How to prove $|z|\cdot |w|=|z \cdot w|$ form complex numbers?

Question

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?

Answer 1

Now, $$(ac-bd)^2+(ad+bc)^2=a^2c^2-2abcd+b^2d^2+a^2d^2+2abcd+b^2c^2=$$ $$=a^2c^2+a^2d^2+b^2c^2+b^2d^2=a^2(c^2+d^2)+b^2(c^2+d^2)=(a^2+b^2)(c^2+d^2)$$

Answer 2

It's easier if you know that you can write $z = r_z e^{i\theta_z}$ where $r_z$ and $\theta_z$ are real.

Answer 3

We have$$|z.w|=|z|.|w|\iff|z.w|^2=\bigl(|z|.|w|\bigr)^2=|z|^2.|w|^2$$and\begin{align}|z.w|^2&=z.w.\overline{z.w}\\&=z.w.\overline z.\overline w\\&=z.\overline z.w.\overline w\\&=|z|^2.|w|^2.\end{align}