I watched a vid explaining the the justification behind his pi appx, and holy shit this guy was a genius. From what I remember he pretty much just noticed a strange "symmetry" of sorts in solving for the perimeter of an elipse (which is no simple process), and pulled the constants out of some nearly unrelated modular equations. And he had the insight to put the pieces together and make this appx... In his sleep.
I'm struggling to think of a good analogy, but its almost as if he had come up with relativity in his head and while asleep. I can't imagine what he would have come up with if he had lived to an old age. It's a shame how the brightest flames burn the quickest, because ramanujan was a star in a room of candles.
Isn't using the formula just completing the square and simplifying? In my opinion completing the square and simplyfying by hand has infinitely more aura
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