Trying to explain this to my buddy, and he has the same conceptual problems with it as everyone who's not a math geek. He's like, is this some kind of math trick that doesn't really mean anything? I told him yes/no.

Someone posted above that the sum of 1+2+3+4.... doesn't have to be anything, but if it is anything it would be -1/12. Would it be similarly accurate to say it really isn't true that this series can have a sum, because well, it's infinite. But if you could sum it, it would be -1/12, and that actually has practical applications. Should I tell my friend this?

In math, whenever someone writes "..." you need to really stop to make sure you understand what is meant.

You can try. When we write 1+2+3+4+... = -1/12, we're NOT saying that if you stop adding terms that you'll end up with a number that is close to -1/12. It means something else entirely.

Help me understand where I'm going wrong. I'm assuming that 1+2+3+4... means add up the infinite series of all positive integers beginning with 1. I'm assuming the equal sign to mean the result when you add up what is before it, or better yet "things on either side of this sign are equal." I also may not understand what a sum is. I thought that just meant the result when you add a bunch of things together.

This is why we can say that 1+2+3+4+... doesn't mean anything, while also saying that 1+2+3+4+... = -1/12.

In the first case, we really are talking about adding up the numbers, and we really are saying that it doesn't add up to anything. But in the second case, we're "extending" the meaning of + to be more than just adding up numbers. The technical details fall under the discussion of "analytic continuation" and specifically the analytic continuation of the Riemann zeta function.

(There are other ways of getting to this result, but this explanation is probably the most complete one without getting overly technical.)

Or as Bill Clinton says, it depends on what the meaning of "is" is. So in that sense it is sort of like a math trick to "extend" the meaning of +. But this is not just some meaningless trick, as there are practical applications to the result.

Can I can tell my buddy that in the sense he and most of us think of adding numbers together, the answer is undefined. But if we look deeper into addition, we can come up with a result that has meaning?

It's not quite about looking "deeper" into addition. Maybe it's best to say a little more about analytic continuation.



Here is a graph of a function f(x). When looking at the graph, you can only see the values between -5 and 5. But there's a sense in which you can reasonably infer what f(6) would be.

For certain types of function (complex analytic functions), there is only one reasonable way of extending a function. This means that if you know the values of the function on a small region, there's only one reasonable value of the function anywhere else that the function is defined.

In this case, there's a specific function (the Riemann zeta function) that has the property that when it's evaluated at a certain value, it gives you 1 + 2 + 3 + ... (specifically, at s = -1, where s the complex input variable). But that way of calculating it doesn't work. Instead, you find a place where the sum DOES make sense, and use analytic continuation to infer what the value will be at that point where the sum doesn't work.

Argh, you had me until this part. Might need to dumb it down some more. Alternatively, I may just need to read up about it more myself. I'll start with the links Masque posted. It will be interesting to see how much more math I need to learn before this clicks. Had 4 semesters of calc and diffi q, god 15 years ago.

I have read this whole thread, yet again I have to give credit to BruzeZ to for explaining this in a way that I can understand.


Backdoor

And now, as sugar topping, what's the sum OF ALL INTEGERS (including zero and negative integers), mentioned in the title? Is it = 0 ?

Is the sum of -1 + -2 + -3 + ... = +1/12? Just want to make sure.

The sum you describe is indeed 0 also.

Pay attention though to the fact that 1+2+3+... is not the same as 0+1+2+3+...as idiotic as this sounds.

The reason is the first is the sequence an=n the second is an=n-1 (n is 1 to infinity).

So the regularization that uses a factor e^(-a*n) (a to the 0 limit eventually) in order to render the sum doable for all a>0 will yield for the first sequence sum
e^a/(-1 +e^a)^2
and for the second
1/(-1 +e^a)^2

hence the difference (-1/12 vs 5/12) (or go by the ramanujan summation link way to get the same)


Which means how you construct your ....-3-2-1+0+1+2+3... thing, matters lol.

I mean do you see it as Sum[an, {n,1,Infinity}]+Sum[bn,{n,1,Infinity}] with an=n,bn=-n
or do you see it as Sum[an, {n,1,Infinity}]+Sum[bn, {n,1,Infinity}] with an=n-1 and bn=-n?



Also for FoldnDark, complex analysis as Aaron W. is suggesting you is the proper way to understand all this discussion. Nobody is seriously arguing that an infinite sum is -1/12 here. The value of the analytically continued function that started from that sum where it was properly defined (finite, convergent) however is indeed-1/12. In particular you need to appreciate how different the concept of derivative (and differentiability) in complex functions is vs real functions. (and therefore how analytically continuing a function outside an initial proper domain is not some absurd arbitrary mechanism)

Here's the link to the Riemann zeta function's wiki page:

http://en.wikipedia.org/wiki/Riemann_zeta_function

But what really matters is this definition:



You may remember something about convergent infinite sums, and the R(s) > 1 simply means that this sum converges when the real part of s is greater than 1. (If you don't want to think about complex numbers, you can just think of s as being some positive number greater than 1).

Anyway, this converges nicely in a certain place, and so by analytic continuation, there's only one reasonable way to extend it to other places. The Riemann zeta function is the function you get when you extend this function.

When you set s = -1 in the formula, you get zeta(-1) on one side, and on the other side you get 1+2+3+... But there are formulas that we can use to get zeta(-1) = -1/12. So when you put the two sides together, you get the formula.

That's interesting because my first reaction would be that this is a good way to prove that those formulas are wrong. You mean when you do that you get 1+2+3... = -1/12? Ahhh, back to the drawing board ;(

But thanks for your help, Aaron. You've brought me a step or two closer to being able explain this to my buddy. Good cocktail conversation among the right company.

No problem. This is non-intuitive stuff when you bump into it the first time. And like with a lot of math, it makes no sense until it makes sense, and then it just makes sense.

Good luck sorting it out.

(Remember that the meaning has been "extended" so that you're not literally adding up the numbers. It's just what you would get if adding up the numbers made sense.)

lol'd

You math people are crazy

So two physicists made an eight minute pop-sci video about an abuse of notation? Why...?

Masochists.

This whole thread is a useless waste of time unless you understand first what a COMPLEX number is. I suggest they all go back to and stick to the Khan Academy.

The very notion that the sum of ANY set of integers is = -1/12 is plainly false. Maybe the video mentioned early on was trying to make the point that so many people are easily mislead or fooled when an ABSOLUTELY IMPORTANT detail is missed or ignored by the careless masses. I did not watch the video and do not need to in order to answer the original question.

Somehow I get the feeling this explains why socks go missing from the dryer, check books never balance, and when someone has 12 outs they river me 100% of the time.

Two posts in ten years must be the record.

Listen to this guy.

It makes me very curious as to what his first post was...

Lol, tried to look it up but it's gone. Anyway, that sentiment, that frustration is very understandable. I shared it quite recently....but now I finally get it!! I finally understand exactly what your advisor meant when he said you don't have to get anything, but if you do it is -1/12.

Not sure why it didn't click before, but watching the latest video playaande posted they start with the sum 1-1+1-1... =1/2, which is actually pretty intuitive. The point is, you can choose to stop at any spot and you'll get either 1 or 0, but then you haven't got the sum because you stopped before reaching infinity. So in one way there is no answer, but in another intuitive way we can see if we really can sum this infinity it will be 1/2. If you accept that, then you can quickly accept the less intuitive -1/12 in the OP.

Finally, people are getting it! This topic is the same as saying that Because Boise is almost midway between Denver and Seattle, the winner of this year's Super Bowl will be Boise!! Any of you REAL PLAYERS out there going to bet BOISE to win in Vegas or wherever???

The Physics Professor's use of 1/2 for the value of S1 is absurd since the Average of a result/value need not reflect ANY of the actual values possible as an earlier poster correctly wrote. AS A LOYAL BOISE RESIDENT and member of BRONCO NATION, I will gladly take ANY and ALL BETS placed for BOISE to win the Super Bowl this next Sunday!! I will EVEN offer INFINITE ODDS for a SUPER, INCALCULABLE PAYOFF!!



PS HECK, I can't even get any two answers to match on all of the various distance calculation websites/services out there on the internet, between the 3 Cities mentioned above, Boise, Denver and Seattle. Some even have the gaul to list the answer in units of TIME and not DISTANCE!! Must be Neuro-Typicals!! (NT's are all non-autistic humans and is a polite way of referring to them whatever their general abilities otherwise.)

Lemme guess, you'll pay out -$1/12? Go Boise!!

Now I'm paranoid Aaron is going to come in here and tell me I'm still thinking about it wrong, but I just think I understand what they mean now when they say the most important property of that series is -1/12.

Here is the final word in a very easy to understand explanation (with proof) of the false claim about the -1/12 result. If any want to remove all fog, haze, doubt, or uncertainty, simply go here:

http://scientopia.org/blogs/goodmath...ad-astronomer/