For any complex number `z` prove that `|R e(z)|+|I m(z)|<=sqrt(2)|z|`

If z!=0 is a complex number, then prove that R e(z)=0 rArr Im(z^2)=0.

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 .

For any two complex numbers z_1 and z_2 , prove that |z_1+z_2| =|z_1|-|z_2| and |z_1-z_2|>=|z_1|-|z_2|

If z is a complex number such that Re (z) = Im(z), then

If z is a complex number such that Re(z)=Im(z) , then

For any two complex numbers z_(1)" and "z_(2) prove that (i) Re(z_(1)z_(2))=Re(z_1) Re(z_2)-Im(z_1)Im(z_2) .

For any two complex numbers z_(1)" and "z_(2) prove that Im(z_(1)z_(2))=Re(z_1) Im(z_2)+ Im(z_1)Re(z_2) .

For any complex numbers z_1,z_2 and z_3, z_3 Im(bar(z_2)z_3) +z_2Im(bar(z_3)z_1) + z_1 Im(bar(z_1)z_2) is

A point P which represents a complex number Z moves such that |Z-Z_(1)|=|Z-Z_(2)| , then its locus is

Fir any complex number z, the minimum value of |z|+|z-1| is