Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number.
I understand that if
$\ z = (a+bi) $ then $\ z - \bar z = 2bi  $ and the denominator $\ 1+z\bar z $ is $\ 1+|z|^2 $ and therefore it is a real number. so need to prove the numerator is imaginary.
I first tried to see what happens if I take $\ z^2 - \bar z^2 $ and it is imaginary $\ (a+bi)^2-(a-bi)^2 = 4abi $
and also $ z^{2010} = (z^{1005})^2$ so if $\ z^{1005} $ is imaginary number... but I have no clue  what is $\ z^{1005}$ and maybe it's a real number..?

$z\bar z=|z|^2$
$(a+ib)^n-(a-ib)^n=2i\sum_{r=0}^{2r+1\le n}\binom n{2r+1}a^{n-(2r+1)}b^{2r+1}(-1)^r$
