Prove that: `R_(e)z=(z+overline(z))/(2),I_(m)z=(z-overline(z))/(2i)`

Prove that: (i) z=overline(z) iff z is real (ii) z=-overline(z) iff z is either zero or purely imaginary.

if z is a complex number belonging to the set S={z:|z-2+i|gesqrt(5)} and z_(0)inS such that (1)/(|z_(0)-1|) is maximum then arg ((4-z_(0)-overline(z)_(0))/(z_(0)-overline(z)_(0)+2i)) is

Prove that: (i) R_(e)(z_(1)z_(2))=R_(e)(z_1)R_(e)(z_(2))-I_(m)(z_(1))I_(m)(z_(2)) (ii) I_(m)(z_(1)z_(2))=R_(e)(z_(1))I_(m)(z_(2))+R_(e)(z_(2))I_(m)(z_(1)) .

Show that arg. overline(z)=2pi-arg.z.

If z is any non-zero complex number, prove that the multiplicative inverse of z is (overline(z))/(|z|^(2)) . Hence, express (4-sqrt(-9))^(-1) in the form x+iy, where x,y inR

Define addition and multiplication of two complex numbers z_(1) and z_(2) . Hence show that: (i) R_(e)(z_(1)+z_(2))=R_(e)(z_(1))+R_(e)(z_(2)) (ii) I_(m)(z_(1)+z_(2))=I_(m)(z_(1))+I_(m)(z_(2)) (iii) R_(e)(z_(1)z_(2))=R_(e)(z_(1))R_(e)(z_(2))-I_(m)(z_(1))I_(m)(z_(2)) (iv) I_(m)(z_(1)z_(2))=R_(e)(z_(1))I_(m)(z_(2))+I_(m)(z_(1))R_(e)(z_(2)) .

For any complex number z, prove that: (i) -|z| le R_(e)(z)le|z| (ii) -|z|leI_(m)(z)le|z| .

For all z in C, prove that (i) (1)/(2)(z+bar(z))=Re(z), (ii) (1)/(2i)(z-bar(z))=Im(z), (iii) z bar(z)=|z|^(2), (iv) z+bar(z))"is real", (v) (z-bar(z))"is 0 or imaginary".

Let S={ Z in C : | z - 1|=1 and (sqrt 2 - 1) (z overline z) - i(z - overline z) = 2 sqrt 2} . Let z_1, z_2 be such that |z_1| = max_(z in S) |z| and |z_2| = min_(z in S) |z| . Then |sqrt2 z_1 - z_2|^2 equals: