Recently a very strange result has been making the rounds. It says that when you add up all the natural numbers

1+2+3+4+...

then the answer to this sum is -1/12. The idea featured in a Numberphile video (see below), which claims to prove the result and also says that it's used all over the place in physics. People found the idea so astounding that it even made it into the New York Times. So what does this all mean?

The maths

First of all, the infinite sum of all the natural number is not equal to -1/12. You can easily convince yourself of this by tapping into your calculator the partial sums

$S_1 = 1$

$S_2 = 1+2 = 3$

$S_3 = 1+2+3=6$

$S_4 = 1+2+3+4=10$

$...$

$S_ n = 1+2+3+4+ ... +n,$

and so on. The $S_ n$ get larger and larger the larger $n$ gets, that is, the more natural numbers you include. In fact, you can make $S_ n$ as large as you like by choosing $n$ large enough. For example, for $n=1000$ you get

  \[ S_ n = 500,500, \]    

and for $n = 100,000$ you get

  \[ S_ n = 5,000,050,000. \]    

This is why mathematicians say that the sum

  \[ 1+2+3+4+ ...  \]    

diverges to infinity. Or, to put it more loosely, that the sum is equal to infinity.

Srinivasa Ramanujan

Srinivasa Ramanujan

So where does the -1/12 come from? The wrong result actually appeared in the work of the famous Indian mathematician Srinivasa Ramanujan in 1913 (see this article for more information). But Ramanujan knew what he was doing and had a reason for writing it down. He had been working on what is called the Euler zeta function. To understand what that is, first consider the infinite sum

$S = 1+1/4+1/9+1/16+ ... .$

You might recognise this as the sum you get when you take each natural number, square it, and then take the reciprocal:

$S = 1+1/2^2+1/3^2+1/4^2... .$

Now this sum does not diverge. If you take the sequence of partial sums as we did above,

$S_1 = 1$

$S_2 = 1+1/2^2 = 5/4=1.25$

$S_3 = 1+1/2^2+1/3^2 = 49/36 = 1.361...$

$...$

$S_ n =1+1/2^2+1/3^2 + ... + 1/n^2,$

then the results you get get arbitrarily close, without ever exceeding, the number $\pi ^2/6 = 1.644934... .$ Mathematicians say the sum converges to $\pi ^2/6$, or more loosely, that it equals $\pi ^2/6.$

Now what happens when instead of raising those natural numbers in the denominator to the power of 2, you raise it to some other power $x$? It turns out that the corresponding sum

  \[ S(x) = 1+1/2^ x+1/3^ x+1/4^ x...  \]    

converges to a finite value as long as the power $x$ is a number greater than $1$. For every $x > 1$, the expression $S(x)$ has a well-defined, finite value. $S(x)$ is what’s called a function, and it’s called the Euler zeta function after the prolific 17th century mathematician Leonhard Euler.

So far, so good. But what happens when you plug in a value of $x$ that is less than 1? For example, what if you plug in $x=-1$? Let’s see.

  \[  S(-1) = 1+1/2^{-1}+1/3^{-1}+1/4^{-1}...  \]    
  \[  = 1+2+3+4+ ... . \]    

So you recover our original sum, which, as we know, diverges. The same is true for any other values of $x$ less than or equal to 1: the sum diverges.

Extending the Euler zeta function

As it stands the Euler zeta function S(x) is defined for real numbers x that are greater than 1. The real numbers are part of a larger family of numbers called the complex numbers. And while the real numbers correspond to all the points along an infinitely long line, the complex numbers correspond to all the points on a plane, containing the real number line. That plane is called the complex plane. Just as you can define functions that take real numbers as input you can define functions that take complex numbers as input.

One amazing thing about functions of complex numbers is that if you know the function sufficiently well for some set of inputs, then (up to some technical details) you can know the value of the function everywhere else on the complex plane. This method of extending the definition of a function is known as analytic continuation. The Euler zeta function is defined for real numbers greater than 1. Since real numbers are also complex numbers, we can regard it as a complex function and then apply analytic continuation to get a new function, defined on the whole plane but agreeing with the Euler zeta function for real numbers greater than 1. That's the Riemann zeta function.

But there is also another thing you can do. Using some high-powered mathematics (known as complex analysis, see the box) there is a way of extending the definition of the Euler zeta function to numbers $x$ less than or equal to 1 in a way that gives you finite values. In other words, there is a way of defining a new function, call it $\zeta (x),$ so that for $x>1$

$\zeta (x) = S(x) = 1+1/2^ x+1/3^ x+1/4^ x...,$

and for $x\leq 1$ the function $\zeta (x)$ has well-defined, finite values. This method of extension is called analytic continuation and the new function you get is called the Riemann zeta function, after the 18th cenury mathematician Bernhard Riemann. (Making this new function give you finite values for $x \leq -1$ involves cleverly subtracting another divergent sum, so that the infinity from the first divergent sum minus the infinity from the second divergent sum gives you something finite.)

OK. So now we have a function $\zeta (s)$ that agrees with Euler’s zeta function $S(x)$ when you plug in values $x>1$. When you plug in values $x \leq 1$, the zeta function gives you a finite output. What value do you get when you plug $x=-1$ into the zeta function? You’ve guessed it:

  \[ \zeta (-1)=-1/12. \]    

If you now make the mistake of believing that $\zeta (x) = S(x)$ for $x=-1$, then you get the (wrong) expression

  \[ S(-1) = 1+2+3+4+ ... = \zeta (-1) = -1/12. \]    

This is one way of making sense of Ramanujan’s mysterious expression.

The trick

So how did the people in the Numberphile video "prove" that the natural numbers all add up to -1/12? The real answer is that they didn’t. Watching the video is like watching a magician and trying to spot them slipping the rabbit into the hat. Step one of the "proof" tries to persuade you of something rather silly, namely that the infinite sum

  \[ 1-1+1-1+1-.... \]    

is equal to $1/2.$

The video doesn’t dwell long on this and seems to imply it’s obvious. But let’s look at it a little closer to see if it makes sense at all. Suppose that the sum $1-1+1-1+1-1....$ has a finite value and call it $Z$. Adding $Z$ to itself you get the infinite sum

  \[ Z+Z = 1-1+1-1+1-....+1-1+1-1+1-... . \]    

But this is just the original sum, implying

  \[ Z+Z=2Z = Z. \]    

Since $Z=1/2,$ it follows that $1/2=1,$ which is nonsense. So the assertion that the infinite sum $1-1+1-1+1-....$ can be taken to equal to 1/2 is not correct. In fact, you can derive all sorts of results messing around with infinite sums that diverge. It’s a trick!

The physics

But how did this curious, wrong result make it into a physics textbook, as shown in the video? Here is where things really get interesting. Suppose you take two conducting metallic plates and arrange them in a vacuum so that they are parallel to each other. According to classical physics, there shouldn't be any net force acting between the two plates.

Casimir effect

Illustration of the Casimir effect. Image: Emok.

But classical physics doesn't reckon with the weird effects you see when you look at the world at very small scales. To do that, you need quantum physics, which tells us many very strange things. One of them is that the vacuum isn't empty, but seething with activity. So-called virtual particles pop in and out of existence all the time. This activity gives a so called zero point energy: the lowest energy something can have is never zero (see here for more detail).

When you try to calculate the total energy density between the two plates using the mathematics of quantum physics, you get the infinite sum

  \[ 1 + 8 + 27 + 64 +... . \]    

This infinite sum is also what you get when you plug the value $x=-3$ into the Euler zeta function:

$S(-3) = 1 + 1/2^{-3} + 1/3^{-3} + 1/4^{-3} + ... = 1+ 8 + 27 + 64 +... .$

That’s unfortunate, because the sum diverges (it does so even quicker than than $S(-1)$), which would imply an infinite energy density. That’s obviously nonsense. But what if you cheekily assume that the infinite sum equals the Riemann zeta function, rather than the Euler zeta function, evaluated at $x=-3$? Well, then you get a finite energy density. That means there should be an attractive force between the metallic plates, which also seems ludicrous, since classical physics suggests there should be no force.

But here’s the surprise. When physicists made the experiment they found that the force did exist — and it corresponded to an energy density exactly equal to $\zeta (-3)$!

This surprising physical result is known as the Casimir effect, after the Dutch physicist Hendrik Casimir.

Take a moment to take this in. Quantum physics says the energy density should be

  \[ S(-3) = 1+8+27+64+... . \]    

That’s nonsense, but experiments show that if you (wrongly) regard this sum as the zeta function $\zeta (x)$ evaluated at $x=-3$, you get the correct answer. So it seems that nature has followed the ideas we explained above. It extended the Euler zeta function to include values for $x$ that are less than 1, by cleverly subtracting infinity, and so came up with a finite value. That’s remarkable!

The reason why we see $\zeta (-1)$ and $S(-1)$ in the Numberphile video and the physics textbook, rather than $\zeta (-3)$ and $S(-3),$ is that when you imagine the Casimir effect as happening in one dimension (along a line rather than in 3D), the energy density you calculate is $\zeta (-1)$ rather than $\zeta (-3)$.

So why did the Numberphile people publicise this strange "result"? They certainly know about the analytic continuation that makes the function well-defined, but that was something that was a little too technical for their video. Knowing they had the analytic continuation method, that would make the final result OK, hidden in their back pocket, they went ahead with their sleight of hand. In doing so they got over a million hits and had the world talking about zeta functions and mathematics. For this they should be congratulated. The mathematics of zeta functions is fantastic and what we described here is just the start of a long list of amazing mathematical properties. In bringing mathematics and physics to the public we always have to make choices about what we leave out and what we explain. Where to draw that line is something we all have to leave to our consciences.


About the authors

David Berman is a Reader in Theoretical Physics at Queen Mary, University of London. He previously spent time at the universities of Manchester, Brussels, Durham, Utrecht, Groningen, Jerusalem and Cambridge as well as a year at CERN in Geneva. His interests outside of physics include football, music and theatre and the arts.

Marianne Freiberger is editor of Plus.

Comments

1+2+3+4+5+... forever

So isn't this series basically sigma of n for which n starts from 1 and ends at infinity? In this case, in my opinion, as every terms in this series is positive, this series can immediately be included in the 'comparison tests for convergence and divergence'. French mathematician Nicole Oresme has proved the divergence of the harmonic series 1/n by proving the divergence of a series which has lesser terms than that of the harmonic series. If, as what the numberphile people are saying, 1+2+3+4+... forever does equals to -1/12 doesn't this make the whole concept of comparison test flawed? Even if the Grandi's series which is 1-1+1-1+1-1... does actually equals to a half, the answer 1+2+3+4+.. forever does not make sense as it makes previous proofs and statements made in infinite series incorrect.

Also, i don't understand what the average partial sums are for. Why is there the need to average the partial sums? The average of the partial sums may converge towards a half but that doesn't mean anything that the 'non-average' (as in real) series converge towards a half...

+1-1+1-1+1-1 . . . in cycles

One of the guys in the vid says there's a proof for Z (the Grandi series) = 1/2. I'll leave that to the more advanced, but to me that makes sense intuitively. In the absence of a graph for now, imagine a horizontal wavy line ascending with every +1 to a peak at 1 on the y axis and descending with -1 again to 0. Repeat, say with six 1's, so for three cycles. Now draw another horizontal straight line skimming along from peak to peak, and clearly you will have a figure falling into two equal halves above and below the wavy line. Each of these two halves will also equal the halves created by yet another horizontal straight line drawn from .5 on the y axis.

Now for the smart bit. Do an Oresme on the series, rearrange the +1's and the -1's into different, though still zero sum cycles: +1+1+1-1-1-1. This time the amplitude will rise to 3 and the horizontal through the mid points will be at 1.5, so the sum of the series will be 3/2.

This hocus pocus, if you want to call it that, also has an echo in nature. I've just been learning about Cepheid variable stars, and the close direct relationship between luminosity and the period over which luminosity varies from minimum to maximum and back. Luminosity, energy emitted per unit of time, could be represented by the area under the wavy line maybe. How the values 1/2 and 3/2 would relate to the natural parameters of luminosity and period I don't know, but any comments gratefully received.

1-1+1-1+1-...=1/2

In order to find the sum of this series, first of all, the sequence of the partial sums must converge. However, the sequence of partial sums does not converge. Rather, it oscillates between 0 and 1.
Thus, insofar as the sequence does not converge there is no reason to talk about a unique sum of the series. In this case we can prove anything we want: 1, 1/2...

Cesaro convergence

Dear David and Marianne,

Thank you for the interesting explanations. I would remark, though, that the summation "1-1+1-1+... = 1/2" is not the source of fallacy in the Numberphile video. That series, while not convergent, is still "Cesaro summable" [which means, that the average value of its first n partial sums converges in the limit as n is taken to infinity].

The Cesaro sum of series has many nice properties, e.g., allowing a certain class of rearrangements of the terms, allowing the sums to be added when Cesaro summable series are added term by term, etc. The fallacies enter elsewhere, in the various manipulations of divergent and non-Cesaro summable series. Such manipulations can be used to arrive at any value at all for the sum of all the natural numbers.

Perhaps others have already made this point, as I've not viewed all of the linked and related sites.

Like others, I too would have wished to see some indication in the Numberphile video of the "tongue in cheek" nature of the "proof," without losing the huge entertainment value. Fallacies (like "proving" that 1 = 0 with a hidden divisioon by zero in the algebra) have always been a source of fun and exploration in school mathematics.

Here, the video is understandable to many middle grades students as well as high school students; this is great! But their teachers then need to know it is "sleight of hand," and encourage their students to question the "authority" behind it. Teaching mathematics by authority rather than through understanding is a continuing, unfortunate trend in our schools, at least in the USA, and I cannot help but feel an opportunity here might be being missed.

Gerald Goldin
Rutgers University

summation of 1 - 1 + 1 - 1 ....

Hi,

I believe you have made a mistake in doing Z + Z = 2Z = Z

for 2Z to equal Z you have to assume that one of the Z sums ends at -1 so that the other can conveniently start at 1. This is a misconception as these are infinite series that have no end.

The only conclusion you can draw from Z + Z is as follows

Z = 1 - 1 + 1 - 1 + 1 - .......
+
Z = 1 - 1 + 1 - 1 + 1 - .......

which is equal to 2 - 2 + 2 - 2 + 2 - .......

Christian Whittaker email me at cpwhittaker@hotmail.co.uk if you are unsure (or disagree) with this

take out a factor of 2 you get 2( 1 - 1 + 1 - 1 + 1 - .....) = 2Z in which case you are back where you started

So because 2Z is not infact Z, your theory is irrelevant

Thank you!

Thanks very much for this post - I feel like this is the follow up that like Numberphile is missing!

I'm not quite sure what Numberphile were thinking... I'm all for promoting discussion and argument, but even I have "limits" (-: Seriously though, I hope that Numberphile come back with a follow up video at some stage explaining themselves!

Closure Property of Natural Numbers

Ok so I have read the article and watched the shorthand proof on numberphile. However don't the natural numbers have a closure property, meaning if you add any of the natural numbers together you must get a natural number? So how can it be that when we add all of them, we get -1/12, which does not belong to the natural numbers? I think my idea of adding all the natural numbers doesn't match with what is going on in the article, so if anyone could fill in the gaps that would be great.

The closure of the natural

The closure of the natural numbers under addition means that the sum of any two natural numbers is a natural numbers. We can apply this principle again and again (finitely many times) to see that the sum of any finite number of natural numbers is a natural number. This however says nothing of the sum of an infinite number of naturals, which we have in the case of an infinite series; in fact we see that this need not exist at all.

Similarly, whilst rational numbers are closed under addition also, the sequence S given in the article for pi/6 gives an example of a series of rational numbers converging to an irrational number (the rational numbers are also not closed in another sense, as a sequence of rational numbers can converge to an irrational number).

Very good, the naturals are

Very good, the naturals are closed under addition. And indeed the point of this article is that the sum of all naturals is not -1/12. Only the so called regularised sum where "an infinity" has been subtracted- that "infinity" is obviously not a natural.

Almost there

Ok thanks for the response, sorta wanna get my head around this incase I tell people about it. I like how Numberphile made a separate video for the sequence S1 = 1/2, cheeky way to get more views. Where this all loses me is where the extended function gets its values from when you plug in values less than 1 (somewhere in the article it says less than -1, is this a mistake, what about -1

yep it should be less than or

yep it should be less than or equal to -1.

So exactly where does it get its value from? This is the technical bit about analytic continuation. There is a way, beyond the scope of this article to extend functions of complex numbers once you know the value of the function elsewhere. I understand why the numberphile people didn't want to bring that up because it is hard to explain nontechnically and indeed to give them credit where it is due, they have a discussion about it in an adjoining video and a written bit about it now too. So basically there is a bit of maths using function of complex numbers that allow you to assign a value to a function once you know its values elsewhere. The point of the artice was that this was too hidden for my taste which is why we still get posts like the one above where they want to still claim that the sum of all the integers is -1/12. it is not. Only a function that is related to that sum has that value. Where by related I mean that function has the same value as that sum when that sum is convergent.

The Numberphile people do explain the 1/2 result

Z = 1 - 1 + 1 - 1 + 1 - 1 ......
1 - Z = 1 -[ 1 - 1 + 1 - 1 + 1 - 1 ....... ] = 1 - 1 + 1 - 1 ..... = Z
1 - Z = Z
2Z = 1
Z = 1/2

1 - Z = Z ; implying: (z+z=1) derives from nonsense

statement is logically faulty ;
substitute : z=zero
result : one minus zero is zero !!!
facit : give me a buck (=1) for each such transaction !

So, I will say it

So, I will say it again..

Writing Z=.. and then assuming it is a well defined quantity is an assumption you cannot make.

The sum is nonconvergent. However you can very simply analytically continue the function that the sum represents beyond the radius of convergence but that is a continuation. The sum itself is not defined, it is not convergent. So if its nonconvergent don't write Z=..
It doesn't equal anything...

The Numberphile folks have a

The Numberphile folks have a video that goes through basically the same area, talks about the sleight-of-hand in the shorter video, discusses analytic continuation and all that good stuff, and it's linked at the end of the short version. Well worth a watch: www.youtube.com/watch?v=E-d9mgo8FGk

yes indeed, this is all good

yes indeed, this is all good stuff and explains what lies behind it all.

The sum S(-3) is wrong

The expression for S(-3) two lines below the two plates draw is wrong. It reads S(-3)= 1/2^3 + 1/3^3 + 1/4^3 +... and should be S(-3)=1/2^-3 + 1/3^-3 + 1/4^-3 +...
Thanks.

Thanks for pointing that out!

Thanks for pointing that out! We've corrected it.

Thank you for explaining the zeta function

It's cooooool.

Infinity or -1/12?

You can't add two infinite series the way you did.

[ Z+Z ≠ 1-1+1-1+1-....+1-1+1-1+1-… .]

However,

Z+Z = 2-2+2-2+2-… ≈ 1,

which is consistent with, Z≈ 1/2.

Not sure how to answer you.

Not sure how to answer you. Non-convergent series are unpleasant and subtle things most 1st year mathematics text books will tell you the convergence properties of this sum. Instead let me just quote from Abel in 1828, "Divergent series are the invention of the devil and it is shamful to base on them any demonstration whatsoever". Hardy's excellent book on the subject explains the rights and wrongs; the good the bad and the ugly.

Congratulated?

I appreciate this piece, and it's helped me make sense of a lot of what's going on here. However, I have to part ways in calling for the Numberphile folks to be "congratulated." As a math teacher, and one who struggles every day to counter the deeply-ingrained notion that math makes no sense whatsoever, I can't stand it when that notion is spread across a wide audience and further ingrained into our culture. Sure, those who are already somewhat mathematically inclined are intrigued and want to know more. But those who are not see a video like this and say to themselves, "Further confirmation that math makes absolutely no sense." So I'm not about to congratulate them.

Matt E.
Sharon, MA

Thank you for your comments,

Thank you for your comments, its something that many math teachers have expressed to me including lecturers at university which is what prompted us to write the piece; Its clear to me where I would draw the moral line and it isn't in the same place they have .