I know this isn’t right, am familiar with complex variable theory, so I know what’s going on with analytic continuation, know about renormalization in quantum electrodynamics, but I can’t get over that intuitive feeling that the sum isn’t right…there’s something more I need to understand about pathways for summation.. Thanks a lot, Matt, for disturbing my day! (…last remark in jest)

  16. Let S= 1 + 2 + 3 + 4 + 5 + 6 + â€¦

    (I stop at 6 since the number 6 signifies smooth sailing in Chinese.)

    Now, letâ€™s calculate S – S, in this case,

    1 + 2 + 3 + 4 + 5 + 6 + …
    – 1 + 2 + 3 + 4 + 5 + …

    And we know that S â€“ S = 0. So,

    0 = 1 + 1 + 1 + 1+ 1 + â€¦

    How fun!

    How about asking an expert? Here is the answer from one of my favorite mathematicians.

  17. Wow! I finally understand why economists have physics envy!

    Given the number of trained theoretical physicists now working as Wall St analysts (and maybe White House economic advisors, Social Security trustees, …) we have a bright future as a nation! If we go deeper into debt every year by ever increasing amounts, and do this long enough, then eventually we will be owed money!

  18. The real question is why haven’t I seen this before? I guess that I’ve lived a sheltered life. Lubos Motl has the best article of the bunch: too much name calling in the rest. Ultimately it may not be any weirder than the claim that the logarithm of a negative number is proportional to the square root of minus one with pi thrown in for good measure, also due to Euler.

    “I learned my math from physicists.” The only way to go. I keep having to train my students out of bad habits that they picked up in math courses. I once got the ultimate compliment from a mathematics professor who said that I was wasted in physics and should have gone into math. At least I think that it was a compliment. Any way that was before my brain ossified.

    Note: Mark CC does not appear to be a mathematician.

  19. It’s a little odd that the Cesaro sum is a smoothed series of partial sums. I don’t know how using this method to find the sum S1 = 1-1+1-1+… is justified without a model that calls for such a sum. But this is more information than what is provided in the videos.

    I imagine that this type of analysis is applicable in physics?

  20. Scotian,

    “Note: Mark CC does not appear to be a mathematician.”

    Either this is an application of the No Good Scotsman Fallacy or is there some authoritative source that is justified in bestowing the title, mathematician, that I am not aware of.

  21. Scotian,

    He does not claim he is not a mathematician. Ironically, he writes a blog called “Good Math, Bad Math.” Given my quick perusal of the site, he certainly appears to be acting as one who most would say — I believe — is “skilled or learned in math” — not that the opinion of most is sufficient either.

    Is the title mathematician self-referencial? If Mark CC claimed be was a mathematician, would that have sufficed? Could Euler have claimed he was not a mathematician?

    A lot of titles get thrown around to get around, so to speak, the question at hand.

  22. Jim,

    I was responding to this statement of Briggs’: “Myers also enlisted the support of his own PhD mathematician”. You are reading way too much into it.

  23. It seems to me that there is a bit of bait e and switch going on here. When you say, 1+2+3+4+5â€¦ it implies to me that the series under consideration is the set of all positive integers from 1 on into âˆž, and does not include any negative numbers nor any fractions. When the gentlemen in the video started breaking up the set into subsets via the brackets, and adding in negative numbers he changes the car actor on the set initially under consideration into an entirely different set of integers.

    Then when he started mixing the 3 different sets of (I did study set theory, back in the day) it reminded me of when doing differential equations and had to solve three equations with three unknowns, by substitution parts of one equation into another as you worked to the solution. But here you have sets not equations, and I am not convinced that 1+2+3+4+5â€¦= – 1/12.

    I recall an algebraic proof going around back then that proved 1=2, however, if you substituted numbers for the letters as the proof was worked out you would see that at one time a number had to be divided by zero to reach the results of 1=2. In the part where he talks of Thomsonâ€™s Lamp I was reminded of one of Zeno’s paradoxes, â€œThat which is in locomotion must arrive at the half-way stage before it arrives at the goal.â€ as recounted by Aristotle, Physics VI:9, 239b10. Suppose you want to catch a stationary bus. Before you can get there, you must get halfway there. Before you can get halfway there, you must get a quarter of the way there. Before traveling a quarter, you must travel one-eighth; before an eighth, one-sixteenth; and so on.

    When I Got my BS in Electronics I took a job right out of school with General Electric in New Berlins, WI, working on integrating all the different components from the various vendors into a working Cat Sacnner (as they were called then). I also soon there after enrolled into Marquitee Universiteâ€™s Graduate program in Electrical Engineering. That turned out to be a big mistake as GE started working me 60-70 hours a week to meet the demand for their whole body scanner.

    I bring this up to tell you about a homework assignment I got in one of my math classes, prove that between 1 and 2 there is an infinite number of positive numbers. Now intuitively I knew that this was true, but I never, back then, could figure out how to express it. I had to drop out of that program to meet my obligations to GE and my growing family. Bty there were 10 students in that class and if just one of us understood what the professor was lecturing on he would not go over it.

    However, last night after reading this post I went to bed and could not sleep for thinking of that math problem from so long ago, and this is what I came up with:

    limâ”¬(1â†’2)â¡ã€–(1+1/1)^âˆž ã€—=2,(1+1/2)=1.5,(1+1/4)=1.25.(1+1/8)=1.1125,â€¦

    Do you think I solved it?

    The symbols did not translate.

  24. I simply don’t trust this “they’re infinite sequences so we can shift them to the right and they are the same.” Why not shift it 100 characters to the right? Is it the same answer? Or a Googol of characters to the right.

    “Look Ma! I made a paradox!”
    “Very nice dear. Wash your hands for dinner.”

  25. Lynn,

    Thank you! I’ve been looking for a new theme to avoid these problems.

    Jim & Scotian,

    Good find.

  26. I think I finally understand (to a degree) what’s going on. JH and Rich have a point…when you shift and associate you’re adding and subtracting infinities. As my sainted grandmother would say, “Is that really kosher?”. If you do experimental mathematics, as Gregory Chaitin would advise (“The Unknowable”), you’ll never get those right-hand values by carrying out the given sums. So then it must have to do with the identification of the integer sums with values in the zeta-function. I tested a web-site that did zeta-function values, and used small steps from -0.999 to -1.001 and the values go smoothly through -1/12 at -1.
    So…let’s look at the individual terms in the zeta-function to see where negative values arise. n^(-z) = n^[-(s+it)] =[n^(-s)]*[n^(-it)] and the second term in brackets is e^(-ilogn*t)= cos(t*logn) -i sin(t*logn).
    Now cos(t*log n) oscillates rapidly between -1 and 1 as n gets bigger and bigger, whence the negative contribution to the sum. Now, will a mathematician tell me why “analytic continuation” justifies the replace of the sum by the zeta function evaluated at an integral value?

  27. @Rich:

    Have not seen the video, so what I am about to say may be completely irrelevant, but there is a theorem due to Riemann that if (a_n) is a conditionally convergent series, then for every extended real number r (that is, including infinity), there is a permutation of the series such that the sum of the permuted series is r.

    In fact, the converse is true in the following sense: say that a series (a_n) is *unconditionally convergent* if every permutation of it converges (to a *finite* value). Then (a_n) is unconditionally convergent iff it is absolutely convergent. This is true for every finite-dimensional space, but breaks spectacularly in infinite dimensions by the Dvoretzky-Rogers theorem, leading to some very deep and beautiful mathematics — see for example, Diestel’s “Sequences and series in Banach spaces”.

  28. Per usual, concerning all this, Shakespeare said it best:

    “‘Tis past the infinite of thought.”

  29. On learning math from physicists, could this explain the difference between Engineering Calculus and the normal garden variety?

  30. If any of you have come across power series, there’s some additional insight to be gained.

    Consider the power series:
    1/(1-x) = 1 + x + x^2 + x^3 + x^4 + x^5 + … forever. Sounds straightforward, right? You can easily multiply both sides by (1-x) to see that it works out correctly.

    Now suppose you set x = 2. “Hang on a minute!” I hear you cry. “That’s outside the circle of convergence!” True. But let it go for a moment and see what we get.

    1/(1 – 2) = 1 + 2 + 4 + 8 + 16 + 32 + … forever. The left hand side is -1, and the right hand side is a ‘number’ that if expressed in binary every digit of which would be a 1, all the way to infinity. Interestingly, if you try to add to it in binary using the usual algorithm, the carry keeps on moving to the right and vanishes into infinity, leaving every digit beyond a certain point zero, and the answer given is exactly what you would expect if it *was* -1. Anyone ever heard of two’s complement?

    How about if we set x = -1?
    1/(1 – -1) = 1/2 = 1 – 1 + 1 – 1 + 1 – … forever. We saw that one discussed above. A lot of the prettier divergent series are just power series for perfectly normal functions like 1/(1-x).

    But unfortunately the trick doesn’t work for the series 1+1+1+1+… because that means setting x = 1 and the left hand side 1/(1-x) goes infinite, too. I’ll let you have a think about whether there’s any way around that.

    However, the trick reveals what’s really going on here. What you’re actually doing is manipulating the perfectly ordinary functions on the left hand side, like 1/(1-x), but expanding it into a power series at a point outside its circle of convergence. The relationship between the terms of the infinite sequence is such that it forces the divergent right hand side to have many of the same algebraic properties as the well-behaved left hand side. They are, in a sense, still the same function. It’s just that the function has been expressed in a sort of “coordinate system” (each coefficient of the power series can be thought of as a coordinate for a vector in “function space” – we’re getting into *really serious* mathematics, now) that doesn’t cover that part of the space properly. It’s a bit like the way all the lines of longitude meet at the north pole, making the question of what is the north pole’s longitude a bit silly. It doesn’t mean there’s anything genuinely weird going on at the north pole – it’s a perfectly normal point on the sphere that looks just like every other one. It’s just that our coordinate system has gone haywire.

    And the same happens in function space. The power series is a coordinate system for functions, but it goes haywire outside a certain part of the space. The function itself is still perfectly valid there, though, and has a sensible value, which the power series still reflects in a warped sort of way.

    Apologies to anyone reading who doesn’t understand power series, but for those who do, this is what’s really going on when you shift series and add them together, and so on.

  31. Rich is correct, the problem is why do you only shift 1 number to the right?

    S,S1,S2 are dynamic numbers and you can only approximate a value of S1 because it has the characteristics of a wave. The same with S2 although it is amplifying – the approximation (using the same technique as S1)would be near 0, but by doing the shift the approximation of 2S2(i.e. S2 + S2shift) is near 1 (not 1/2).

    But, as can be demonstrated by changing the value of the shift, S2 and S2shift are not the same and to use them as the same in an algebraic analysis will lead to a false result.

    What is truly difficult to understand though is that S-S2=4S

    Whether S2 approximates to 0 or 1/4, S ends up being a very small negative number. Or S2 is something other than it appears.

    Don’t play around with the infinite.

  32. The simplest (and correct also, I think) explanation is that the sequence of partial sums of those infinite series indeed diverges.

    However, values can be usefully assigned to those divergent series by summation methods in specialized mathematics contexts.

    For the sake of people’s sanity!

  33. Mark CC is not a “PhD mathematician”, but the lovely chap Dr Grime (in the second video) certainly is.

    The point is that the “sums” being discussed (call them something else if it suits you) are properties of the series as series – you can’t just treat them as a collection of numbers to be reordered or shifted. As it happens, I think the two particular series S2 and S2 shifted by one zero are equal in this sense, but this is soemthing that needs to be proved rather than assumed.

  34. @ Matt

    Shifting is just fine — it’s an algebraic operation. There’s no property of the series that says you can’t reorder it. S is S even if you randomly transpose all of the integers from 1 to positive infinity instead of taking them in order. We often reorder equations in algebra to get simpler answers, and we can often use the same algebra to make simple equations very nasty looking. Shifting just helps reveal a simpler form of the equation; it doesn’t change or add new meaning, it just makes it clear.

    For a cleaner example, consider
    (1) S1=1-1+1-1+1-1….

    I could rewrite that as
    (2) S1=1+(-1+1-1+1….)

    I’d like to call the part in parenthesis S1′, so that

    (3) S1′ = -1+1-1+1-1+1…

    (Amusingly, this is the same as -S1).

    That done, I can add S1 to both sides and get:

    (4) 2S1 = 1+ S1′ + S1 = 1 +(-1+1-1+1-1…) + (1+1-1+1-1….)

    S1′ and S1 cancel, leaving

    (5) 2S1 = 1
    (6) S1 = 1/2

    I just shifted the series to the right and added together, in effect, by doing nothing more than adding in a parenthesis. There’s no trickery here, outside of normal algebraic sophistry, and the series has no properties that require it to remain intact against this kind of normal operation.

    Shifting is entirely okay. Here, at least.

  35. That’s not right, Spellbound. What you call “normal algebraic sophistry” relies on properties of addition such as commutativity and associativity , which apply to finite sums and absolutely convergent infinite series. It’s well known that in general reordering infinite series can lead to contradictory “sums”. So, if you want to work in a world with all the normal algebraic moves, you have to accept these sequences as not having a sum.

    To be clear: by definition, if you reorder/shift a series, it is a different series. 5 + 3 is a a different series from 3 + 5, and
    A = 1 + 1/2 + 1/4 + 1/8 + …
    is a different series from
    A’ = 0 + 1 + 1/2 + 1/4 + 1/8 + …
    In both cases, the series have a property (absolute convergence, or even finiteness) that means the sums of the two different series are the same, and so we can think of them as the same as sums.

    Other series which converge but not absolutely have sums in the usual manner, but it is changed by reordering. Also, the sort of values discussed in the original video can be sensibly assigned to even non-convergent series and in some sense correspond to sums. The catch is that a shifted sequence can have a different sum from the original sequence. The example in Terry Tao’s blog linked above by JH is
    0 + 2 + 3 + 4 + 5 + … = -13/12
    2 + 3 + 4 + 5 + 6 + … = -7/12

    In fact all the series the video said were the same do have the same sum, but they glossed over the fact that this wasn’t because shifting never changes things.

  36. Saw this video a couple of years ago. Watched the follow ups. Discovered a massive error in his proof. When you adjust for it, the limit comes out as infinity. Which is of course what it obviously is.

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