Why is this not a homomorphism?

Question

F : C → R such that f (z) = |z|. R and C are groups under addition.

This seems to be a homomorphism to me... can someone explain or provide a counterexample?

Answer 1

It is a homomorphism between the two additive groups if and only if $f(a+b)=f(a)+f(b)$

Thus, for two complex numbers $z,w$, it must follow that $|z+w|=|z|+|w|$.

This is not the case for example if $z=i,w=1$, as $|i+1|=\sqrt{2}$, $|i|+|1|=2$, $2\not= \sqrt{2}$

We can prove this algebraically, with two complex numbers $a+bi, c+di$ $$|a+c+(b+d)i|=\sqrt{(a+c)^2+(b+d)^2}$$

$$|a+bi|+|c+di|=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}$$

These are not equivalent