" (18) Prove that : "R_(e)z=(z+z)/(2),I_(m)z=(z-bar(z))/(2i)

Prove that: R_e z= (z+bar(z))/2, I_m z= (z-bar(z))/(2i) .

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

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

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".

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)) .

If z_(1) = (1+i) and z_(2) = (-2 + 4i) , prove that Im ((z_(1)z_(2))/(bar(z)_(1)))=2 .

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| .

If z_(1)= 3 + 5i and z_(2)= 2- 3i , then verify that bar(((z_(1))/(z_(2))))= (bar(z)_(1))/(bar(z)_(2))