Having immense fun reading John D. Barrow's The Infinite Book. A sampler in which Barrow borrows from Frank Morgan (yes, you've heard versions of this before, tis a classic, but none so cunningly put):
Three guys go into a hotel, each with $10 in his pocket. They book one room at $30 a night. A short while later a fax from the headquarters directs the hotel to charge $25 a night. So the receptionist gives the bellhop $5 to take to the three guys sharing the room. Since the bellhop never got a tip from them and because he can't split $5 three ways, he decides to pocket $2 and give them each one dollar back. So each of the three guys has now spent $9 and the bellhop has $2 for a total of $29. Where's the extra dollar?"
Woo-hoo! (Yes, I can be a nerd.)
Ok, I'm throwing this in for Joseph who mentioned it. It's the case of the terminator 0, 1/2 and 1:
We have thus proven that when dealing with infinity, S can be any number.
We begin with the infinite series S = 1 - 1 + 1 - 1 + 1 - 1 ...
which we can rewrite as: S = (1-1) + (1-1) + (1-1) + (1-1) +.. meaning S = 0
but grouped differently: S = 1 + (-1 + 1) + (-1 + 1) +.. or S = 1 + 0 + 0 + .. thus S = 1
so we have proved that S = 0 and S = 1 thus if S = S then 0 = 1 !
not only that, S = 1 - (1 - 1 + 1 - 1 + 1 - 1...) or S = 1-S or 2S = 1 meaning S = 1/2!
What we have here is a beautiful mathematical explanation to the philosophical and metaphysical presumption that ultimately, we are all one and the same, and that, just as we are part of the universe, the universe is within us.