# $Amp(z)+Amp(w)=\pi.\;$ Find a relation between $\;z\;$ and $\;w.$

The question says:

Let $$\;z\;$$ and $$\;w\;$$ be two non-zero complex numbers such that $$|z|=|w|$$ and $$amp(z)+amp(w)=\pi,\;$$ then find a relation between $$\;z\;$$ and $$\;w.$$

In the solution they turn $$amp(z)+amp(w)$$ into $$amp(Z)-amp(\overline w)\;$$ and equate it to $$\pi$$. What was the need to do so ? I think I may be missing some concept.

The final answer is $$\;z+ \overline w=0.$$

• What is $amp$? Is it argument of the complex number? – user376343 Jan 22 at 9:52

$$w=|w|e^{i\angle w}=|z|e^{i\pi-i\angle z}=e^{i\pi}|z|e^{-i\angle z}=-\overline z.$$