Oh, I have an another one! Sum of all natural numbers as in: 1 + 2 + 3 + … = - 1/12 Yes! Minus one twelfth! Here are two videos crucial for getting it: Explanation of why 1 - 1 + 1 - 1 + … = 1/2 (important partial result): Explanation of the equation for all naturals:
While definitely cool, I refuse to believe that somehow adding natural numbers makes a negative fraction. I think the weirdness comes where it's just assumed that 1 0 1 ... = 1/2 which I don't buy.
I guess that I have succeeded with delivering mind-blowing fact. And that one bit was 'assumed' as in "we don't want to make this video a boring 30 minute slog so just trust us that we know what we are doing" not "we have just arbitrarily decided on that one result and will continue onward without regard if that's true or not". But it is true. It's not all that counter-intuitive as it might seem, but it forces you to stop assuming that what you know about finite operations hold true. Article. Consider this: Sum of all numbers from k = 0 up to k -> Infinity (k is assumed a natural number) in form of: 1/k! (inverse of a factorial of k). You are adding 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 + … and you are getting Euler's number. Think about it. You are adding rational numbers but get an irrational number as the result! It defies normal finite-element reasoning. I mean, it's not like I can add two rational numbers and get something like square root of 3, right? Mathematics is full of such situations where you can't allow yourself to just go on intuition, but need to have a strong formal backing. I can do a formal reasoning for you if you are actually interested.