An Astounding Result: All Positive Integers Equal -1/12th

by | Jan 31, 2020


Sometimes advanced math and physics makes no sense to the layperson. One such instance is this crazy assertion: all positive integers equal -1/12th. Or:

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

Amazingly, this result is used in physics in various calculations.

How can an infinite series of integers possibly be equal to -1/12th? Below is a video from Numberphile that simply walks through an explanation. It’s quite interesting and compelling:

The key conceptual point is this: as you add more and more integers together you get bigger and bigger numbers as the sum, tending towards infinity. That is clear and without dispute. This -1/12th stuff is a different kettle of fish than adding up finite numbers. Rather it’s assigning a finite value to an infinite series of numbers. This is a different thing than thinking about adding together a finite series of integers. Few of us ever deal with the concept of an infinite series; our lives only deal with the finite.

Another point: the technique used in the video to explain the -1/12th result isn’t the proof, but rather a trick to explain the result. The end of the video links to another longer video which actually gets into the higher mathematics that is the proof.

Not all mathematicians agree (and actually it seems to be physicists who more readily agree with the -1/12th assertion, whereas mathematicians generally do not). Here’s a post from Physics Central explaining that the sum of all integers doesn’t equal -1/12th, but rather all integers and -1/12th are associated with each other.

Did you know that Chuck Norris counted to infinity . . . twice?

Want more math? Here’s the IFOD about Euclid’s proof of infinite primes: Good News! Primes Go All the Way Up!


  1. This post diverted my attention for close to one hour.

    The sums in the video are divergent so the steps shown are impermissible. Interestingly, the famous equation n(n+1)/2 which produces the sum of a finite series of integers, when plotted, has an area under the x-axis equal to -1/12.

    You can see this by entering “plot x(x+1)/2” in the google search bar. Then, if you go to WolframAlpha and compute the definite integral (-1to 1) you’ll see it’s -1/12.

    This is a good explanation:

    This reminds me of one of my favorite movies, “The God’s Must be Crazy.”

    • that is so impressive that you dug into this as you did.
      the link to the physics central post goes into this point that if you graph the diverging infinite series that they are associated by an area below the x-axis that is equal to -1/12th.

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