The old fashioned notion of dividing a whole slice for N people is not going to do us good in this, we need to break the problem down to its core. When you are slicing a cake, you are taking and series of angles of a circles in parts and distributing them one by one. The procedure is the following: You take the whole 360 deg angle and divide it by the number of people you want to give it to, then you trace that much from the starting point around it, and give that much you traced to the first person. Then repeat until you got no cake left. So consider this:
abraker wrote:As a workaround, take the limit of the thing you are trying to divide the thing by 0 and divide it by x as x approaches 0The limit, as the number of people I'm dividing cake for approaches zero, the number of slices approaches...
You have 2pi of cake and need to divide it by one person. Ok: (2/pi)/1 = 2/pi, a whole cake. Here you go one person!
What about half a person? (2/pi)/0.5 = 4/pi or 2 wholes. Right? Wrong! You just go around once, and obviously the rest doesn't matter since you traced the whole, but for the sake of math we will trace a full circle again. The whole cake has been traced around anyway and the half person gets the full cake
What about a third of a person? Same you just go around and stop where you went around already, effectively making no difference.
What about Infinity? Same no difference. Whole cake.
Does this work for 2 people, you, a skeptical reader might be now wondering, now concerned about the nonsense I wrote. Well yes, idiot reader, it does work. Here is how: (2/pi)/2 = pi or half of a whole. So you give the traced amount to the first person and continue tracing (2/pi)/2 for the second person. Another half!
And so, quod erat demonstrandum
Edit: What happens if you try dividing by infinity: