Complex numbers multiplication I know how to multiply complex numbers (with formula ), but i can't figure out what is really happening on them , I was able to understand that if $z_1z_2=z_3$ then $z_3$ will have the argument of $z_1$ + argument of $z_2$ (its kind of rotation). My question is, what is happening with modules of this $z_3$. I want an intuitive answer not mathematical proof , wanted to understand the phenomenon. 

Denote by $|z|$ the modulus of the complex number $z$. Let $z,w\in \mathbb{C}$. Then $|zw|=|z||w|$ and $\text{arg}(zw)=\text{arg}(z)+\text{arg}(w)$. This completely determines a complex number of you think in terms of the polar coordinates of such a number.
So multiplication by a complex number can be seen as first rotating and then rescaling.
