Axiom 2: iff
Axiom 3: for
Axiom 4: The triangle inequality.
How do I get started on any of these axioms?
is |z| ≥ 0, for any complex number z? certainly for any real a,b (why?). thus the positive square root is defined for such a number. why is this non-negative? these are just basic properties of real numbers.
when can |z| be 0? draw a picture. formalize your picture with a proof.
for axiom 3, write out α and z as α = c+id, z = a+ib. do the multiplication. take the norm of the product, and compare with the product of the norms. what do you find?
you have to use your definitions to prove axioms.
you're just guessing here. when is √x = 0? how many real numbers have this property? if (aČ + bČ) = 0, what could a and b possibly be? could a be -4? 6? 117? how about b?|z| can be 0 only when the function z is 0, ie is the 0 function...?
z is not a "function". it's just a plain ol' complex number.
there's no "i" terms in the definition of modulus, look again.I get