<s>[INST] How to prove $|z|\cdot |w|=|z \cdot w|$ form complex numbers? I am trying to prove $|z|\cdot |w|=|z \cdot w|$
$|z\cdot w|=\sqrt{(ac-bd)^{2}+(ad+bc)^{2}}$
and
$|z|\cdot|w|=\sqrt{a^{2}+b^{2}}\cdot\sqrt{c^{2}+d^{2}}=\sqrt{(a^{2}+b^{2})\cdot(c^{2}+d^{2})}=\sqrt{(a^{2}c^{2}+a^{2}d^{2}+b^{2}c^{2}+b^{2}d^{2})}$
And I'm stuck
Help?
 [/INST] Now, $$(ac-bd)^2+(ad+bc)^2=a^2c^2-2abcd+b^2d^2+a^2d^2+2abcd+b^2c^2=$$
$$=a^2c^2+a^2d^2+b^2c^2+b^2d^2=a^2(c^2+d^2)+b^2(c^2+d^2)=(a^2+b^2)(c^2+d^2)$$
</s>