Friday, January 17, 2014

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

I haven't been blogging recently because my new startup is taking up all my time, but someone needs to stand up and say the emperor has no clothes so it might as well be me.  About a week ago, two British mathematicians named Tony Padilla and Ed Copeland, who produce a video blog called Numberphile, posted a video that purports to prove that the sum of all the positive integers is -1/12.  It was making the usual geek rounds where I would have been content to let it circulate, but today the story was picked up by a naive and credulous reporter at Slate, where the story stands to do some real damage if not challenged.

(Aside for mathematicians: yes, I am aware that the claim is true under Ramanujan summation.  That is not the point.)

Let me start by recapping the argument.  Finding the flaw in the reasoning makes a nice puzzle:

Step 1: Let S1 = 1 - 1 + 1 - 1 ...

Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)

 = (1 - 1 + 1 - 1 ...) +
   (0 + 1 - 1 + 1 - 1 ...)

 = (1 + 0) + (1 - 1) + (1 - 1) ....

 = 1 + 0 + 0 ... = 1

So 2xS1 = 1.  So S1 must equal 1/2.

Step 2: Let S2 = 1 - 2 + 3 - 4 + 5 ...

So S2 + S2 = (1 - 2 + 3 - 4 + 5 ...) + (1 - 2 + 3 - 4 + 5 ...)

 = (1 - 2 + 3 - 4 + 5 ...) +
   (0 + 1 - 2 + 3 - 4 + 5 ...)

 = (1 + 0) + (1 - 2) + (3 - 2) + (3 - 4) + (5 - 4) + ...
 = 1 - 1 + 1 - 1 + 1...
 = S1 = 1/2

So S2=1/4.

Step 3: Let S = 1 + 2 + 3 + 4 + 5 ...

So S - S2 = (1 + 2 + 3 + 4 + 5 ...) - (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) -
   (1 - 2 + 3 - 4 + 5 ...)

 = (1 + 2 + 3 + 4 + 5 ...) +
  (-1 + 2 - 3 + 4 - 5 ...)

 =  0 + 4 + 0 + 8 + 0 + ...

 = 4 x (1 + 2 + 3 + 4 + 5 ...)

 = 4S

So S - S2 = 4S.  But S2 = 1/4.  So:

S - 1/4 = 4S
3S = -1/4
S = -1/12

Seems like an ironclad argument, doesn't it?  Like I said, finding the flaw in the reasoning (and there most assuredly is one) makes an interesting puzzle.  Here's a clue:

Let S3 = 1 + 1 + 1 + 1 ...

So S3 - S3 = (1 + 1 + 1 + 1 ...) - (1 + 1 + 1 + 1 ...)
 = (1 + 1 + 1 + 1 ...) - (0 + 1 + 1 + 1 ...)
 = (1 - 0) + (1 - 1) + (1 - 1) + ...
 = 1 + 0 + 0 + 0 ...
 = 1

But S3 - S3 must also equal 0, so we have just proven that 0=1.

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.

16 comments:

Chris said...
This comment has been removed by the author.
Jeremy Stieglitz said...

My friend, Ray Sidney, with a PhD in mathematics from MIT explained away this nonsense rather succinctly.

infinity + 7 = infinity

If you minus infinity from both sides, you can tell the world 7 = 0!

Don Geddis said...

Your "real math" link has another interesting example. S3 (1+1+1...) "sums" to -1/2. S (1+2+3+4...) "sums" to -1/12. Now add S3+S together. Term by term gets us 2+3+4+5... Adding the "sums" gets us -1/2 + -1/12 = -7/12. But, alternatively, you can just start with S (1+2+3+4...) = -1/12, and subtract 1 from both sides, to get (0+2+3+4...) = -13/12. Trying to do "normal algebra" on the divergent series suggests that 2+3+4+5... = both -7/12 and also -13/12, at the same time.

Just echoing your own conclusion, divergent series are not something you should be doing regular algebra on.

Corby said...

I've seen the video and the addendum and researched this on Wikipedia and found that it doesn't work on divergent series pretty quickly:
http://en.wikipedia.org/wiki/Reimann_zeta_function

Pretty shoddy research from Slate.

However, I did find it interesting that Numberphile mentioned that -1/12 shows up in quantum field theory fairly regularly (Casimir effect, 26 dimensional field theory).

Why, if the series only works in a very specific complex plane (s=-1), can this be applied to physics and backup a measurable theory (Casimir http://en.wikipedia.org/wiki/Casimir_effect )?

Though, to be totally honest, a 20 minute search could not verify that -1/12 is a component in the Casimir calculations.

Jon Rimmer said...

"A grave disservice to numerical literacy" Good grief, do you think you could sound any more absurdly pompous if you tried? Most people's numerical literacy doesn't extend past high school mathematics, and doesn't need to. They certainly won't be harmed by having a confused understanding of infinite series. Given the amount of debate this video has generated, I'm not convinced mathematicians themselves understand infinite series.

ash said...

Jon, I think Ron's problem is not about infinite series themselves. The problem is about the care and rigor one has to maintain when doing mathematics properly. The video failed in this regard.

Literate people know what are "+" and "=". We use them every day. But the guys in the video overloaded "=" without even giving the name to it. It's not "=". It's something else. You don't silently replace "=" with something else at the cashier, don't you? The equation should really be "1 + 2 + 3 + ... blabla -1/12". It now looks boring, doesn't it? How cares about "blabla"?

I think the same problem applies to convergent series too.. For some reason it's common to overload "=" even for convergent series. Why do people do it? An example from Wikipedia article: 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ... = 2. But it's wrong! This equal sign is not the "=" we are used to. You have to assign some new meaning to it (and to "..."). Cauchy criterion, for example. Why not use different sign even for convergent series? Lim notation, for example.

And I don't think Numberphile meant Cauchy criterion when they "proved" their equation.

Nuno José said...

We could also assume that 2S'' is realy 1+1/2 so = 3/4

So:

S = -1/4

So -1/12=-1/4

Nuno José said...

Made a mistake :P

2S''=1+1/2
S=3/4

vertical rock said...

They actually do mention riemann zeta functions

https://www.youtube.com/watch?v=w-I6XTVZXww&t=1m3s

and they have another video explaining the correct proof using them.

https://www.youtube.com/watch?v=E-d9mgo8FGk

Unknown said...

Finally someone... Thanks!

tensaix2j said...

I think the author only watched one of the 2 videos of numberphile.

Georg said...

Hmmm,
this"wrong" manipulations of diverging series remind me of complete induction.
Is there some connection?
Georg

Neil Bates said...

Indeed, it does not. I have found a direct contradiction in the treating of the infinite sum of positive integers (1 + 2 + 3 + 4 + 5 + ...) = -1/12. Yes I know, it can be related to that number by the zeta function etc. but the concept is not algebraically consistent. I don't just mean regarding rearranging associative parentheses either. The infamous proof in the circulating Youtube makes use of offset series additions to support the idea that 1 - 1 + 1 - 1 + 1 - ... = 1/2 and that 1 - 2 + 3 - 4 + 5 - ... = 1/4. Combining those with another offset, there is a "consistency" argument that 1 + 2 + 3 + 4 + 5 + ... = -1/12. Actually I think it's a clever paradox that apparent "algebra" supports the apparently unrelated result from higher math. However, there is a true contradiction as I show below (hoping for decent formatting but write it out as needed):

1 + 2 + 3 + 4 + 5 + ... "= -1/12"
-2 - 4 - 6 - 8 - ... "= 1/6"
---------------------------------------------------
1 - 1 - 2 - 3 - .... = 1 1/12 OR 1/12?

Sliding the -2x of the original series over more and more creates ever larger discrepancies. It is inconsistent.

Neil Bates said...

Sorry, let me redo that offset series addition:


1 + 2 + 3 + 4 + 5 + ... "= -1/12"
......-2 - 4 - 6 - 8 - ... "= 1/6"
---------------------------------------------------
1..........-1 - 2 - 3 - .... = 1 1/12 OR 1/12?

Sliding the -2x of the original series over more and more creates ever larger discrepancies. It is inconsistent.

Ralph said...

What possible harm can it do if a few (million?) people come to believe that the sum of all the positive integers is a small negative rational number?

Vinícius said...

I don't either agree with the two British mathematicians. I think their theory is as strange as wrong. I guess Ron's analyses are great and clear, mainly that part of sum of non-converging infinite sums.
I've got something else to prove their theory isn't correct.

Let S1 = 1 + 1 + 1 + 1 + 1 + ...
This sum logically goes to infinit.
Let S2 = 1 + 2 + 3 + 4 + ... = 1 + (1 + 1) + (1 + 1 + 1) + ... =

Telling that S1 has more terms than S2 or vice versa doesn't make any sense. Thus it's possible to state S2 is just another way of writing S1.
If S2 = S1, so 1 + 2 + 3 + 4 + ... = S1.
According to the considerations above, it's proved S2 goes to infinite.