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Comment by _v7gu | Hacker News Reader

An infinite number of mathematicians walk into a bar, where a beer costs 3$.

The first mathematician orders a beer.

The second mathematician orders two beers.

The third mathematician orders three beers.

"Oh, I see where this is going" says the bartender and pays a quarter to the first mathematician. "This should cover your check, then."

Is this an antijoke or did I just need to know where it's going, like the bartender? Maybe the bartender is confused?

This joke relies on the listener knowing that the regularized sum of 1+2+3+... = -1/12.

That's what makes it funny!

The joke is that the Ramanujan sum of 1 + 2 + 3 + 4 + ... is equal to -1/12, so the check should be $3 * -1/12 = minus one quarter. [0]

Thanks, that's a bit of math trivia I either forgot or never learned.

Can you explain?

The sum of all positive integers can be somewhat described as negative one twelfth: https://www.youtube.com/watch?v=E-d9mgo8FGk

It's referencing a theorem from string theory, as discussed in this Numberphile video: https://www.youtube.com/watch?v=w-I6XTVZXww

The catch is that this isn't valid in what we consider standard mathematics, and you can find many discussions of this online, but this one is fairly short and straightforward: http://curiouscheetah.com/BlogMath/infinity-and-string-theor...

1 + 2 + 3 + ... = -1/12 total beers

-1/12 beers * $3 / beer = -$0.25

negative -> pay them for taking the beer

The -1/12 result is pretty widely known, but not particularly true. It's called "analytic continuation." There's a nonrigorous proof that's accessible at a highschool level - see [here](https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B...)

Delightfully, this comes up in string theory, in computing the dimensionality of spacetime.