I understand why for n > 0 n/0 is indeterminate but I'm not convinced 0/0 isn't simply 0. If you have zero cookies and split them amongst zero friends then nobody gets anything so zero?

This confuses the hell out of Hector Zeroni

http://videosift.com/video/Numberphile-Problems-with-the-Number-Zero

Skip to 10 minutes for 0/0 if you don't want the background. A quick spoiler clue is that x/x = 1; Also, 0 x 0 = 0, which implies the answer is 0. Indeed, there are a number of different mathematical constructs one can make to demonstrate that the answer is any number one likes.

Mathematics needs to be uniform and consistent, and an operation that can return infinite legitimate values depending on your approach is a disaster as far as this is concerned. Suddenly the whole enterprise of mathematics is ruined. Better to call 0/0 undefined.

Into English and cookies. How many cookies does nobody get when you don't divide cookies among no people? Does this make sense?

Into speed. If I am travelling at speed and you want to measure it you need distance travelled and time taken. If I travelled 100 kilometres in 1 hour you can say I'm travelling 100kph. If I travelled 50 kilometres in half an hour (50 / 0.5) , you can say the same thing. 25km in quarter of an hour (25 / 0.25) and so on and so forth. All come out at 100kph. But what if you decide to measure how far I travel in 0 seconds? I travel 0 metres. 0 / 0 = ? 0kph? And if you took this narrowly focussed measurement every few seconds you would see that my speed remains at 0kph but I manage to cover 100km in an hour anyway. Maths is now broke