Sadly the sum of all natural numbers is not actually -1/12. It’s a divergent sum. You could call it infinity.

There’s a little parlor trick that maths teachers like to perform where they do algebraic manipulation on a simple formula, eventually cancelling out the variables and arriving at some absurd statement like 1=2. The game is for the students to figure out what went wrong.

The trick is always that at some point, snuck into the progression, you ended up dividing something by “(X-X)” before moving on, seemingly without violating any algebraic rules. Very astute students (or ones who were warned by students from earlier classes haha) will notice that right at that point in time, you are in fact attempting to divide by zero, which is not possible.

So the reason you ended up with 1=2 is because you applied rules to something which they definitionally cannot apply to. At that point, the equation became undefined.

…the funny thing though, is you were able to just…continue. And get something to come out. Now, in this case, that thing was utter nonsense. An amusement for children to help teach them of various pitfalls they might fall into when playing with numbers.

But what if you were one of the most brilliant mathematicians who ever lived, and you were concerning yourself with questions such as,

“What would happen if I took [1 + 2 + 3 + 4+ 5….], and subtracted [1 + 1/2 + 1/3 + 1/4 + 1/5…]”

Now, those are both divergent sums. So we could just call them both infinity. But the second infinity just kind of…feels smaller, doesn’t it? It feels like you should be able to just…perform some type of operation and get…something to come out.

It wouldn’t be “correct” to do so, but this is basically what Ramanujan did. Illegal math. With a nonsense output of negative -1/12.

The funny thing though is that this “nonsense output” is actually now a cornerstone of quantum mechanics. It turns out we subtract diverging infinites from each other literally all the time, even just by walking around, and that pesky little -1/12 trick has proven to be consistently useful. Astonishing.

He died a preventable death at 32. Imagine what the world might have looked like today if only he could have dreamed a little longer, asked more impossible questions, and broken more rules. Specifically he died after a bout of dysentery, in case anyone is confused why they’re reading all this in the shit posting community

Makes sense though.

The complex Riemann-Zeta function (Zeta(x) = sum from 1 to infinity of 1/n^x) only converges for Re(x) > 1, else it is infinite. We can use analytic continuation - which I understand as extending the function such that it remains holomorphic (basically continuous but in the complex numbers and for some reason well-defined) and find the value of Zeta(-1) that way.

The value of Zeta(-1) when viewed this way is -1/12.

It's not really saying the sum of all natural numbers is equal to -1/12. It's really just a mathematical trick, just like: lim (x -> 0) x/x = 1 but this doesn't mean 0/0 = 1.

This meme is brought to you by math nerds! "I swear we have a sense of humor"

I forget which yt dumb fuck said this. Oh yeah, numberphile.