Solve the following equation:


Where $\bar{z}$ is the complex conjugate of z, and n is a natural number such that $n\neq 2$.

I have tried to write z in rectangular form and polar form. I have tried to play with De Moivre's formula.

But I still do not see where to proceed from here.

Could you please point me to the right direction?


  • $\begingroup$ Multiply both sides by $z$, and taking absolute values on both sides gives $|z|^2=|z|^n$, so either $|z|=1$ or $n=2$, which is contrary to your assumption. $\endgroup$ – Adam Hughes Sep 9 '14 at 3:39
  • $\begingroup$ Does this imply that the only complex number satisfying this equation is any complex number such that its magnitude is 1? $\endgroup$ – user101998 Sep 9 '14 at 3:52
  • $\begingroup$ Not all of them work either, this is a necessary but not sufficient condition. You only asked how to get started, so this is a push towards the full solution. $\endgroup$ – Adam Hughes Sep 9 '14 at 3:53
  • $\begingroup$ Hint: $z^n = z^{n-1}z = \bar{z}z = |z|^2$. $\endgroup$ – achille hui Sep 9 '14 at 3:54
  • $\begingroup$ Okay, let's see what I can do. Thanks! $\endgroup$ – user101998 Sep 9 '14 at 3:59

First, multiply both sides by $z$ to find $$ |z|^2 = z^n $$ So, in particular, $z^n$ needs to be a non-negative real number. So, we can say that the argument of $z$ must be a multiple of $2 \pi/n$.

Now, how about the magnitude? Taking the magnitude on both sides of the original equation gives us $$ |z| = |z|^{n-1} \implies|z|^{n-1} - |z| = 0 \implies\\ |z|\cdot (|z|^{n-2} - 1) = 0 $$ So, what are the possibilities for $|z|$?

Together, the two pieces of information should be enough to figure out all possibilities for $z$.


Use polar coordinates, $z = re^{i\theta}$: $$re^{-i\theta} = r^{n-1}e^{i(n-1)\theta},$$ $$r = r^{n-1}\implies\cdots,$$ $$\cdots$$


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.