"A number is nothing but the proportion of one magnitude to another arbitrarily assumed as the unit."
My main argument against this, philosophically, is that some units are more logical than others, i.e. those which are the atoms of a system.
One person brought up the complex numbers as a counterexample.
I took the quote to mean: x = ur , where x is the unknown magnitude, u is the unit of measure, and r is a real number.
Taking 'number' in the quote above to mean any type of number, in what cases is the above quote invalid?