# 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.

• Think about rotation matrix $R_a.R_b=R_{a+b}$ Does it make a sense ? – Khosrotash Feb 26 '17 at 9:56
• In my experience Rotation matrix doesn't change the length (If not send me to reference) – Ursescu Ionut Feb 26 '17 at 10:01
• You are right , it doesn't change the length ,but add the angles . – Khosrotash Feb 26 '17 at 10:09

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.
• 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. – Ursescu Ionut Feb 26 '17 at 10:14
• Multiplication in $\mathbb{R}$ rescales stuff, the argument of a complex number determines the rotation. – Mathematician 42 Feb 26 '17 at 21:52