1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Think about rotation matrix $R_a.R_b=R_{a+b}$ Does it make a sense ? $\endgroup$ – Khosrotash Feb 26 '17 at 9:56
  • $\begingroup$ In my experience Rotation matrix doesn't change the length (If not send me to reference) $\endgroup$ – Ursescu Ionut Feb 26 '17 at 10:01
  • $\begingroup$ You are right , it doesn't change the length ,but add the angles . $\endgroup$ – Khosrotash Feb 26 '17 at 10:09
1
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.