Define addition and multiplication of two complex numbers `z_1` and `z_2`. Hence show that :
`R_e (z_1+ z_2)=R_e (z_1)+R_e (z_2)`.

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1 z_2)=R_e (z_1)I_m (z_2)- I_m (z_1)R_e(z_2) .

Define addition and multiplication of two complex numbers z_1 and z_2 . Hence show that : I_m (z_1+ z_2)=I_m (z_1)+I_m(z_2) .

For any two complex numbers z_1 and z_2 , prove that Re (z_1 z_2) = Re z_1 Re z_2 - Imz_1 Imz_2

If z_1,z_2 in C , show that (z_1+ z_2)^2= z_1^2+2 z_1 z_2+ z_2^2 .

If two complex numbers z_1,z_2 are such that |z_1|= |z_2| , is it then necessary that z_1 = z_2 ?

Find the difference of the complex numbers : z_1=- 3 + 2i and z_2= 13-i .

If the complex number z_1 and z_2 be such that arg(z_1)-arg(z_2)=0 , then show that |z_1-z_2|=|z_1|-|z_2| .

For any complex number z, prove that : -|z| leR_e (z) le |z| .