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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.

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    Think about rotation matrix $R_a.R_b=R_{a+b}$ Does it make a sense ? – 2017-02-26
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    In my experience Rotation matrix doesn't change the length (If not send me to reference) – 2017-02-26
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    You are right , it doesn't change the length ,but add the angles . – 2017-02-26

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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.

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    Hm, should I consider this operation of rotating - rescaling an axiom (the way it was defined ) or it has some relation with $\mathbb{R}$ multiplication. – 2017-02-26
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    Multiplication in $\mathbb{R}$ rescales stuff, the argument of a complex number determines the rotation. – 2017-02-26