It's also on Wikipedia. We're not talking about operating on infinities, we're talking about operating on a sum of numbers, where the numbers go on to infinity, we do this all the time with calculus and other branches of math.
Alright, let's see then, here's a problem: There's a hotel with an infinite number of rooms, however each room is filled. A new resident enters and asks for a room. Clearly he can't walk to the end of the hallway, as he'll never reach an unoccupied room, since there are an infinite number of them and you can never get to the end of infinity. Instead, the hotel owner tells the man in Room 1 to move down to Room 2 and tell the man in Room 2 to go to Room 3 and so on. The new resident then takes Room 1, as it is clear. The hallway is still infinite, even after we've added one new resident. That means that ∞+1=∞. Now, this defies all mathematics. Surely if you add one to a number it will not equal the same amount, right? Now let's say an infinite amount of new residents comes and asks for a room for each of them. The hotel owner simply asks each resident in the hotel to multiply their room number by two, and then move to that room. This means that rather than rooms 1,2,3,4,5,etc being occupied, rooms 2,4,6,8,10,etc are occupied, leaving an infinite numbers of free rooms in between. The infinite number of new residents then slides in nicely with the already present number of residents. There are still an infinite number of residents in the hotel, mind you. That means ∞*2=∞. Now that doesn't make any sense either! Surely doubling a number will not yield the same exact number, will it? That's the kind of logic you get when you work with infinities. Also, quoted from Wikipedia: "Because the sequence of partial sums fails to converge to a finite limit, the series is divergent, and it does not have a sum in reality. ... Generally speaking, it is dangerous to manipulate infinite series as if they were finite sums, and it is especially dangerous for divergent series. If zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step <snip for anyone who wants to solve the problem> is not justified by the additive identity law alone. For an extreme example, appending a single zero to the front of the series can lead to inconsistent results." so yeh
4 pages damn, should have done it when it was at 2 pages, thankfully 90% of it can be skipped as maths is bad. I'll do it when I'm not sick.
