Ramanujan summation: How does 1+2+3+4... = -1/12?

Factual Questions
41

Of course. Although I didn’t express it well, I was commenting on the general cultural tendency to resist novelty which conflicts with current wisdom, and not as any particular comment on you in particular.

It’s reasonable to say “I have trouble assimilating this idea because it clashes with my intuition” (implicitly accepting that the intuition in question is actually learned). And that explains a lot of why learning advanced mathematics can be difficult: you have to accept a new operating mental model which doesn’t match the operating models you already know and trust, and sometimes uses the same language as what you already know to express something you don’t currently understand.

42

Just as obvious as 1+2+3+4+... \neq -1/12

Definitions are created. Axioms are created. Rules of inference are created. Theorems are the results of mechanical application of rules of inference.

I really wish as many things were obvious to me as they are to other people.

43

I agree terminology is a problem, and one that is widespread through science.

Maybe the best example is the use of “dark” in dark energy and dark matter. Originally the term merely meant “unknown.” (Although dark matter is also not electromagnetic or “bright” matter.)

That everything we can see is a tiny percentage of the universe is totally counter-intuitive. Bad enough that mass and energy are interchangeable, when they are “obviously” different. That mass and matter need different definitions? That the stars and galaxies that fills all those images that scientists are puffing so much are being pushed around by invisible nothingness is ridiculous. That the universe is “expanding” but into more nothingness and it’s the same distance to every farthest point but not round and there is no “center” defies language.

Scientists have tried hard to made science more understandable by the public. That might have backfired. Einstein was adored because he invented something he said that only twelve people in the world could understand. (He didn’t, but the NY Times said so, which is a different rant.) Wow. Awesome. Don’t even try to explain it to me. My head might explode.

And when people heard the simplified explanation anyway, using terms they thought they understood but meant something totally different, they figured that scientists were either crazy or lying to them.

Science is math, not words. Math especially is math rather than words. Since only a tiny percentage (here we go again) of the world has any conception of this advanced math they want explanations in words. Words fail me. And everyone.

44

Here’s another one: 1 + 2 + 4 + 8 + 16 + … = -1.

That one seems just as counterintuitive, and obviously it doesn’t mean anything in the finite case… and yet, if you naively try to do that sum on a finite computer, you will in fact get -1. Or at least, you might, depending on how the computer implements integers, but this one is a fairly common one, since it makes ordinary integer operations much easier.

45

Do you mean the OP (me in this case) should not ask such questions or ask for a simplified, understandable answer? I understand such answers cannot be truly complete and that’s ok.

46

Wise comments all. Plus @gnoitall’s other comments along this line. And a couple of @Chronos.

Putting my gloss on it all …
Said another way IMO, schooling in general, and American schooling for sure, does a really lousy job with “lies to children”. They tell the lies, but condition the children that they are absolute truth. Which is a dangerous mental habit to form, and the earlier it gets locked into kids’ heads, the more willfully ignorant the student becomes later in life.

There is nothing wrong with simplifying stuff to teach to youngsters. Including up through about undergrads. What’s wrong is not including caveats all the way along like

What we’re teaching you here today is only true in particularly simple cases, and there’s lots more you’ll learn later. Don’t lock yourself into the idea that what I’m teaching today is THE truth. It’s a small piece that is true enough for itself, but is part of a much larger more complicated truth you can’t yet see or understand.

Always think of your knowledge as provisional and limited to only the areas you’ve explored. Amazing things you’ve never heard of or thought of are out there. And next year you’ll learn about some of them.

If THAT was the Prime Directive for how we taught, and we did not have a culture where half of all parents would go apeshit if they heard such a thing, maybe we’d get past the difficulty with people accepting relativity, QM, more advanced / obscure notions of summation, etc.

47

As others have said, this doesn’t happen for any finite set of numbers. But there is an example of one where, in a very broad hand-wavey sense, you can kinda see what’s happening.

Consider the sum:
9 + 90 + 900 + ...

Apply Ramanujan (or various other kinds of) summation to that and you get an answer of -1. How does that make sense? Well, if we have this:
9 + 90 + 900 + ... = -1
Then this is also true:
9 + 90 + 900 + ... + 1 = 0
So we get a sequence something like:
9 + 1 = 10 \approx 0
99 + 1 = 100 \approx 0
999 + 1 = 1000 \approx 0
...

The digits on the right are all zero. There’s just that pesky 1. And it’s getting farther and farther away as we add more terms. Eventually, we can’t see it any more, and just have 0 = 0. Which makes perfect sense.

Of course this doesn’t correspond to any normal notion of equivalency in numbers. But there is some hint of what it might mean for the sum to “become apparent.” And in fact there’s a way to formalize this whole process, called the p-adic numbers, though it only works for primes and not powers of 10.

48

An old thread of mine on the subject.
​​​ ​1+2+3+4+…infinity = -1/12? - Factual Questions - Straight Dope Message Board

49

Bad doggy! I have a minor in math from years ago, and started reading the description of the Ramanujan Series, and all I can say is “meow”. :smiley:

50

I am happy to say I get both references.

Some things are more important to know than math. :wink:

51

It seems there were more than a few threads on this over the years.

I relied on the Discord auto We’ve-Done-This-Before popup thingy to let me know. I thought surely it would catch something as obscure as this pretty easily. It didn’t.

I’ll have to be sure to do my own search before posting in the future.

52

As I said in one of those earlier threads, it’s not particularly easy to search on “1 + 2 + 3 + 4 … = -1/12.”

53

I think @LSLGuy already gave a good explanation. Just a few words. (Who, me?)

Of course people want to understand more about science. I love popular science books. I subscribe to a pile of science magazines. I look at stuff online, whatever the term is for a variety of stuff. Through that interaction I’ve become frustrated.

Stephen Hawking famously said he was told that every equation included in a book cut sales in half. True or not, A Brief History of Time has one equation. The rest of the book, just like every other good popular book on modern physics, is a series of metaphors and analogies and sortas and kindas. Even so, it is also famously called the most unread bestseller. The metaphors Hawking used 35 years ago didn’t connect with willing readers. Since then, other physicists have abandoned them for different sets, probably to no better result.

Perhaps the best book on the subject I’ve read is 101 Quantum Questions, by Ken Ford, who is 97 this year and has seen it all. He starts the book with basic physics concepts and then builds upon them, layer by layer, so that weirdness of QM emerges logically out of simpler physics. It all suddenly seems remarkably clear. Things that most popular science books handwave past become lucid, in less than 300 pages of text. But that’s only because question 101 is built on top of questions 1-100. If you skip around, say, just reading the odd numbers, you’ll probably be lost. The reader needs to make a commitment to be a partner of the writer. The last, deep questions make no sense otherwise. And then the book should probably be reread with the ending in mind.

His book was not a bestseller. I never heard of it until I stumbled upon it in a library. Not many equations, either, and no advanced ones, but some are absolutely necessary.

People in general always want to skip to the ending and be given “the” answer. I don’t believe it can be done. In anything, but especially so in math and physics. Telling people that mass works like a bowling ball on a rubber sheet gets the two sides only so far. The picture is there, but the concepts behind it are left out.

In short (hey, I kept this under 500 words!), education has failed people. No simple answers exist to complex questions. Almost every math and physics thread here winds up with experts shooting equations back and forth as they argue the nuances of the issue between themselves. I just don’t think that’s helpful to beginners, although I greatly appreciate the time and effort they’ve devoted to teaching us over the years. Somebody started a thread here not long ago, asking what they need to do to understand QM. The answer boiled down to getting a couple of degrees in physics. I have no idea how to move past that in the modern world.

54

Let me to explain what is going on. I will start by saying that
1+2+3+\cdots=-1/12 is obviously absurd. But the claim does have some meaning, just not the obvious one. What Ramanujan did was start with the function
\zeta(s) =\sum_{n=0}^\infty n^{-s} , for a complex number s. It converges whenever the real part of s is >1 and diverges when the real part is \le1. In particular, it is obviously divergent when s=-1. Now through the magic of a process called analytic continuation it turns out that this \zeta function can be “continued”, that is extended to all complex numbers, except \zeta=1. Moreover the extension is unique as long as you stick to analytic functions, which basically means that is defined by a power series around in a disk in the complex plane around each point at which it is defined. Finally, when \zeta(s) is calculated at s=-1, the result is -1/12.

The function, incidentally, is called the Riemann \zeta function and I don’t think it is any exaggeration to say that the locations where it is 0 is the greatest unsolved problem in math. The Riemann hypothesis is that all the zeroes lie on the line the s=1/2+ai where a is real. Thousands, maybe millions, of zeroes are known and they are all on that line. So far. It is known that they all lie on the strip where the real part is between 0 and 1.

55

10 trillion:

Though that figure is from 2004. I’m somewhat surprised that it hasn’t gone up since then. There have been some efforts to compute patches of zeros much further out (like at 10^22), skipping in the ones in between.

56

I found 3Blue1Brown quite helpful in understanding this:
Visualizing the Riemann zeta function and analytic continuation

57

I would argue that math is both created and discovered. Specifically, mathematicians will create rules, but then discover the implications of those rules.

I’d also argue, however, that there are some rules that were created before math was formally studied, being seen as just intuitive. And when we discover the math that follows from those intuitive rules, the discovery feels more “real.”

58

How do you perform that calculation, though? That brings us to the specific procedure Ramanujan actually used in this case, which seems to be to write the partial sum \sum_{n=a}^Nn and examine its asymptotic expansion. For illustration, if I may steal a bit that Terry Tao posted on his blog, if we introduce an appropriate cutoff function \eta(x) that approaches 1 near zero and is compactly supported, bounded, and smooth enough, then the Euler–Maclaurin formula can be used to show that

\sum_{n=1}^\infty n \eta\bigl(\tfrac{n}{N}\bigr) = -\frac{1}{12} + C_{\eta,1}N^2 + O\bigl(\tfrac{1}{N}\bigr)

and the constant term is the Ramanujan sum.

59

To be honest, I have no idea how that calculation is performed. I have a superficial idea of how the analytic continuation is performed, but no idea of the detailed computation. The zeta function has been studied within an inch of its life by people chasing the Riemann hypothesis.

60

Hi, guys!

To, ahem, sum up.

“Summation” is a term applied to a multitude of totally different mathematical processes, with the 2+2=4 sum of arithmetic being just one of them. Arithmetic applies only to finite groups. Infinite summation requires complicated advanced math that is sometimes difficult to calculate and cannot give partial answers to finite parts of the string. The Ramanujan summation requires an additional set of carefully laid out definitions and limitations about what it can be applied to and how to proceed.

I think that’s the base, in Ken Ford fashion, for understanding the leap to what our mathematicians are arguing about. I reverse-engineered the thread to get that. Trying to figure out what beginners don’t know and the easiest way to convey the needed pre-knowledge is a skill unto itself. An OP on a message board seldom provides any clues about what knowledge is lacking. I think often a presumption is made that if an OP can frame a question the OP already has some knowledge to apply. That also seldom works. The result is a jumble of explications at various levels of difficult and with various presumptions that lead to confusion and can result in acrimony.

I think that’s saying it better than my previous try above. No blame is attached to either side. Both are victims of not knowing what they don’t know and of making assumptions that are skewed because of that.

Science communications is a hugely difficult problem without math. Nobody has come close to solving it yet.