The sum of the natural numbers is -1/12?

Carson Chow Mathematics, Physics

This wonderfully entertaining video giving a proof for why the sum of the natural numbers  is -1/12 has been viewed over 1.5 million times. It just shows that there is a hunger for interesting and well explained math and science content out there. Now, we all know that the sum of all the natural numbers is infinite but the beauty (insidiousness) of infinite numbers is that they can be assigned to virtually anything. The proof for this particular assignment considers the subtraction of the divergent oscillating sum S_1=1-2+3-4+5 \dots from the divergent sum of the natural numbers S = 1 + 2 + 3+4+5\dots to obtain 4S.  Then by similar trickery it assigns S_1=1/4. Solving for S gives you the result S = -1/12.  Hence, what you are essentially doing is dividing infinity by infinity and that as any school child should know, can be anything you want. The most astounding thing to me about the video was learning that this assignment was used in string theory, which makes me wonder if the calculations would differ if I chose a different assignment.

Addendum: Terence Tao has a nice blog post on evaluating such sums.  In a “smoothed” version of the sum, it can be thought of as the “constant” in front of an asymptotic divergent term.  This constant is equivalent to the analytic continuation of the Riemann zeta function. Anyway, the -1/12 seems to be a natural way to assign a value to the divergent sum of the natural numbers.

6 thoughts on “The sum of the natural numbers is -1/12?”

  1. i came across this awhile back (divergent series—there are quite a few results of this type)—-i use it to confuse people. from what i gather, there really is only one consistant way of doing this, so it has to be -1/12. wikipedia has an article on it (ramanajun did it using zeta functions, and i’m weak there (though i was supposed to know this at one time).

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  2. This is the value you get if you analytically continue the Riemann zeta function to s=-1 so it is unique in that way. But you could probably assign the sum to the analytical continuation of some other function and get some other result.

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  3. Who are these guys? An Aussie and a Mancurian? I like the way they dis physics “these are the kinds of sums that occur all the time in physics”

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  4. do you have any examples of what the analytical continuation of some other functions would be. zeta is pretty general (i consider it to be the partition function of the prime numbers, and you can get their configurations to fill in the gaps between them–‘god is in the gaps’ (god made the integers, everything else was on wikipedia (kronecker)—i even have a (not even wrong ) proof that the countable numbers are uncountable.), though i also like weirstrauss-stone theorem (the world is piece wise linear, such as via taylor’s theorem—course you may end up with the problem of ‘i’ such as fourier series, or born’s rule (which some argue is a mystery as to where it comes from, when its obvious it comes from max born—hitler even said he could come back to nazi germany, but not his wife.) . i actually do agree (even if consensus opinion is its -1/12) —there’s alot of ways of doing those sums.

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  5. @Alan Yes, I find it amusing that they believe this assignment is a miraculous truth of nature rather than a choice that is well motivated.

    @ishi I’ll give an example in the next post.

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