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.