So there you are living your life, content in your grasp on how the world works: up is up, down is down, the Sun rises in the east and sets in the west. Then, out of nowhere, a bunch of mathematicians try to tell you that the sum of all positive integers, that is, 1 + 2 + 3 + 4 + 5 + 6 +... and so on to infinity is equal to... -1/12.
Well, that's clearly ridiculous, right? How can increasingly big numbers, when added together, make a small number? How can whole numbers make a fraction? How can positive numbers make a negative?
So you watch the rest of the video (above), made by Numberphile, a bunch of math wonks with a popular (and generally trustworthy) YouTube channel about math. (Watch it. It's worth it. We'll wait.) Their proof seems rock solid. A bunch of really smart people, from astronomer Phil Plait to the folks at Physics Central, double down on the claim. "It turns out that the conclusions they draw in that video are literally correct. You can add an infinite series of positive numbers, and they’ll add up to a negative fraction,” said Plait.
So that's that, right? Math doesn't make sense, and the world is weird.
But that's not the end of the story.
A second set of the mathematically inclined people, including Scientific American blogger Evelyn Lamb and physicist Greg Gbur, took to the web to show that while the sum of all positive numbers can kind of sort of equal -1/12 (a result that, they explain, is used all the time in accurately solving physics problems), this mind-bending answer really only works if you totally redefine some core concepts of mathematics.
Phil Plait and the Physics Central crew eventually came around, and it was the follow-up from Physics Central that most helped us get our minds around this quandary.
According to Physics Central, 1 + 2 + 3 + 4 + … only equals -1/12 because the mathematicians redefined the equal sign. In this style of mathematics, called analytical continuation, "=" stopped meaning “is equal to” and started meaning “is associated with.” Tricky mathematicians.
This mathematical trick goes way back, says Physics Central. It's in the work of pioneering Indian mathematician Srinivasa Ramanujan, for instance:
Ramanujan is to blame a bit too. After all, how are we supposed to understand what he was trying to say here?
"I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal."
-S. Ramanujan in a letter to G.H. Hardy
Are we supposed to realize that "under my theory" means that "=" doesn't mean equal?
I haven't found his original work, but several people have reproduced a calculation by Euler that uses an equal sign in the same way. If the two sides aren't equal then, as I recall from second grade math, you can't use an equal sign.
So, does 1 + 2 + 3 + 4 + 5.... = -1/12? Yes, but only if, to you, an equal sign means something other than “is equal to.”
Now, that's not to say that the Numberphile team were just straight up messing with our heads. The -1/12 value can be proven in a number of ways, and the result is certainly useful. But 1 + 2 + 3 + 4 + ... definitely does not "equal" -1/12 in any way that a person would normally think about it.
That the Numberphile team took such a leap without explaining it to people, says physicist Greg Gbur, is sort of a shame:
The video makes it seem so simple, and uncontroversial, almost obvious. But there are some big mathematical assumptions hidden in their argument that, in my opinion, make it very misleading. To put it another way: in a restricted, specialized mathematical sense, one can assign the value -1/12 to the increasing positive sum. But in the usual sense of addition that most human beings would intuitively use, the result is nonsensical.
To me, this is an important distinction: a depressingly large portion of the population automatically assumes that mathematics is some nonintuitive, bizarre wizardry that only the super-intelligent can possibly fathom. Showing such a crazy result without qualification only reinforces that view, and in my opinion does a disservice to mathematics.
We haven't even attempted to tackle to long proofs involved in sorting out this debate here, but if you want more, check out:
Correction: Does 1+2+3+4+ . . . =-1/12? Absolutely Not! By Physics Central
Does 1+2+3… Really Equal -1/12? by Evelyn Lamb