Prove the following properties:
z is real if and only if z = `bar(z)`

arg z= pi if and only if z is a

barz= z if and only if z is a

barz = -z if and only if z is a

Prove the following properties: Re(z) = (z + bar(z))/(2) and Im(z) = (z - bar(z))/(2i)

For complex numbers z and w , prove that |z|^2w- |w|^2 z = z - w , if and only if z = w or z barw = 1 .

The points z_(1)z_(2)z_(3)z_(4) in the complex plane are the vertices of a parallelgram taken in order if and only if :

Prove that following inequalities: |z/(|z|)-1|lt=|a rgz| (ii) |z-1|lt=|z||argz|+||z|-1|