Why is the Euclidian norm used to measure complex numbers?
The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the Euclidian geometry. As far as a I can see, the only connection between Euclidian plane and pairs of numbers is that the last can be conjugated to the plane in very "neutral" way. But as much as i understand, the euclidian plane is part of a more general notion of manifold, and you can conjugate pairs of numbers to any 2 dimensional manifold, including non euclidian planes.
Are we using the euclidian norm ("modulus") only because we want some kind of way to measure complex numbers that "grows bigger as the complex numbers grows, no matter in what direction"? a way of measure which is "free of direction" (not taking into account the direction of which the quantity grows, but only its growth)?
This is my way to understand it so far. If I am right, we can basically use any similar norm to measure complex numbers "size" (independent of direction), not only the Euclidian norm, and they'll do the same work and fulfill their destiny.
Obviously there would be another but similar way to measure their "direction"..
So, am I right, and its just an convention to use that way to measure "size" and "direction" of complex numbers by the so called modulus and argument? Or am I wrong? thx for answering! :)