Prove that for any complex number z, the product `z bar(z)` is always a non-negative real number.

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

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

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

Prove that : the product of a complex number and its conjugate is real.

For a complex number z=x+ iy find the modulus of zbar(z) where bar(z) is the conjugate of z

If z is any complex number, prove that : |z|^2= |z^2| .

Consider four complex numbers z_(1)=2+2i, , z_(2)=2-2i,z_(3)=-2-2iandz_(4)=-2+2i),where i=sqrt(-1), Statement - 1 z_(1),z_(2),z_(3)andz_(4) constitute the vertices of a square on the complex plane because Statement - 2 The non-zero complex numbers z,barz, -z,-barz always constitute the vertices of a square.

Express in the form of complex number if z=i^(-39)

Prove that the complex numbers z_(1) and z_(2) and the origin form an isosceles triangle with vertical angle (2pi)/(3),ifz_(1)^(2)+z_(2)^(2)+z_(1)z_(2)=0.

If Z is a complex number, then |Z-8| lt= |Z-2|implies: