Does 1+2+3+... Really Equal -1/12?
blogs.scientificamerican.com
blogs.scientificamerican.com
You can define a function f from a subset of the complex numbers to the complex numbers where f(z) = \sum_{n=1}^\infty 1/n^z. Be careful with the domain of this function: the series does not converge for all z. You can plug in -1 and see that symbolically, f(-1) = 1 + 2 + 3 + .... But the series does not converge for z = -1, and it is simply not true that \sum_{n=1}^\infty n = -1/12; the series does not converge; equating it to something is a nonsensical thing to do.
What is going then? Even though f is not defined for all complex numbers, there exist functions from the complex numbers to the complex numbers that -- restricted to the domain of f -- are equal to f. They "continue" f to all of the complex numbers. And if one imposes a restriction on these continuations (namely that they are analytic), then it turns out that there is a unique analytic continuation of f: the Riemann zeta function. And zeta(-1) = -1/12.
Don’t confuse the definitions here. The Riemann zeta function can be defined as the analytic continuation of the series, the series is not defined in terms of the Riemann zeta function!
Thus g(-1) is 1 + 2 + 3 + ... while g(z) is otherwise 0 + 0 + 0 + ... = 0. And then the zero function is the unique analytic continuation of g.
The reason people care about the Riemann zeta function is because of its deep connections to analysis, number theory, and physics.
(a <*> b)(n) = \sum_{k | n} a(k) * b(n/k)
However, it is difficult to get asymptotic estimates for the coefficients of a series by purely algebraic means. This is the first and last time that complex valued functions enter the picture, but it's a very neat trick. Let's consider two series a, b and define the functions
A(s) = \sum_{n >= 1} a(n) * n^-s
B(s) = \sum_{n >= 1} b(n) * n^-s
A(s) * B(s) = \sum_{n >= 1} (\sum_{k | n} a(k) * b(n/k)) * n^-s = \sum_{n >= 1} (a <*> b)(n) n^-s
Unfortunately, the world is not quite this simple. The functions we are mapping into typically aren't very well behaved and usually aren't defined on large parts of the complex plane (consider a(n) = n^n). This means that all of our nice tools from complex analysis actually won't work very well!
The whole idea behind "analytic continuations" is that we aren't actually using this mapping! We are constructing a different (partial, injective) ring homomorphism from arithmetic functions to meromorphic complex functions.
The idea behind this is that meromorphic functions are rather restricted in what they can do. In particular, there is at most one meromorphic function A with A(s) = \sum_{n >= 1} a(n) n^-s for s with Re(s) > k, for some k. We define our mapping from sequences to functions by mapping the sequence a(n) to the meromorphic function A(s) with A(s) = \sum_{n >= 1} a(n) n^-s for Re(s) > k for some k, if this function exists.
By the same argument as above, this is a ring homomorphism, and since it is injective we can still use information about the functions to gain information about the underlying sequences.
For example, the Riemann zeta function is not really defined by the equation Zeta(s) = \sum_{n >= 1} n^-s. It is defined to be the unique meromorphic function with Zeta(s) = \sum_{n >= 1} n^-s for all s with R(s) > 1. In particular, Zeta(-1) has nothing to do with \sum_{n >= 1} n. The latter expression doesn't define a complex number at all, but for the former it is not too difficult to show that Zeta(-1) = -1/12.
The main advantage, though, is that meromorphic functions are very well behaved. This allows us to use Cauchy's residue theorem and Mellin transforms to get very deep results about the underlying sequences. If you play this game with the "von Mangoldt" sequence you can, for instance, derive an asymptotic bound on the density of the prime numbers. This is a surprisingly simple derivation, given that this problem had the worlds greatest mathematicians stumped for a hundred years!
Summing up, the mapping or "continuation" you use is choosen (!) so that multiplication of functions corresponds to your chosen multiplication in the ring of sequences and so that you get functions which are as well-behaved as possible. There is a large design space here, and you can find different "analytic continuations" for a given sequence.
It makes one wonder if there can be a better theory then that doesn't require magic. There is at least one book I know of (before I became a plasma physicist) which apparently avoids the regularization altogether, but people still do regularization like this in particle physics.
A bunch of people had issues with the Numberphile episode where these restrictions where elided (due to time, I'd imagine), but they had another episode featuring the zeta function and a math professor (rather than physics professors, who were featured on the one that went viral) explaining the summation with more context: https://www.youtube.com/watch?v=0Oazb7IWzbA.
lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),
that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form
sum = 1/epsilon - 1/12 + ...
Crucially, there was another term in the calculation that naturally appeared that canceled the 1/epsilon. Without that other term, the sum would of course be infinite when epsilon -> 0.
This is much simpler than analytic continuation through the complex plane, and again, this is how the physics calculation appears in QFT courses. There is no need to appeal to complex analysis here, which leads to all of this mysticism and confusion.
So how exactly is it that by multiplying a regular sum of positive numbers by a decaying exponent do you get a negative number when you take the limit?
Funny, I thought the entire point of this 'simplification' was not to be mystical and hand wavey.
1) You introduce a family of series, parametrized by epsilon, whose terms, for small epsilon, closely approximate your original series. Sort of; obviously for large n they don't. But the idea is that if you pick any N and delta you can pick epsilon such that for n < N the approximation is within delta of the actual terms.
2) You show that the series in this family can all be summed and the sum of each one is 1/epsilon - 1/12 + O(epsilon).
Now of course what this means is that the sums blow up as you approximate your original series better and better, since 1/epsilon gets large. That's good, because your original series totally diverges off to infinity. ;)
The part after this point I'm less clear on, but it sounds like in the computation involved what you actually have is your (divergent) series 1+2+3+... plus some _other_ (also divergent, going off to negative infinity) stuff. And that you might be able to arrange things such that the other divergent stuff looks like -1/epsilon, cancels out the 1/epsilon from your approximation, and you come out with the sum of the two things being -1/12.
The obvious issue here is that once you start adding up divergent things by rearranging terms and telescoping you can come up with whatever answer you want: see <https://en.wikipedia.org/wiki/Riemann_series_theorem>. So this procedure all only makes sense if there are some sort of fundamental reasons to think that this particular rearrangement is the "right" one in some sense.
The issue is, of course, the illegitimate manipulation of a diverging series, which was the exact issue that prompted the original article (due to Numberphile doing it) in the first place.
Now e^(-x) is a totally bogus "cutoff function" per the definition in Terry's blog post, since it is not compactly supported, but it _is_ bounded, _does_ equal 1 at 0, and drops off fast enough that for practical purposes it can be used to do smoothed sums. In particular the smoothed sums will converge for most cases (e.g. anything where the sequence we're "summing" has at most polynomial growth will do so), which means you can at least try to do the rest of the analysis. I suspect, but have not checked, that the other places where compact support is used in his presentation also work out for the sorts of sequences we're talking about.
Either way, the upshot is that in some sense you have some sequence of approximations to your "actual" sum, indexed by N, and you show that for large N they all look like "power series" in 1/N which allows some finite number of negative exponents and all the approximations have matching coefficients for the negative exponents and the same constant term. And then you compute that constant term. Calling that the sum of the series is nonsense, of course, but it can still give you interesting information about something, maybe.
Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to.
edit: For fun, a short story:
If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.
My mountain example obviously couldn't happen in the physical world. I suppose in that case you might as well substitute the infinite value for an arbitrary large one. Which is not really what infinite values are about in mathematics, as they are more about describing the (imaginary?) limit of a divergent series.
I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. But for a layman infinite values and infinite series are interpreted as larger than any value you can come up with, and more than any repetition you can write down. So any explanation for this equality should, I think, involve first deconstructing that.
> I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value.
I'm a mathematician and I might agree (not entirely sure what you mean). But -1/12 is a concrete value so it doesn't apply here.
In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate).
https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C...
and
https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...
lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),
that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form
sum = 1/epsilon - 1/12 + O(epsilon).
The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.
Nevertheless, there are infinite sums of "real" things in physics too. I have put "real" in scare quotes because it turns out they aren't real :)
In quantum electrodynamics, the charge of an electron turns out to be infinite. And it turns out that in Real Life, the charge of an electron is indeed infinite. Ish.
... but we know it isn't, right?
So what happens is that the real electron gets surrounded by positively charged "virtual" particles. Virtual particles are basically quantum probabilities of a particle appearing out of nowhere with its antiparticle (among other things). So you can say that with some probability, that particle is there. Since there's an electron nearby, the positively charged particle is attracted to the electron, while the negatively charged antiparticle is repelled. This screens the electron charge. With an infinite number of virtual particles, the electron's charge is screened enough to become finite again. Basically, we subtracted two infinities and got something finite. The subtraction done here is called renormalization -- and a similar thing is being done in the -1/12 sum. While mathematics tells us that divergent series can be rearranged to get any "sum", this trick is often used in physics -- provided you can justify that rearrangement.
In fact, if you probe an electron hard enough (by bombarding it with other charged particles with tons of energy), its apparent charge increases since the particles used to measure its charge "pierce" the shielding.
Of course, this is all really a fancy way of saying that charge itself is energy-dependent, and what we call charge is actually the 0-energy charge.
But for modelling purposes, virtual particles work better, and thinking about things in those terms gives a physicist a cleaner abstraction boundary to deal with. You get infinities everywhere, though.
This is basically an example of the pattern I'm talking about. Abstractions in the model may have all kinds of infinities popping up. In the real world, these don't really manifest themselves because they're not directly linked to observables. You can apply your model to your detection mechanism to get values for non-observables and say "hey, look, an infinity", but that's really circular logic. The "Real Charge" of an electron isn't something we see. Virtual particles aren't something we see; unless we make them into real particles, but you can't do that to the infinite virtual particles around, so you'll never see an infinity.
There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. The answer is definitely not positive infinity, as that is not a number and the sum of integers must be an integer.
The article points out a few times that such a "sum" is undefined.
This number system was used to invent calculus, and worked just fine for over 150 years despite theoretical unsoundness. And it turns out that it's possible to formally define a provably consistent number system that obeys our intuition, the hyperreal numbers. See:
I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete.
I also still claim that the result cannot be positive infinity due to the definition of addition on integers. The result of addition of integers must be another integer and positive infinity is not an integer.
I do agree with the article however, that one can take the limit of a well-defined infinite series; but the limit is not the sum, only the bound that will never be exceeded no matter how long you are able to continue adding numbers for.
I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected.
This not something you can agree or disagree on. You can make physical calculations with this result and get a prediction that you can measure and confirm. This result is sound.
In this case you might find the -1/12 useful, but have the opinion that they really should not be using '=' as a shorthand for what they're doing with the zeta function.
Also, yes, you can't literally compute an infinite sum but among any crowd that has likely taken calculus 1, you can place implied limits. :P (which, arent actually needed in the hyperreals because it HAS infinity, but whatever)
That same argument gives you Zeno's paradox.
You can sum a pattern of numbers in O(1) time if you use logic instead of brute force. It doesn't matter if physically spending O(n) time on something is impossible when you only need O(1).
I believe Zeno's paradox is on the rationals, not integers.
>You can sum a pattern of numbers in O(1) time if you use logic instead of brute force.
There may be closed solutions for finite summation patterns, but infinite summation patterns of integers have no solution (by definition).
I agree. Infinity is a process that can yield a number but is not an actual number and, Cantor et. al. notwithstanding, there is no such thing as a "completed infinity" other than terminating it at a finite step. If you are careful and in certain contexts you can use the "limit" of an infinite converging process but you must make that assumption explicit to avoid errors.
All these bizarre math tricks rest on treating it as a number when its undefined. Its like those puzzles I read as a kid that "prove" 1=0 and they typically depend on an implicit division-by-zero step which is also undefined just like infinity. Once you start working with the undefined you have to very careful and even Gauss made errors when he was laying the groundwork for infinite series. To the degree that this math has ANY validity it is in the context of some esoteric and specialized area of math and it is NOT appropriate to foist it on the general public as a general result. The motive in such attempts is to impress or intimidate or destroy math (nihilism) which I find despicable.
for example on a circle of circumferance 15, 14 is also equal to -1, 13 is equal to -2 etc.
which says the sum of 1+2+3... averages to -1/12
Why do you think otherwise?
But now you've moved away from talking about the integers mod N, which is what zeroer was talking about[0] which was in response to you talking about the numbers wrapped around a circle[1]. Their response to you seemed reasonable, but I don't understand why you leaped to talking about real numbers, nor why you claim that 1.2 is not equal to -0.8 when working modulo 2.
The point of the construct is it applies to a circle of arbitrary/unknown length 14,12.2, 500,000 million billion point 6 light years.
and the average of the sum of 1+2+3+4+... will be/tend to -1/12
and its easy to test, just use a signed x bit number.
> ... to get 0.5 for the sum of
> 1-1+1-1+.... requires real numbers.
But it's clear now that you're not really talking about maths at all, so the comment about existing, established theory about modular arithmetic doesn't really help. You seem to be doing something, well, different.
And regardless, in the long-established theory of modular arithmetic, 1.2 is equal to -0.8 mod 2, regardless of you claiming that it's nonsense.
So at this point I have no idea what you're talking about.
There are two ways of constructing a number line from -inf to +inf
The first, is that "nothing exists" to the left of -inf or to the right of +inf The other, more useful, is that -inf=+inf+1 and +inf=-inf-1, (or -inf=+inf, never remember which is the more useful) and they form a loop. Such as that constructed by a signed integer. e.g. e.g. with an 8 bit number 127+1 = -127
This has nothing to do with modulo afaik. (but all the basic construct stuff is related) and is more to do with every dimension being curved in another (meaning they always form such loops)
It's not surprising, you can pretty much get any result you want like this.
Proof:
S1 = 1 + 2 + 3 + 4.....
S2 = 0 + 1 + 2 + 3 + 4...
S1 - S2 = (1 + 2 + 3 + 4 + ...) - (0 + 1 + 2 + 3 + ...) = 1 + 1 + 1 + 1....
S2 == S1 (by definition, since all you're doing is adding a 0) => S1 - S2 == 0
Therefore 0 = 1 + 1 + 1 + .....
Obviously this is pure nonsense. You can't just "shift things around" and use elementary mathematics when dealing with infinite series that don't converge. Maybe there's a more convincing proof out there, but the one they presented in the video is bogus.
This is begging the question. Why can't 1 + 1 + ... = 0?
Also, I wouldn't be so sure that S2 == S1. You can't re-arrange infinitely many terms in an infinite series and still be guaranteed the sum is the same.
Let S1 = 0 + 0 + 0 + ... = 0 Then surely, S1 = (1 - 1) + (1 - 1) ... = 1 - 1 + 1 - 1 + ... -1 + S1 = -1 + 1 - 1 + 1 - 1 ... = (-1 + 1) + (-1 + 1) ... = 0
But then -1 + 0 = 0
In first series, there are 2 different elements (1 and -1) and series can end at any one of them rendering the end result of the sum uncertain. On second one there is only one element - (-1 + 1) which is 0, so wherever you end it the result is always the same.
To be clear, yes
(1 - 1) + (1 - 1) + ... != 0 + 0 + ... either
> (1 - 1) + (1 - 1) + ... != 0 + 0 + ... either
I don't see why. I understand the sets themselves are not equal, but the sum of the elements of those sets is at any given index.
I don't know if I would go that far... but I agree with the general spirit of your comment. Which is also the point of my original post. If you think that my appending a zero calls my proof into question, the proof presented in the video takes far more dubious and horrific liberties.
I can see some differences: for any finite N > 0, it's false that the sum of the first N terms of S2 is the same as the sum of the first N terms of S1. And what do we know about the infinite sum? Maybe you're right, but you'd have to prove it; they are definitely not equal "by definition"!
However, to the normal viewer this video probably made maths look incredibly interesting and more than likely even caused them to research it a bit more. I would hazard to say an article like the one Dr. Lamb wrote would not have that same effect, though it is technically more correct. Numberphile to me is more about reinstating the interest in maths in a society where you are usually introduced to the topic by doing repetitive, seemingly impractical calculations and this video of theirs as referenced in the article has definitely done that.
Just like a good host at a party, a good educational video should leave the viewer felling positive and better about themselves. So clearly, in this case, it depends on the audience.
Is the purpose of education to make the other person feel positive and good about themselves?
The people in the Numberphile video may well have misjudged their audience, but I do not believe there is a "second level of ignorance" here, or even a first level of ignorance. At least, not about the mathematics.
The people in the Numberphile video know full well all the underlying details of the mathematics, rigorous and informal. Like may others, I believe they badly misjudged the content and approach. I agree entirely with others that there are many people who have watched this and thrown their hands up and declared "Just proves I'm crap at maths and will never understand it." And that is unfortunate, which is why I personally believe it was a poor decision to make the video and explain it the way they did.
But I don't believe it's fair to level an accusation of ignorance at them.
In any case, I have talked to many physicists who have expressed to me their desire to better understand a topic like, say, differential geometry, but abhor the idea of actually reading a proof or studying a precise definition. In their words, "I only want an intuition for the subject." An understandable desire, but now my bar for believing a physicist when they make a mathematical claim is quite high. This video isn't helping them any.
To be sure, there is a right way to do it, a simple practice that the men in the video could have employed to appease everyone: when you do something egregiously false, mention that strictly speaking you're not allowed to do it, and that you're actually sweeping a lot of complexity under the rug, and then continue anyway. This is the opposite of the blind appeal to authority in the video, and maintains their integrity.
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.From there it's not hard to imagine that the -1/12 result could be useful and justifiable as an intermediate step in other calculations despite being an "impossible" destination on its own.
1 + x + x^2 ... = 1/(1-x)
Plug in x = 2 to get:
1 + 2 + 4 + 8 ... = -1
There's a million of these series in your dusty old Calculus textbook. Or you could look in 'generatingfunctionology' to find others.
Complex analysis makes it more interesting, because the additional Cauchy-Riemann constraints make the solution unique. And so people are more willing to say that the unique solution is the "true" answer.
Really, I say this is just another example of the complex numbers being weird. I took a couple different courses in it(one in college, 2 online) because it was clear something interesting was happening with them, but there was never a unifying theme. I can kind of spot a rule like "analytic functions preserve 90-degree angles" in the Cauchy-Riemann equations, but it hardly explains all the crazy theorems.
That series is just another example of someone taking a well-behaved series, analytically continuing it into the complex numbers, and now it's clear something interesting is happening but it's not clear what.
Anybody who understands the definition of convergent series.
> That series isn't just a cute bit of trivia, it is the basis for signed integer mathematics in your CPU. "True" enough for you?
Eh. That's only half true. It's related, certainly, by means of a similarity between 2-adic numbers and two's complement representation, but I'd hardly call it the basis for signed integer mathematics.
* Define the series over the 2-adic numbers rather than the reals
* Associate every 2-adic integer with its corresponding sequence in the inverse limit Z/2^nZ for n = 1..infinity
* Truncate the sequence to a certain precision(an index i), and make an equivalence class a ~ b if a and b agree on the first i elements. You get a field Z_(2^n), and identities in 2-adic analysis should carry down to the equivalence classes. Here 1 + 2 + 4 ... = -1 in the 2-adics.
That isn't the way I would've thought about signed integers, but it seems like it would work.
I.e. physics.
http://www.claymath.org/sites/default/files/ezeta.pdf
This series is ζ(-1). I don't know about the Youtubers or Scientific American, but mathematicians studying Complex Analysis know exactly what it means.
It happens a lot in fairly serious technical computing blog posts and I've been trying to wrap my head around why people do it.
Aaarrrggghhhh !!!
WALL OF TEXT !!!
Aaarrrggghhhh !!!
Disappointed, but not surprised.
Nevertheless, I feel the same way. I wish people would use simple diagrams relevant to the discussion (like the 1/2 + 1/4 + ... = 1) instead of overused macro images.
Source: https://en.wikipedia.org/wiki/Betteridge%27s_Law_of_Headline...
EDIT: Some further elaboration: I'm sick of the question. The answer is: not in any sense that would be meaningful to the people to whom this stuff is being told. You're just being misleading by implying that the sum of positive integers can converge. I don't want to hear any "But if you take this analytic continuation..." or "But in a certain sense...", they're just misleading as the thousands of proofs that 1=0.
Shifted version:
1-2+3-4+5-6 ...
1-2+3-4+5 ...
sum: 1-1+1-1+1-1 ...
Multiplied version:
2-4+6-8+10-12 ...
What if we triple shift?
Triple shifted version:
1-2+3-4+5-6+7-8+9 ...
1-2+3-4+5-6 ...
sum: 2-3+3-3+3-3+3 ...
Your triple shifted version would be between -1 or 2 depending on the cut right? So, still 1/2.
What about the multiplied version? That is how I would intuitively understand 2S_2, and I still don't accept that shifting is the same.
To me it's one of those things where you just go 'damn' because of the perplexing relations that exist between math and physics. If anything hints at what the hell goes on in this universe, to me it's stuff like this.
Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.
What does it even mean to exist?
Far better to talk about whether a number is defined in a particular numerical system.
In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird.
The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable.
Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information.
In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination.
If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.
Similarly weird feeling is encountering zeta(-1) = -1/12 after years of calculus teachers telling you to ignore divergent sums because they are infinite.
When you're first learning about imaginary numbers in 8th or 9th grade, the answer to "what is sqrt(-1)?" _should_ be undefined. If you claim otherwise, you're pulling the rug out from under their feet, because the number system that they are familiar with indeed has sqrt(-1) undefined.
Instead, the teacher should go on to introduce a new system of mathematical objects that have certain rules, and the students could play around with them and see how they have two components, how you can plot those two components in 2 dimensions, how you can think of them as arrows sticking out of the origin, how you can combine their components to rotate each other, etc. Then work backwards into showing that we can call these objects complex numbers for short, because those operations are similar to addition, multiplication, etc. And finally, just as a curiosity, you can see that sqrt(z) = i for z = -1 + 0i.
There's no need to introduce this whole concept of an imaginary number line that points off in a direction nobody can see or measure. The whole takeaway should be that you can't just square real numbers get negatives. If you have something that can "multiply" by itself to get its own inverse, then you have either overloaded the multiplication operator with something very very different, or you're dealing with an object that can "rotate" through another dimension. It's an ordinary two dimensional space, and the only difference between the two axes is their name, just like "x" and "y". In my opinion, this lesson should actually be reassuring to a young mathematical intuition: there's only so many ways to skin this cat.
And here's some "ancient" HN commentary on that article: https://news.ycombinator.com/item?id=2712575
I think it's Very Not Good to see a totally unplausible mathematical result in physics and take it as anything other than an open problem that needs more work. It's okay to be amazed by it, but it's not okay to say "sure, okay, let's just leave this as it is".
There are stuff like this https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7..., but writing 1+2+3+... = -1/12 is purely formal and pretending to deduce it from physics is silly because you are not performing any sum at all.
The trick step of assuming that 1+0+1+0.... converges is also not Cauchy.
It's interesting and string theory and some other physics get results by using -1/12 but it's not strictly correct. No more than the omni proof
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("The" is the appropriate article to use because the analytic continuation of a function is unique.)
Let f: N -> N, f(x) = x be a function on the natural numbers. Then I could define two functions g(x) and h(x) on N u {foo} that behave just like f for natural numbers. However, g(foo) is 42 while h(foo) is 666. Wouldn't g and h both be valid analytic continuations of f in N u {foo}, according to the definition of analytic continuations explained in the article?
I'm wondering about that as the uniqueness seems to be an important property for the rest of the explanation, yet it is simply assumed here without any further explanation.
That is a non-trivial theorem/lemma though, and the proof is typically gone through at a graduate complex analysis course.
The fact that the unsound reasoning in this particular case led to a conclusion that superficially resembles a conclusion that can also be arrived at by sound reasoning just makes it that much worse. It encourages people to think: because this mode of reasoning led to a "correct" conclusion in that case, then it will probably lead to correct conclusions in other cases.
If the problem were confined to mathematics I might not make such a big deal out of it, but it's not. The problem of people uncritically accepting conclusions drawn by unsound methods of reasoning pervades our society and causes real damage.
http://blog.rongarret.info/2014/01/does-it-matter-if-sum-of-...
Because as far as I can tell (with my admittedly limited understanding of mathematics), that's basically what's going on here: they define the result of 1-1+1-1+1-1... to be 1/2, which it can of course never be. The result is never 1/2; it's either 1 or 0. Taken to infinity, the only reasonable definition for that sum is undefined. If I can say that 1/2 is fine too, then I should also be able to attach my own definition to 1/0.
Also, if string theory really relies on such questionable mathematical steps, then that would make me question string theory even more. As far as I understand, string theory makes no testable predictions, which suggests to me that no results based on this questionable mathematical trick have been experimentally verified. If there is some real, experimentally verified physics that relies on the sum of all natural numbers to be -1/12, then I'd love to be corrected (though I doubt I'll understand it).
https://terrytao.wordpress.com/2010/04/10/the-euler-maclauri...
It goes into detail and shows the derivation and internal consistency of the method.
I would go as far as to say that just as the Uncertainty Principle prescribes a limit on what is knowable in quantum physics, so does philosophy suggest limits on what will ever be rationally proven through human perception, ideas, and knowledge.
I don't think it's bad to say that 0^0=1 when you hear the whole story, which is that there are multiple right answers given how we define exponents and operations on 0. But it's misleading to say 0^0=1 and stop there. Similarly, after reading about this today, it's making more sense to me that we can, if we choose, define 1+2+3+...=-1/12, and it makes sense in some contexts. It's just that this isn't the only answer and it isn't the whole story.
I liked the article, and ended up reading a bunch of Evelyn's column earlier today. I see some people complaining about the pictures... I thought they were funny and relevant, my only nit pick is I can barely read the light blue text on the wink gif, the colors are so hard for me to look at.
> "The sum of the series 1+2+3+4+5+6... = -1/12" is patently false, without a previous assertion that we have assumed the Cesàro sum of a series is equal to the series. Even mathematicians working with Cesàro sums surround such statements with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..." [0] Precisely none of the times I've heard the "1+2+3+4...=-1/12" bullshit has the person stating it prefaced their statement with "this holds only if we interpret the infinite sum defining Z to be the Cesàro sum..."
> If you say that "1+2+3+4...=-1/12" without stating your prior assumptions, you suddenly allow anyone to make any assumption whatsoever, no matter how obscure it is. In your imaginary world, someone could walk into a store and claim that "this 95 cent pack of gum is free" because they just made the unstated assumption that all non-integers do not exist, and seconds later they could return it for a full refund of $0.95 after making the unstated assumption that in fact the rational numbers do exist. Numbers, and in fact the entire system of mathematics fail to work at all once you allow arbitrary, unstated assumptions no matter their obscurity. And in fact, the assumption that non-integer numbers do not exist is made far, far more frequently than the assumption that the infinite sum defining the sequence is the Cesàro sum.
> The only difference is that assuming the non-integer numbers do not exist is a defensible assumption in many, many scenarios... but Cesàro summations are only invoked about twelve times a year, in pure math or advanced physics papers.
> [0] Madras, Neal. "A Note on Diffusion State Distance." arXiv preprint arXiv:1502.07315 (2015).
My favorite post on the subject still has to be this: http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-...
Why does an average of the two values that you can converge on help?
Also, what about something like:
sin(π/2) + sin(3π/2) - sin(5π/2) + sin(7π/2) - ...
what would this be?
More precisely, the sume is 0.5+/-0.5, which is greater than 0 and less than 1 and ambiguous in between.
I'm going to copy and paste the explanation I originally wrote at Quora (https://www.quora.com/Whats-the-intuition-behind-the-equatio...), because I think it captures well everything I'd like to say about this at every level of the discussion:
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 so, in some sense, 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 so, in some sense, 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 so, in some sense, 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.
That's it. It's that simple. Everything else I'm going to say is just to comfort those who are uncomfortable with the game we've just played.
Note that I've said "in some sense" several times in the above argument. That's because, while we all know how to add and subtract a finite collection of numbers in the ordinary way, when it comes to adding and subtracting an infinite series of numbers, there are many different ways of interpreting what this should mean. Just knowing how to add finitely many numbers doesn't automatically tell us what it means to add a whole infinite series of them. And when it comes to summation of infinite series, it turns out there's not just one nice notion of "summation"; there are many different ones, which are nice for different purposes.
One such notion is "Keep adding things up, one by one, starting from the front, and see if the results get closer and closer to some particular value; if so, that value is the sum". On that account of what summation means, you clearly won't get any finite answer for 1 + 2 + 3 + 4 + ...; since the terms never get any smaller, the partial sums will never settle down to a finite value (and certainly not a negative one like -1/12!). They instead, in a natural sense, should be understood as summing to positive infinity.
And there's nothing wrong with this! You are not wrong to feel that 1 + 2 + 3 + 4 + ... is positive and infinite, and math does not deny this; there absolutely is an account of summation corresponding to this intuition.
It's just not the only account of summation worth thinking about.
We could instead consider other notions of "summation", including ones designed precisely so that arguments like the one we made at the beginning (which are very natural arguments to make!) counted as legitimate ways to reason about such "summation". And then, by definition, we will have that 1 + 2 + 3 + 4 + ... = -1/12, on such accounts of "summation". (In doing so, we will lose certain familiar properties such as "A sum of positive terms is always positive". But this is how generalizations work; generalizations very often lose familiar properties. Even the textbook, limit-based account of infinite summation loses familiar properties like "The order of summation doesn't matter". Even finitary summation of integers loses the familiar property "If a sum is zero, so are all the summands" from basic counting. But there is a web of resemblances to more familiar kinds of summation which can justify, in certain moods, thinking of each of these generalizations as a form of summation itself.)
If you insist that "Keep adding things up and see if the results get closer and closer to some particular value" is the only account of summation you're interested in, you'll object to the argument we gave at the beginning, saying "You're not allowed to do that kind of shifting over and adding to itself reasoning all willy-nilly; look at what nonsense it produces!".
But it can be made sense of, and is even fruitful to make sense of, in certain contexts in mathematics, and there is no need to blind ourselves to this insight.
Again, that's it. It's that simple. Everything else I'm going to say is just to comfort those who are still uncomfortable. For those who want a more systematic, formal account of series summation of a sort which validates the above manipulation, read on:
[Comment too long, will be continued in reply]
We can look at it this way: We can try to assign values to a non-absolutely convergent series by bringing its terms in at less than full strength, producing an absolutely convergent series, and then increasing the terms' strengths towards full strength in the limit, observing what happens to the sum in the limit as well.
This is the idea behind the traditional account of series summation, mind you: at time T, we bring in all the terms of index < T at 100% strength and all other terms at 0% strength. This gives us our partial sums, and as T goes to infinity, each term's strength goes to 100%, so we can consider the partial sums as approximating the overall sum.
But we don't have to be so discrete as to only use 100% strength and 0% strength. We can try bringing in terms more gradually. For example, rather than having strengths discretely decay from 100% to 0% at some cut-off point, we can instead have the strengths decay exponentially in the index. (So at one moment, we may have the first term at 100% strength, the next term at 50% strength, the next term at 25% strength, etc.). Then we consider what happens as the rate of exponential decay slows, approaching no decay at all.
In symbols, this means we assign to a series a0 + a1 + a2... the limit, as b approaches 1 from below, of a0 * b^0 + a1 * b^1 + a2 * b^2 + .... Put another way, the limit, as h goes to 0 from above, of a0 * e^(-0h) + a1 * e^(-1h) + a2 * e^(-2h) + ..., where e is any fixed base you like. (Let's take e to be the base of the natural logarithm for convenience, and call this function of h the characteristic function of the series).
Again, this is not so different than the traditional account of series summation; we're just using exponential decay rather than sharp cutoff in our dampened approximations to the full series. (Actually, for the results we're interested in, it's really just the smoothness of the decay that's of interest. We could use other forms of smooth decay as well, and get the same results, but exponential decay is so convenient, I won't bother discussing in any further generality right now)
Now we've turned the question of determining the value of a series summation into the question of determining the limiting behavior of some function at 0.
Well, it's easy to determine limiting behavior at 0. Just write out a Taylor series centered at 0, and drop all the terms of positive degree, leaving only the term of degree 0. Boom, you've got the value of the function at 0.
Except... suppose the Taylor series has a few terms of negative degree as well. (As in, say, 5h^(-1) + 3 + 4h^2). Then the behavior at 0 isn't given by the degree 0 term; rather, the behavior at 0 is to blow up to infinity!
And, indeed, we'll find that this is precisely what happens when we look at the characteristic function of a series like 0 + 1 + 2 + 3 + ...; we get that f(h) = 0e^(-0h) + 1e^(-1h) + 2e^(-2h) + 3e^(-3h) + ... = e^(-h)/(1 - e^(-h))^2 = h^(-2) - 1/12 + h^2/240 - h^4/6048 + ....
Note that there is a negative degree term there. So in a very familiar sense, we can say that the behavior of this series is to blow up to infinity.
However, since any time a series DOES converge in the ordinary sense, the value it converges to is the degree 0 term of this characteristic function, it is very tempting and fruitful to think of the degree 0 term as the sum even when there are those pesky negative degree terms.
And in this more general sense, we see that the value of 0 + 1 + 2 + 3 + ... is that degree 0 term of f(h): -1/12. [In fact, we can understand the argument at the beginning of this post as outlining a rigorous calculation of this degree 0 term. (See https://www.quora.com/Mathematics/Theoretically-speaking-how... to see this spelt out)]
Now, you can propose other manipulations to produce other answers for this series in other ways, but this is one particular systematic account of summation which leads to this value alone and no other. [That is, for the series whose nth term is n. I should warn that, in the presence of negative degree terms in the characteristic function, this method is sensitive to index-shifting, so we would get different results if, for example, we considered 1, 2, 3, ... to be not the 1st, 2nd, 3rd, ..., terms, but rather the 0th, 1st, 2nd, ..., terms, respectively.]
Why should you care about this particular account of summation? Well, you don't have to; I can't force you to care about anything. But it's fairly natural and comes up with some significance in mathematics. It is, in a certain formal sense, precisely the account of summation which allows one to interpret the sum 1^n + 2^n + 3^n + ... for general complex n, yielding the Riemann zeta function (of great significance in number theory, and whose behavior (specifically, the Riemann hypothesis concerning its zeros) is generally considered one of the most important open problems in mathematics). So, you know, there's reason for some people to care about it, even if you don't.
Young man, in mathematics you don't
understand things. You just get used
to them.