Two physicists explain: The sum of all positive integers equals −1/12
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How far should scientists go in simplifying complexity to engage the public imagination? Boosted by a Dennis Overbye article in the New York Times’s Science Times section, the YouTube version of the video “ASTOUNDING: 1 + 2 + 3 + 4 + 5 + . . . = −1/12” had drawn well more than 1.5 million hits as of the morning of 4 February.
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?
Tony Padilla and Ed Copeland, physicists at the University of Nottingham in the UK, appear in and narrate the eight-minute video. Minute by minute, what they explain requires from the viewer no mathematical grounding beyond simple addition and a smattering of the most basic algebra. But in the aggregate, what they argue goes well beyond that simplicity.
The American Physical Society’s PhysicsCentral website offers the Physics Buzz Blog posting “Correction: Does 1 + 2 + 3 + 4 + . . . = −1/12? Absolutely Not!” It begins by pointing to a graph: “Brief Summary: 1 + 2 + 3 + 4 + . . . is not equal to −1/12, but both the infinite series and the negative number are associated with each other in a way that can be seen in this graph.”
At the Times, Overbye quips, “After watching the video myself, I checked to make sure I still had my wallet and my watch.” But he also reports something that Padilla and Copeland adduce in the video. They open the textbook String Theory, by Joseph Polchinski of the University of California, Santa Barbara, and point to a page showing his use of the mathematics in question.
On the web, Padilla has posted a write-up for physicists. He begins with this explanation:
It’s by no means obvious, but this is the only sensible value one can attach to this divergent sum. Infinity is not a sensible value. In my opinion, as a physicist, infinity has no place in physical observables, and therefore no place in Nature. David Hilbert, one of the founding fathers of quantum mechanics, described infinity as “a mathematical abstraction that does not have a physical content.” I think most physicists would firmly agree with this sentiment. The trouble is that divergent sums like the one we discuss in the video do appear in calculations of physical observables, such as the Casimir energy, or in the dimensionality of the Universe in bosonic string theory. Therefore, only a very brave individual would dream of attaching the value infinity to sums like this. Minus a twelfth is far less crazy a value when you start talking about physics.
Padilla notes that the result is “utterly counterintuitive,” but he explains it at some length and in some mathematical detail, and he offers links for further reading.
He also defends the license he and Copeland have taken to reach out to the public. Here’s his write-up’s final paragraph:
There is an enduring debate about how far we should deviate from the rigorous academic approach in order to engage the wider public. From what I can tell, our video has engaged huge numbers of people, with and without mathematical backgrounds, and got them debating divergent sums in internet forums and in the office. That cannot be a bad thing and I’m sure the simplicity of the presentation contributed enormously to that. In fact, if I may return to the original question, “what do we get if we sum the natural numbers?”, I think another answer might be the following: we get people talking about mathematics.
Steven T. Corneliussen, a media analyst for the American Institute of Physics, monitors three national newspapers, the weeklies Nature and Science, and occasionally other publications. He has published op-eds in the Washington Post and other newspapers, has written for NASA's history program, and is a science writer at a particle-accelerator laboratory.