So I'm taking some courses in calculus, and I am surprised by how little explaining there is to the definition of the euclidean norm. I have never understood why you want to define the length of a vector through the pythagorean way. I mean sure, it does seem that nature likes that measure of distance, and sure classic geometry is based on it. But in reality when defining continuity and all the other things specific to analysis you shouldn't in my view really be restricted by geometry. Couldnt you choose other kinds of norms which would create an equally well described calculus, or is there some kind of special property to the euclidean norm that I should know of? I mean all I can see it really doing in analysis is that acts as a numerical value. But to fews surprise, you can find a lot of other algebraic relations between vector coordinates that would have the same property. So please just explain why the euclidean norm is so important, and don't be confused by all my talk, some of it probably doesn't make sense.