In the End, It All Adds Up to -1/12 (1+2+3+4+ ··· ∞ = -1/12)

reply to post by ImaFungi



Why -1/12?

Because that's the answer you get by considering the sum as only one example of an analytical process.

en.wikipedia.org...

en.wikipedia.org...

en.wikipedia.org...

reply to post by mbkennel

The wikipedia link shows this graph which illustrates a y-axis intercept of -1/12, which is pretty clear to me:



I don't have any problem saying -1/12 is a "zeta regularized sum", but it doesn't make sense to me to call -1/12 a "sum".

What seems to be going on in the graph and the math is something like this:

1 + 2 + 3 + 4 +.... = y + (infinite expression)

where y= -1/12

-1/12 is what you get if you toss the infinite expression you want to get rid of, but to express the equation as

1 + 2 + 3 + 4 +.... = y

disregards the infinite expression which was discarded, and is misleading at best.

mbkennel
reply to post by ImaFungi

 

Why -1/12?

Because that's the answer you get by considering the sum as only one example of an analytical process.

en.wikipedia.org...

en.wikipedia.org...

en.wikipedia.org...

" The divergent series 1 + 2 + 3 + 4 + ⋯. The first few partial sums of the series are 1, 3, 6, and 10, which appear as the height of the first few stair-steps. The terms 1, 2, 3, 4, and a corner of the term 5 are shown in alternating gray and yellow, to emphasize their individual contributions to the partial sums. The green parabola is the asymptote of the smoothed sum. Its y-intercept is -1/12, which is the regularized sum of the series"

Im asking what is not arbitrary about the choice to only use those specific partial sums? If you went further would that answer be different?

I am stating that -1/12 is completely arbitrary and it might as well be -1/2 -1/3 -1/4 -1/5 etc. You can give no absolute reason why the mode of averaging was stopped at the number it was, that in turn provided the -1/12.

ImaFungi
Im asking what is not arbitrary about the choice to only use those specific partial sums? If you went further would that answer be different?

So go further and tell us what happens. I don't see how it will be any different.

Arbitrageur

ImaFungi
Im asking what is not arbitrary about the choice to only use those specific partial sums? If you went further would that answer be different?

So go further and tell us what happens. I don't see how it will be any different.

Well I disagree with what they had to do to get to that point in the first place, taking different sets and subtracting them from each other, its bs. The sum of all numbers is not -1/12. That is a fraction you get when you do some weird fun number play, but not when you add all positive integers. There is no non arbitrary relation to that fraction and the totality of positive numbers.

Meh. I call that "fuzzy math" and/or an accounting trick.

First off, the answer to 1-1+1-1+1-1... will ALWAYS be either zero or one, depending on where you stopped adding. The answer will never be -1/2, no matter where you stop adding. You can't average them out, because you won't get one of the two possible real answers by averaging them -- you get a third "fake" answer. That's why it's just an accounting trick.

Therefore, their first example is wrong right off the bat -- and when they try to apply that failed logic to the 1+2+3+4+5..., that's why the get the -1/12 answer.

One overriding proof of this would be that adding only positive integers together can only result in a positive sum. Therefore, that right there is evidence that something is wrong with the -1/12 answer. That "something wrong" is the fallacious idea that you can simply "average out" the answer to the 1-1+1-1+1-1... question.

Arbitrageur
reply to post by mbkennel

 

The wikipedia link shows this graph which illustrates a y-axis intercept of -1/12, which is pretty clear to me:

I don't have any problem saying -1/12 is a "zeta regularized sum", but it doesn't make sense to me to call -1/12 a "sum".

More advanced mathematics universally progresses by analogy and extension. All of abstract algebra extends e.g. "addition" and "multiplication" to less elementary domains. Calculus extended "sum" as well, and this is just another one. Linear operators on a Hilbert space extend 'matrix multiplication' beyond finite dimensionality.

What seems to be going on in the graph and the math is something like this:

1 + 2 + 3 + 4 +.... = y + (infinite expression)

where y= -1/12

-1/12 is what you get if you toss the infinite expression you want to get rid of, but to express the equation as

1 + 2 + 3 + 4 +.... = y

disregards the infinite expression which was discarded, and is misleading at best.

edit on 23-2-2014 by Arbitrageur because: clarification

It's not misleading if you understand the mathematics, where the symbolic notation is not literally confined to mathematics invented before the 1700's and taught in high school (or elementary school if you were smart and russian).

Soylent Green Is People

One overriding proof of this would be that adding only positive integers together can only result in a positive sum. Therefore, that right there is evidence that something is wrong with the -1/12 answer. That "something wrong" is the fallacious idea that you can simply "average out" the answer to the 1-1+1-1+1-1... question.

That's the whole point. Mathematicians have recognized the many forms of meaning of the "+ ..." part of the concept that you have not clearly examined, and assume meant only one particular thing.

The complexity of the conceptual extension to infinites is a huge theme in post Newtonian mathematics and central to all forms of analysis (meaning the class of mathematics).

That "something wrong" is the fallacious idea that you can simply "average out" the answer to the 1-1+1-1+1-1... question.

What do you mean by "fallacious"? Mathematics is not 'wrong' or immoral if you don't understand it, it's only wrong if it is actually wrong.

Every mathemetician will agree that partial sums are finite and do and do not have certain limiting properties.

It works like this:

a) ok, these things don't have limits under classical definition X.
b) so what do we do with them?
a) how about this, in an analogy to other circumstances we can define a different kind of transformation and limiting process in which they do
b) alright, but why should I choose this one instead of another
a) because in other cases, the classical and new ones merge very well, and this permits us to use new tools to reason about whole classes of things we previously had to give up on
b) alright, cool new math, let's hunt theorems!

ImaFungi
I am stating that -1/12 is completely arbitrary and it might as well be -1/2 -1/3 -1/4 -1/5 etc. You can give no absolute reason why the mode of averaging was stopped at the number it was, that in turn provided the -1/12.

This isn't true.

The picture is just an illustration, not the underlying actual mathematics.

mbkennel
...It works like this:

a) ok, these things don't have limits under classical definition X.
b) so what do we do with them?
a) how about this, in an analogy to other circumstances we can define a different kind of transformation and limiting process in which they do
b) alright, but why should I choose this one instead of another
a) because in other cases, the classical and new ones merge very well, and this permits us to use new tools to reason about whole classes of things we previously had to give up on
b) alright, cool new math, let's hunt theorems!

I agree that it can be "cool new math" and it's fun to consider these things, but the answer to 1-1+1-1+1-1+1-1... is still EITHER -1 or 0. it is never -1/2, unless mathematicians decide they will allow themselves to play a bit "fast and loose" and allow for accounting tricks, such as saying the answer is NEITHER -1 NOR 0, but is actually the average between the two.

I'm all for mathematicians coming up with these interesting proofs and digging down deep into the way they like their science to manifest itself to the world, but the "real" answer to 1+2+3+4+5+6+7... can never be negative. It's only negative if mathematicians decide to allow the answer to manifest itself as one of these weird and interesting proofs.

i.e., it is the act of providing a manner in which to prove the math itself that is clouding the issue, not the actual real-world answer to what happens when you add positive integers together. Like I said, it's fuzzy math.

If I decided I would flip a coin, a statistician would tell me that I hand a 1/2 chance (50%) of getting a heads, and a 1/2 chance of getting tails. But the reality of the situation is that I will get EITHER a heads OR a tails; one or the other....

...Therefore, it's not relevant what the odds are for flipping a heads or a tails. The odds don't matter to me, because I will truly get one or the other; that's all I care about, not the odds. There is a 100% chance I will get either a heads or a tails; it's binary (unless it lands on its side, but neither I nor the statistician are arguing that point, so let's remove that from this case).

The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.

Similarly, When I add together 1-1+1-1+1-1+1-1..., depending on where I stopped, I will get a -1 or 0 100% of the time. There is no other answer. The proof that tells me I get a -1/2 is simply an approximation (granted -- it is an approximation that used the scientific principles behind math, but so what?). Therefore, applying that concept to answer 1+2+3+4+5...will only give an approximation (and a poor one at that, because the inherent problem with using the approximation is being compounded), and not a real answer.

Soylent Green Is People


The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.

I guess you don't accept Heisenberg's uncertainty principle then.

mbkennel
It's not misleading if you understand the mathematics, where the symbolic notation is not literally confined to mathematics invented before the 1700's and taught in high school (or elementary school if you were smart and russian).

It's helpful to clarify using different notation that two expressions which look similar are not actually the same. For example, look at the notation Wikipedia uses to distinguish a "Ramanujan summation":

Ramanujan summation


Do you see anything wrong with using such clarifying notation? Note the comment says they are not sums in the usual sense.

Alternatively, I would also think using some type of modified equals symbol would be an alternative if you didn't find this notation acceptable. If the equals sign is being used to mean different things in different instances, why use the same symbol? Why not modify the symbol?

mbkennel
More advanced mathematics universally progresses by analogy and extension. All of abstract algebra extends e.g. "addition" and "multiplication" to less elementary domains. Calculus extended "sum" as well, and this is just another one.

I don't see where calculus summations distort the meaning of the equals symbol like summations of divergent series do. Divergent series aren't really sums, as explained above and below:

www.slate.com...

You may have noticed that all through this post, I have avoided writing “This series equals -1/12,” or “the value of the sum of the series is -1/12.” This is due to my conversation with Ellenberg, which was fascinating to me. We talked about different methods, different rules, how new concepts were not accepted at first, and that things we think are simple now (like using fractions) were at one point in history heatedly debated as to their reality and usefulness. He put it very well:

"It's not quite right to describe what the video does as “proving” that 1 + 2 + 3 + 4 + .... = -1/12. When we ask “what is the value of the infinite sum,” we've made a mistake before we even answer! Infinite sums don't have values until we assign them a value, and there are different protocols for doing that. We should be asking not what IS the value, but what should we define the value to be? There are different protocols, each with their own strengths and weaknesses. The protocol you learn in calculus class, involving limits, would decline to assign any value at all to the sum in the video. A different protocol assigns it the value -1/12. Neither answer is more correct than the other."

Here's an explanation by another mathematician:
math.ucr.edu...

One reason bosonic string theory works best in 26 dimensions is that

1 + 2 + 3 + .... = -1/12

and 2 x 12 = 24. Of course, this explanation is unsatisfactory in many ways. First of all, you might wonder what the above equation means! Doesn't the sum diverge???

Actually this is the least unsatisfactory feature of the explanation. Although the sum diverges, you can still make sense of it. The Riemann zeta function is defined by

ζ(s) = 1-s + 2-s + 3-s + ....

whenever the real part of s is greater than 1, which makes the sum converge. But you can analytically continue it to the whole complex plane, except for a pole at 1. If you do this, you find that

ζ(-1) = -1/12.

Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12.

Why do you suppose he says "Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12"?

saneguy
I guess you don't accept Heisenberg's uncertainty principle then.

Heisenberg's uncertainty principle has nothing to do with coin tosses.

saneguy

Soylent Green Is People


The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.

I guess you don't accept Heisenberg's uncertainty principle then.

I do, but not for a toss of a coin.

I suppose if I used some version of the technology behind Douglas Adams' Infinitive Improbability Drive to toss my coin, my coin toss may result in "ostrich", but I'm simply talking about tossing a coin in the everyday world.

Again, I understand the importance of the use of mathematical proofs in the science of mathematics, and I understand why a mathematician might say that the answer to 1-1+1-1+1-1+1-1... could be said to be -1/2, and similarly that the answer 1+2+3+4+5+6+7... could be said to be -1/12, but in reality, the answer to the first sum will always either be -1 OR 0 (and will never be -1/2), and the answer to the second sum will always be positive (and will never be -1/12)...

...just like my coin toss in the real world will always be a heads or a tails, and NOT some probability that involves both possibilities.

Soylent Green Is People

mbkennel
...It works like this:

a) ok, these things don't have limits under classical definition X.
b) so what do we do with them?
a) how about this, in an analogy to other circumstances we can define a different kind of transformation and limiting process in which they do
b) alright, but why should I choose this one instead of another
a) because in other cases, the classical and new ones merge very well, and this permits us to use new tools to reason about whole classes of things we previously had to give up on
b) alright, cool new math, let's hunt theorems!

I agree that it can be "cool new math" and it's fun to consider these things, but the answer to 1-1+1-1+1-1+1-1... is still EITHER -1 or 0. it is never -1/2, unless mathematicians decide they will allow themselves to play a bit "fast and loose" and allow for accounting tricks, such as saying the answer is NEITHER -1 NOR 0, but is actually the average between the two.

To repeat. Every finite partial sum is -1 or 0, which is undisputed. The question is the varying meanings of infinite limits.

You seem to feel that something which you don't understand is immoral or unethical, but it isn't. It's just strange.

Why should a partial sum sequence which has no classical limit, when extended with a more inclusive definition of limit necessarily have a value equal to any of its partial sums?

And yes, sometimes mathematics isn't intuitively obvious, which is why it took hundreds of years of the most preposterously intelligent people to walk the earth to figure it out, instead of taking a quiz at the local pub.

This is one of those cases. In fact, I think it's awesome that geniuses figured out a way to make useful mathematical structure out of what would otherwise be undefined uselessness, and then have that be useful for quantum physics decades later.

I'm all for mathematicians coming up with these interesting proofs and digging down deep into the way they like their science to manifest itself to the world, but the "real" answer to 1+2+3+4+5+6+7... can never be negative.

Mathematics isn't about "real". In any case, this sort of mathematics has been useful and used for making nontrivial physical predictions in quantum field theory, where the appropriate treatment of infinities---meaning considering the finite residuals disputed above as the "real" answer---did turn out to give the physically useful ones. So there's a data point in favor of indulging mathemeticians for a while.

If I decided I would flip a coin, a statistician would tell me that I hand a 1/2 chance (50%) of getting a heads, and a 1/2 chance of getting tails. But the reality of the situation is that I will get EITHER a heads OR a tails; one or the other....

...Therefore, it's not relevant what the odds are for flipping a heads or a tails. The odds don't matter to me, because I will truly get one or the other; that's all I care about, not the odds.

The philosophical underpinnings of probability have been examined for hundreds of years. You're not going to invent a new idea.

The statistician could tell me "Well, the science of probability tell us that it is 50% chance for heads and 50% for tails", but I really don't care about what his science called "mathematics" may tell us. I know it is 100% certainty that I will get one or the other; that's what matters.

And indeed that's why mathematicians long ago developed the notion of random variables and precisely have distinguished events from trajectories from probability spaces etc.

Similarly, When I add together 1-1+1-1+1-1+1-1..., depending on where I stopped, I will get a -1 or 0 100% of the time. There is no other answer. The proof that tells me I get a -1/2 is simply an approximation

You're assuming that an infinite limit in certain kinds of mathematics is trivially related to partial sums, when it isn't, especially when the ordinary elementary limits don't exist.

And yes, mathematics has firmly distinguished events from random variables from expectations from partial sums from limits.

Arbitrageur
I don't see where calculus summations distort the meaning of the equals symbol like summations of divergent series do. Divergent series aren't really sums, as explained above and below:

The bodies are buried in the "+ ..." part of it. The 'crimes' committed by the "=" are minor in comparison.

In the ramanjuan summation (presumably an alternative to the zeta function business) the

divergent series = C (Ramanjuan sumamtion)

really means C + some peculiar kind of infinity, but the interesting part is in C.

Why do you suppose he says "Thus we may jokingly say that 1 + 2 + 3 + .... = -1/12"?

Jokes are about the absurd and unexpected. And of course the guy knows it's an extension to the elementary definitions.


Anyway I found this on another message board about this very same question:

boards.straightdope.com...

This thread having been bumped recently, I thought I would do two things.

Thing #1: I'll recap the simplest, most layperson-friendly answer to the titular question (which, although present in the previous discussion, might not ever have been made as explicitly clear as possible):

Specifically, the sense in which 1 + 2 + 3 + 4 + ... = -1/12 is this:

First, consider X = 1 - 1 + 1 - 1 + .... Note that X + (X shifted over by one position) = 1 + 0 + 0 + 0 + ... = 1. Thus, in some sense, X + X = 1, and thus, X = 1/2.

Now consider Y = 1 - 2 + 3 - 4 + ... . Note that Y + (Y shifted over by one position) = 1 - 1 + 1 - 1 + ... = X. Thus, in some sense, Y + Y = X, and thus, Y = X/2 = 1/4.

Finally, consider Z = 1 + 2 + 3 + 4 + ... Note that Z - Y = 0 + 4 + 0 + 8 + ... = (zeros interleaved with 4 * Z). Thus, in some sense, Z - Y = 4Z, and thus, Z = -Y/3 = -1/12.

In contexts where the above reasoning is applicable to what one wants to call summation, we have that 1 + 2 + 3 + 4 + ... = -1/12. In other contexts, we don't.

Soylent Green Is People
I agree that it can be "cool new math" and it's fun to consider these things, but the answer to 1-1+1-1+1-1+1-1... is still EITHER -1 or 0. it is never -1/2, unless mathematicians decide they will allow themselves to play a bit "fast and loose" and allow for accounting tricks, such as saying the answer is NEITHER -1 NOR 0, but is actually the average between the two.

mbkennel
To repeat. Every finite partial sum is -1 or 0, which is undisputed. The question is the varying meanings of infinite limits.

You seem to feel that something which you don't understand is immoral or unethical, but it isn't. It's just strange.

Again -- I said I understand why the science of Mathematics needs to look at problems the way they do. I didn't ever say it was immoral or unethical. I just said it was not the practical way of looking at it for the rest of the world (non mathematicians). I'm not saying it is BAD thing for the science of Math to think of the problem in this manner -- but not everyone needs to use math like a math scientist (a mathemetician) does.

Soylent Green Is People
I'm all for mathematicians coming up with these interesting proofs and digging down deep into the way they like their science to manifest itself to the world, but the "real" answer to 1+2+3+4+5+6+7... can never be negative.

mbkennel
Mathematics isn't about "real". In any case, this sort of mathematics has been useful and used for making nontrivial physical predictions in quantum field theory, where the appropriate treatment of infinities---meaning considering the finite residuals disputed above as the "real" answer---did turn out to give the physically useful ones. So there's a data point in favor of indulging mathemeticians for a while.

I agree, and that's why I said a few times that I understand why the science of mathematics uses these sorts of proofs and thought experiments. It can be useful.

Soylent Green Is People
If I decided I would flip a coin, a statistician would tell me that I hand a 1/2 chance (50%) of getting a heads, and a 1/2 chance of getting tails. But the reality of the situation is that I will get EITHER a heads OR a tails; one or the other....

...Therefore, it's not relevant what the odds are for flipping a heads or a tails. The odds don't matter to me, because I will truly get one or the other; that's all I care about, not the odds.

mbkennel
The philosophical underpinnings of probability have been examined for hundreds of years. You're not going to invent a new idea.

I'm not trying to invent a new idea. I'm simply stating that while a mathematician may say that the answer could be said to be -1/12 (and I fully understand WHY a mathematician may say that) it is still not the practical answer.

There is the science of Mathematics, and then there is math for practical everyday use. I'm just stating that the practical answer tells me that adding only positive whole integers will never result in a negative sum (or a whole number)...AND the sum of 1-1+1-1+1-1... will either result in a -1 or a 0, and not something in between. There is no in-between state when considering the practical answer. The practical everyday answer is either -1 or 0, just like the practical result of a coin toss is heads or tails, and not some hybrid probable function between those two states.

If a mathematician wants to say "yeah, but there are ways of approaching this problem that will result in the answer being -1/12, and those approaches are useful to science", I'll say fine -- I agree. By all means explore approaches to math that can be useful. However, I will still argue that adding only whole positive integers together will never -- in practicality -- result in an answer of -1/12.

WeAre0ne
So let me get this straight...

If you give me $1, and someone else gives me $2, then another person gives me $3, and other people keep giving me money in the same pattern, forever (infinitely), and I don't spend any of it...

I would end up $-1/12 in debt?

That is the most idiotic thing anyone has ever postulated.

YUP! that's the logic
(nope, not really)

This stupid equation defies the very concept of what infinity is in the first place. Basically what they've said is that infinity is equal to -1/12. So they've gone ahead and made infinity finite, and not only that- but a negative! Since when do we add positive whole integers and get negative fractions for results! That's David Blaine stuff right there! And everyone is in on it, even the camera man and the numbers themselves.

It has to be a goof right?

mbkennel
Jokes are about the absurd and unexpected. And of course the guy knows it's an extension to the elementary definitions.

Words have meanings, as do symbols. Jokingly means one thing. Absurd means something else. Unexpected means something else.

A Ramanujan sum is not the same as a zeta regularized sum which is not the same as a sum in calculus. If we want to communicate clearly we can use different symbols to indicate when we are talking about different things which I illustrated with the wikipedia notation of Ramanujan sums. The context of Baez's comments includes references to several ways in which the handling of the divergent sum is unsatisfactory, so I think if he meant unexpected he would say unexpected, not "jokingly".

Anyway I found this on another message board about this very same question:

I'm not sure if you're aware of the ruckus that occurred after the first video numberphile was posted, but part of the reason a second video was posted is that the first video wasn't right, and that straight dope except has the same problem, it promotes mathematical illiteracy:

No, the sum of all the positive integers is not -1/12

The flaw in both cases is the same: the algebraic rules that apply to regular numbers do not apply to infinity. Actually, it's more general than that: the algebraic rules that apply to regular numbers do not apply to non-converging infinite sums. All of the sums above are non-converging infinite sums, so regular algebraic rules do not apply. It is no different from using regular algebra when dividing by zero. It doesn't work.

Now, there are ways to define the sums of non-converging infinite series so that they do not lead to contradictions. The one that leads legitimately to the conclusion that 1 + 2 + 3 + 4 ... = -1/12 is called Ramanujan summation, which in turn is based on something called an analytic continuation. But the problem is that the Numberphile video makes no mention of this. They present the result as if it is legitimately derivable using high school algebra, and it isn't. Telling people that it is does a grave disservice to the cause of numerical literacy.

The second numberphile video that was posted was intended to explain the result more rigorously, and not using the type of "math" that can "prove" that 0=1 like the first video did.

sum = 0;
x = 0;
while(true)[
x++;
sum += x;
]

Tell an experienced programmer that the value of "sum" will become a signed decimal at some point during runtime, and not remain an unsigned integer, and he will have a good laugh in your face.

WeAre0ne
sum = 0;
x = 0;
while(true)[
x++;
sum += x;
]

Tell an experienced programmer that the value of "sum" will become a signed decimal at some point during runtime, and not remain an unsigned integer, and he will have a good laugh in your face.

edit on 24-2-2014 by WeAre0ne because: (no reason given)

So you have shown that programming is not mathematics.