It’s been almost a year since I last blogged, and I’ve spent much of that time feeling guilty about not blogging enough. So here we are. I was lured out of my state of blog-apathy by a recent post by mathbabe; in fact, this will be the second of my very few blog posts inspired by that blog. If you’re not already reading that blog, you really should – it’s a brilliant mix of math, politics, economics, data, and sex.

In any event, mathbabe was commenting on a video which has apparently been making its way around the internet. In this video, some mathematicians (Or perhaps physicists? What are string theorists calling themselves these days?) attempted to explain the mind-boggling “fact” that

1 + 2 + 3 + 4 + 5... = -1/12

Watch the video if you like, but by now a number of other mathematicians have rightfully pointed out that most of the fishy manipulations in the video amount to fraudulent nonsense which can be used to justify just about anything. This infuriates me, because the people who made the video could have used the opportunity to legitimately blow people’s minds by placing the equation above (which does make sense, from the right point of view!) in its proper context and explaining some beautiful mathematics.

I don’t have the apparatus to make a cool video, but I do have a blog. So I’m going to make an attempt to do what I think the video should have done (I am not optimistic that my attempt will get picked up by Slate, of course). Instead of adding up all of the positive integers, I’m going to start by adding up all of the powers of two:

2 + 4 + 8 + 16 +... = -2

We still get a negative number, so this equation should be just as counter-intuitive as the original one (though admittedly -1/12 is pretty bizarre). Our strategy for making sense of both equations will be the same:

  1. Write down an equation which makes sense (both logically and intuitively) in a narrow context
  2. Observe that the right-hand side of the equation actually makes sense in a much larger context than the left-hand side
  3. Use the right-hand side as a proxy for the left-hand side in the larger context

This strategy, called analytic continuation by mathematicians, is extremely powerful. But the basic idea is really quite simple, and it is even familiar in the context of language. When you “log in” to your e-mail account, your messages are likely organized into various “folders”, some of which are in your “inbox” and some of which are in your “outbox”. Perhaps you have a list of your friends’ e-mail “addresses” in your “address book”. The words that I put in quotes all began life in the narrow context of physical reality but have been extended to the new context of the internet; your e-mail inbox is not a physical box anywhere in the world, nor does your e-mail address refer to an actual place you can go. Someone at some point in the history of the internet realized that physical mail is a good metaphor for the electronic messages people send each other, and thus the language surrounding physical mail actually makes sense in the context of the internet.

The strange equations that I wrote above are mathematical counterparts of taking a word such as “hyperlink” which only really makes sense in the context of the internet and applying it to real world mail. You would end up with a sentence which looks pretty bizarre, but there would nevertheless be a certain logic to it.

Let’s see how this all plays out mathematically. We’ll start with something that isn’t likely to stir up much controversy:

1/2 + 1/4 + 1/8 +... = 1

This is the mathematical counterpart of the observation that if you walk across half of a room, then a quarter of the room, then an eigth, and so on then you will have crossed the whole room. (Of course, there are some philosophical questions to be raised by the fact that the phrase “and so on” took the place of an infinite number of actions. Even non-controversial infinite series deserve serious thought.)

You might also convince yourself that

1/3 + 1/9 + 1/81 +... = 1/2

It might not be obvious that the answer is 1/2, but this answer is at least plausible: we start with a number which is smaller than 1/2 and add increasingly tiny numbers to it. And if you plug numbers into a calculator you will get good numerical evidence that this equation makes sense; the further out you go in the sum, the closer you get to 1/2. In general, if the absolute value of s is a number smaller than 1, we have:

s + s^2 + s^3 +... = \frac{s}{1 - s}

Of course, the only context in which the left-hand side really makes sense is when |s| < 1; this ensures that the powers of s get very small very vast and thus the settles near a particular value. If |s| \geq 1 then there is no such guarantee: the powers of $s$ do not get smaller, and you can get a number as large as you want by adding up enough terms in the sum.

The right-hand side, on the other hand makes sense in a much larger context: we can plug in any number except s = 1! In particular, we can plug s = 2 into \frac{s}{1-s} to obtain \frac{2}{1-2} = -2. Since the expressions s + s^2 + s^3 +... and \frac{s}{1-s} agree when |s| < 1, it makes sense to use the latter expression as a proxy for the former at other values of s, such as s = 2. In othe words, it is not entirely stupid to write:

2 + 4 + 8 +... = -2

There is some theory which makes this equation even less stupid: \frac{s}{1-s} is (in a sense which can be made precise) the only sensible way to extend s + s^2 + s^3 +... beyond the set |s| < 1. Properly justifying this requires techniques coming from one of the most beautiful subjects in all of mathematics: the calculus of complex numbers. It should not be at all obvious, but in the end this whole discussion is really all about the mysterious powers of complex numbers.

The same techniques allow us to analyze the sum 1 + 2 + 3 +... which got this post started; this time, our starting point is the Riemann Zeta function:

\frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} +... = \zeta(s)

This time the sum on the left-hand side makes sense as long as |s| > 1, but the same tools described above imply that the Riemann Zeta function can be “analytically continued” to allow any input except s = 1, and its value at s = -1 can be calculated to be -1/12. This calculation could occupy another entire blog post, so I will not go any further than that at this time.

Now that I have explained the sense in which it is not completely stupid to say that the sum of all the positive integers is -1/12, I would like to conclude by arguing that it still is pretty stupid. Notice that according to the reasonging described in this post we did not assign the sum a value by thinking about it intrinsically as we can with, for instance 1/2 + 1/4 + 1/8 +...; instead we related the sum to the Riemann Zeta function and analyzed that function. But there are infinitely many other possible functions which have a similar relationship to 1 + 2 + 3 +..., and many of them will assign different values to the series following the steps outlined here. In fact, you can use these steps to justify giving the sum any value you want. Still, the Riemann Zeta function enjoys a privileged position in mathematics (and physics) so -1/12 is a pretty good choice.

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