the time it was not properly used was to show it did not hold when rules where altered (so I am very aware that it was not correct). I agree with fredrik let's close the thread ... it is drifting...
if you take radius 1:
##e^{i2\pi} = cos(2\pi)+i sin(2\pi)=1+ 0i## since you "walked" an entire circle (just filling in the Eulers identity).
walking an entire circle is the same as walking 2 half circles mathematically :
##e^{i2\pi}=e^{i\pi}e^{i\pi}##
walking half a circle gives...
you could see them as ratios yes and you can use them to calculate lengths of triangles and then indeed the radius does not matter but
the equation I gave is an equation which is very helpful and would be lost if you take another radius (as shown in the example).
I think the unit circle is taken not because of physical reasons but mathematical ones.
The main reason I think is because of the
##e^{i\theta}=\cos{\theta}+i\sin{\theta}## (how do i add latex in my reply...I am new).
Suppose it had radius 2 then we would have...