Observe the sequence $(i^{n})=(i,-1,-i,1,\dots).$

The notation $\sqrt{-1}$ is ugly and unnecessary: use $i$ instead; some engineers prefer $j$.

Note that $z=\bar{z}$ if and only if $z$ is real.

$2\Re z=z+\bar{z}$;   $2i\Im z=z-\bar{z}$