If z be a complex number, show that the minimum value of |z| + |z - 1| is 1.

For any complex number z show that zbar z=|z|^2

If z1=x1+iy1 and z2=x2+iy2 be two complex number,then show that bar(z1,z2)=bar z1.bar z2 .

If z = x + iy, show that the locus represented by |z - 3 |= |z + 6 |is a straight line

If z_1 , and z_2 be two complex numbers prove that z_1barz_1=|z_1|^2

If 31z5 is a multiple of 9, where z is a digit, what is the value of z? You will find that there are two answers for the last problem. Why is this so?

Write any two complex numbers,then show that |z1+z2|^2+|z1-z2|^2=2(|z1|^2+|z2|^2)

If z_1 , and z_2 be two complex numbers prove that |z_1+z_2|^2+|z_1-z_2|^2=2[|z_1|^2+|z_2|^2]

For any two complex numbers z1 and z2 ,prove that |z1.z2|=|z1|.|z2|

For all two complex numbers z1 and z2,prove that Re(z1z2)=Re z1 Re z2-Im z1 Im z2

If z=1+i ,then mention the value of arg(z) .