Science Sunday: Does the Sum of All Natural Numbers Equal -1/12?
They say not to write a post with a question if the answer is “no” but I’m going to do it anyway.
You may have seen this video floating around the intertubes that claims to show that summing all the natural numbers (1 + 2 + 3 + 4 + 5 + … all the way to infinity) is equal to -1/12:
As you may have guessed by the first sentence of my post, the sum of all the natural numbers does not, in fact, equal -1/12. (Usually. Math is weird.) The people in the video employ some slightly underhanded tricks to make it seem that the math is sound, but it’s really not. So what’s wrong?
First, let’s talk about what a series is. Very simply, a series is any sum of numbers (technically, a series is a sum of the numbers in a sequence, which is an ordered list of numbers). 1 + 2 + 3 + 4 + 5 + … is a series, but so is 1 + 2 + 3 and 1 + 1 + 1. A series doesn’t need to be infinite, and it can have as few or as many terms as you want.
All finite series have solutions, because no matter how large the numbers are, or how many you’re adding together, you’re still just adding numbers together, and it’s just like regular math. But if you’re adding an infinite series, things get more complicated.
Your first impression might be that every infinite series equals infinity, but that’s not quite right. For instance, let’s look at this series:
1/2 + 1/4 + 1/8 + 1/16 + …
This is a very famous series. You may have heard of it as Zeno’s Paradox. Half the distance to 1 is covered, then half the remaining distance, then half the distance after that, and so on to infinity. As the number of terms approaches infinity, the value of the sum gets closer and closer to 1. In math, we say that this series converges to 1.
But not every infinite series converges. Take a look at this series:
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …
This series looks a lot like the one above, but it’s different in a very important way. This series does go to infinity. It might seem strange because the terms keep getting smaller, but you can test this for yourself. Write a simple code that adds, say, the first million terms of this series together and see what you get.
Mathematicians call this kind of series a divergent series. There are divergent series that go to infinity, there are divergent series that go to negative infinity, and there are divergent series like this:
1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + …
This series is called Grandi’s divergent series. It’s divergent because the series never approaches a single value. At any point in the summation, the series is either 0 or 1, but a series must converge to a single value in order to be convergent.
But what if we force it to take a single value? For instance, what if we write Grandi’s series like this:
(1 – 1) + (1 – 1) + (1 – 1) + …
Now the value of the series is 0, right? Well, not quite. We can also write the series like this:
1 – (1 – 1) – (1 – 1) – (1 – 1) – …
Now the value of the series is 1. Wait, what?
So here we get into the mathematical trickery. The part that I skipped over here is that the series 1 – 1 + 1 – 1 + … doesn’t actually equal anything. It’s undefined. And the problem with undefined series is that you can’t do math to them.
This is a tricky concept, but it’s important. An undefined value isn’t a number, any more than, say, an apple is a number. Saying “I’m going to add 5 to this undefined value” makes as little sense as saying “I’m going to add 5 to this apple.” 5+apple doesn’t mean anything. It’s a nonsense equation.
If you go back through the video, you’ll see the people in the video frequently add divergent series, multiply divergent series, and subtract divergent series, all of which can’t actually happen. They’re doing what looks like math, but it’s not math. They might as well be adding apples.
Now, if you watch the second video Numberphile made on this topic, you’ll see what is almost an actual proof showing that the sum of the natural numbers does equal -1/12. They mention something called analytic continuation, which is a mathematical technique that basically makes it okay to do math to divergent series. Kinda. Sometimes. Math is weird. So if you do it right, the sum of all natural numbers can equal -1/12. Why? No idea.
I should point out here that I’m a physicist, not a mathematician (Dammit Jim, I’m a physicist, not a mathematician!), and my knowledge of this topic ends right around analytic continuation. If anyone with more math experience than I have wants to tackle this topic in the comments, be my guest.