50g - Exponent of zero

[AlexFekken]

Perhaps I should expand, at the risk of boring everybody again.

When you build the number systems and arithmetic from the ground up, you define everything very precisely. But as you expand your context (I like that word), previously defined concepts or results may become undefined (e.g. ordering when going from real to complex numbers), or vice versa (e.g. extending/compactifying the real or complex numbers).

While I don't think that there is a context in which 42^0 is anything but 1, 42/0 is a perfectly valid infinity in the extended complex plane but not on the real line or the (two point) extended real line.

To bad for the suggestion that you do not want to consider limits, because in many cases that is just how things (including the numbers themselves) are defined. Or at least that's one standard way of defining for example the real numbers (it's actually a bit more complicated than that). And even when a definition does not necessarily require you to use limits, limiting behaviour is very relevant to the question of whether a certain definition makes sense and will be useful. If you want to, you can build your own mathematical world around the definition that 42/0 = 7, but good luck getting any value out of such a system.