Calculate z+1/z if z = (-1+i*3^1/2)/2.

To  calculate z+1/z if z =
(-1+i*3^1/2)/2

z =
(-1+6*3^(1/2)/2

z = (-1/2) + ((sqrt3)/2)
i.

1/z = 1/{(-1/2+
((sqrt3)/2)i}

1/z = {[ (-1/2)- ((sqrt3)/2)i)]/{[ (-1/2) +
((sqrt3)/2)i)]*{[ (-1/2)* ((sqrt3)/2)i)]}. We multiplied both numerator and denominator
by {[ (-1/2)- ((sqrt3)/2)i)].

1/z = {[ (-1/2) -
((sqrt3)/2)i)]/{1/4 - 3/4}.

1/z = (-2){[ (-1/2) -
((sqrt3)/2)i)].

1/z = 1+
(sqrt3)i.

Therefore z+1/z =  (-1/2) + ((sqrt3)/2) i + 1+
(sqrt3)i.

z 1/z = (1-1/2) +( sqrt3/2
+sqrt3)i.

 z+1/z = (1/2)
+(3/2)(sqrt3)*i.