# Ramanujan summation: How does 1+2+3+4... = -1/12? **URL:** \ **Category:** Factual Questions\ **Created:** [July 11, 2023, 4:14pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903 "2023-07-11T16:14:07Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 11, 2023, 6:02pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/21 "2023-07-11T18:02:15Z")
Previous thread (with contributions from some of our mathematical Dopers who haven’t been around in awhile): > [@1+2+3+4+... = -1/12. In what sense?](https://boards.straightdope.com/t/1-2-3-4-1-12-in-what-sense/403329): > > I am currently reading a book called The Music of the Primes which is about prime numbers and in particular the Riemann Hypothesis. It discusses the work of the Indian mathematician Ramanujan and, almost in passing, mentions that his equation: 1+2+3+4+…=-[sup]1[/sup]/[sub]12[/sub] appeared at first to be the ramblings of a madman. It doesn’t really elaborate, so I can only assume that the equation is in fact cromulent. I looked on Wikipedia and found a page on [Ramanujan summation](http://en.wikipedia.org/wiki/Ramanujan_summation), but that d… ---
**Author:** ![RitterSport](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/rittersport/32/6326_2.png) [@RitterSport](https://boards.straightdope.com/u/RitterSport)\ **Post date:** [July 11, 2023, 6:23pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/22 "2023-07-11T18:23:04Z")
Unfortunately, that other thread is almost unreadable, since most of the formatting is screwed up. > [@Great\_Antibob](#): > > How’s this for some mathematical trickery (about this particular case, not Ramanujan summation in general)… That post shows how it works nicely, but it seems like it’s that silly thing where 1 = 2, as long as you have a division by zero in there. What I mean is, is the Ramanujan summation just really showing that you can’t actually sum up divergent series like you can with convergent series? Or, is there really some deep mathematical meaning in there somewhere? Also, in the other thread, someone asked whether that series has ever found a use elsewhere, say in physics. Has it? ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [July 11, 2023, 6:43pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/23 "2023-07-11T18:43:43Z")
> [@RitterSport](#): > > That post shows how it works nicely, but it seems like it’s that silly thing where 1 = 2, as long as you have a division by zero in there. Yes and no. Summing an infinite number of elements is not really straightforward, so you can get some weird and contradictory results if you go in blindly. It really depends on how you define things, while the 1=2 “proof” relies on a relatively subtle and oft overlooked error involving division by zero. > [@RitterSport](#): > > Or, is there really some deep mathematical meaning in there somewhere? Yes. But the problem that regularly occurs (this thread is a great example) is people think of it as equivalent to a standard summation. It’s clearly not. It has validity but, as with a lot of math, you have to take a lot of care and make sure you are well aware of the assumptions, definitions, etc. This is true for absolutely convergent series as well, but those tend to follow our intuition more closely, so we don’t tend to think about them as deeply. Think of it as math bait and switch. You lead with a shocking statement (1 + 2 + 3 + … = -1/12) but then later explain the type of ‘summation’ involved is not the summation they learned in grade school. > [@RitterSport](#): > > Also, in the other thread, someone asked whether that series has ever found a use elsewhere, say in physics. Has it? In string theory, apparently it can be used to help explain the Casimir effect. I’m not a practicing mathematician and definitely not a quantum physicist, so explaining what that is about is better left to smarter folks than me. > **[Quantum mechanics/Casimir effect in one dimension - Wikiversity](https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension)** ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [July 11, 2023, 6:58pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/24 "2023-07-11T18:58:48Z")
nm accidentally sent unfinished post/ ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [July 11, 2023, 7:17pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/25 "2023-07-11T19:17:23Z")
Like most posters in this thread, most people throughout history have objected to any innovation in math. Obviously there cannot be numbers less than zero! Ok so you can invent negative numbers but there is no number greater than one and less than two! Ok so you can invent rational numbers but I can prove that no number multiplied by itself equals two! Etc. Similarly it’s trivial to add up a finite set of numbers but you can’t sum an infinite set. Ok so you can invent infinite summation that uses partial sums and limits to define a new type of summation that is defined on some infinite sets of numbers. It’s clearly not the same definition of summation that you used for finite sets but it gives the same result as finite summation for all cases where finite summation is defined and extends in a natural and self-consistent way to other objects for which finite summation is not defined. But now we’re done because we only allow mathematics to be extended in ways that I was taught in school. There is no way to sum divergent sets! I think the problem is that most of us believe that math exists somewhere somehow independently of us instead of being something we created and can continue to create. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 11, 2023, 7:19pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/26 "2023-07-11T19:19:12Z")
> [@Saint\_Cad](#): > > For an infinite summation to make sense, it needs to converge, that is adds up to a specific finite number. You might as well say that for a summation to make sense, it needs to be of a finite set of numbers. Or say that for a summation to make sense, it must be two numbers added together. All of those are true, for various notions of summation. If you’ve gotten through eighth grade math, you’ve already gone through a half-dozen different notions of summation, each of which only works in some cases, and each of which agrees with the previous notions in the cases where the previous notion works. Ramanujan summation is just another extension of the notion of summation. I mean, you can say that it’s “not really summation”, but if you go that route, then you might as well say that “1+2+3 = 6” isn’t really summation, because real sums only have two operands in them. Also note that Ramanujan summation gives a unique answer, and if you use it on any sum that is defined by other methods (such as convergence), that unique answer is the same as the one given by that other method. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [July 11, 2023, 7:27pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/27 "2023-07-11T19:27:27Z")
I don’t disagree but remember that the OP wanted the explanation as simple as possible. You’ll note I left out the entire issue of absolute vs. conditional convergence as it would muddy the waters. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [July 11, 2023, 7:31pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/28 "2023-07-11T19:31:59Z")
My interpretation as a math guy is the following. There are two mappings that takes in a function, f, and returns a value. The first with we will call S, has the following definition. S(f)=\sum\_{k=1}^{\infty}{f(k)}=f(1)+f(2)+f(3)+\ldots The second one which we will call R, has a value equal to R\left(f\right)=\ \frac{f(0)}{2}+i\int\_{0}^{\infty}{\frac{f\left(it\right)-f(-it)}{e^{2\pi t}-1}dt} Now the first mapping is only defined for those function that decay fast enough that their series converges. So it is undefined for f(x)=x, where S(f) = 1+2+3+… But for any of the functions for which it is defined it turns out that S(f)=R(f). But there are many function for which R(f) exists even though S(f) is not defined. For example if f(x)=x, then R(f)=-1/12. Any of those “proofs” that you see on youtube, effectively make the hidden assumption that the series does in fact converge, at which point they can prove that it must be equal to -1/12 since all functions that do have convergent series have the S(f)=R(f) property. Its similar to those [1=0 proofs](https://math.hmc.edu/funfacts/one-equals-zero/) that involve a hidden divide by zero. To think of it another way, imagine that I wrote a program that given any two points on earth in terms of X,Y and Z coordinates from the earth center, it will give you the surface travel distance travel between them according the the [great circle](https://en.wikipedia.org/wiki/Great_circle). I then give it one X,Y,Z coordinate that is my house, and the next X,Y,Z coordinate that is on the moon. My program might spit out an answer, but that doesn’t mean that I can get to the moon by following a great circle of that length around the earth. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [July 11, 2023, 7:34pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/29 "2023-07-11T19:34:31Z")
Arithmetic is not the same as mathematics. Sure, when people are taught a certain type of arithmetic in school it’s called math (maths for you UK weirdos). But it’s just a tiny little corner of a vast universe. What’s true in that dot doesn’t have to be true elsewhere, and a huge percentage of the time isn’t. American schooling (and seemingly other country’s as well, but I’m no expert) does a terrible job on this distinction. Yet it appears everywhere in science. Newtonian mechanics covered everyday experience fine for hundreds of years before Relativity and Quantum Mechanics told us that Newton is just arithmetic, something that doesn’t apply to large parts of the universe. People still howl at this finding. It violates common sense, just as the Ramanujan summation does. And so does everything else about infinities, which made even mathematicians howl until the next generation took over. What we “know” is common sense. Everything else is doubtful, and probably evil. That’s as old as mankind. But we should “know” better. ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [July 11, 2023, 7:56pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/30 "2023-07-11T19:56:10Z")
> [@74westy](#): > > I think the problem is that most of us believe that math exists somewhere somehow independently of us instead of being something we created and can continue to create. There has been a long running philosophical debate on whether math was discovered or invented. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [July 11, 2023, 8:05pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/31 "2023-07-11T20:05:24Z")
> [@Chronos](#): > > Also note that Ramanujan summation gives a unique answer, and if you use it on any sum that is defined by other methods (such as convergence), that unique answer is the same as the one given by that other method. I am not sure even that is true (depending on precisely how one wants to define Ramanujan summation) ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [July 11, 2023, 8:10pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/32 "2023-07-11T20:10:49Z")
> [@Whack-a-Mole](#): > > There has been a long running philosophical debate on whether math was discovered or invented. Indeed. And being a philosophical debate, there is no answer either way but this thread shows the problems people get into when they assume platonism is true. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 11, 2023, 8:15pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/33 "2023-07-11T20:15:09Z")
> [@74westy](#): > > this thread shows the problems people get into when they assume platonism is true. I’m not convinced that it makes a difference to this thread whether we think of Ramanujan summation as discovered or created. ---
**Author:** ![74westy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/74westy/32/3950_2.png) [@74westy](https://boards.straightdope.com/u/74westy)\ **Post date:** [July 11, 2023, 8:17pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/34 "2023-07-11T20:17:41Z")
Is there some other reason people can’t accept it? ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [July 11, 2023, 8:21pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/35 "2023-07-11T20:21:26Z")
I think that it runs so much against our intuition it makes it difficult to accept. Just my $0.02 ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [July 11, 2023, 8:23pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/36 "2023-07-11T20:23:38Z")
> [@74westy](#): > > Is there some other reason people can’t accept it? @Exapno_Mapcase gave the example of people having trouble accepting Relativity and Quantum Mechanics. They’re non-intuitive, but that doesn’t mean they were created rather than discovered. ---
**Author:** ![gnoitall](https://avatars.discourse-cdn.com/v4/letter/g/bb73d2/32.png) [@gnoitall](https://boards.straightdope.com/u/gnoitall)\ **Post date:** [July 11, 2023, 8:57pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/37 "2023-07-11T20:57:51Z")
As was pointed out, mathematics has systematically progressed beyond its contemporary intuitive limitations, leap after leap after leap. So intuition is not a valid reason to reject a sufficiently consistent, expressive, and powerful mathematical expression or concept. Intuition is a valid reason to say “I can’t understand this” or “it doesn’t make sense to me”, but those statements don’t invalidate the value and correctness of mathematical discoveries. Editing to add: for most people, “summation” means “arithmetic summation” only, because it’s strictly arithmetic and fits with “intuitive” (commonly-taught) ideas of arithmetic operations. But “summations” is actually a wide class of mathematical operations applied to different types of series, and many of which defy the simplicity of an arithmetic summation. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [July 11, 2023, 9:01pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/38 "2023-07-11T21:01:11Z")
> [@Whack-a-Mole](#): > > I think that it runs so much against our intuition it makes it difficult to accept. I don’t think that’s it. I don’t think it’s fundamentally much less intuitive than summing an infinite series by convergence. I think it’s literally just that so many folks who talk about it start off by saying “Well, it’s not REALLY summation…”. EDIT: Well, that, and “If it were REALLY true, I would have learned about it in school”. ---
**Author:** ![glowacks](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/glowacks/32/5548_2.png) [@glowacks](https://boards.straightdope.com/u/glowacks)\ **Post date:** [July 11, 2023, 9:08pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/39 "2023-07-11T21:08:42Z")
The question of whether mathematics is created or discovered is actually very simple to answer. I’m surprised it’s still a matter of dispute, because once I tell you the answer, it’s obvious. Definitions are created. Theorems are discovered. ---
**Author:** ![Whack-a-Mole](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/whack-a-mole/32/141_2.png) [@Whack-a-Mole](https://boards.straightdope.com/u/Whack-a-Mole)\ **Post date:** [July 11, 2023, 9:27pm UTC](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903/40 "2023-07-11T21:27:01Z")
FWIW: I do not reject anything here or any other math that is largely accepted by mathematicians. I get I do not understand most of it and I and I’m happy to rely on experts to tell me what’s-what. I may not understand it all but I do not need to. I don’t understand all the things a doctor may tell me or the engineer who builds a bridge I may drive over. I trust their profession to keep most of it on the up-and-up. I am not saying you thought I was pushing back on any of this but I want to be clear. [Previous page](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903.md?page=1) [Next page](https://boards.straightdope.com/t/ramanujan-summation-how-does-1-2-3-4-1-12/986903.md?page=3)