I have been having a hard time to prove these two questions

I have been having a hard time to prove these two questions.

I would appreciate if you can help!!!

Thank you!!

(1) If 2, w are complex numbers show that zu wz. (2) If 2, w are complex numbers show that Iz wl s El +lul

Solution

(1). Let z = a+ib and w = c +id. Then zw = (a+ib)( c +id) = ac +iad +ibc +i² bd = (ac-bd) +i( ad+bc) as i² = -1. Further, wz = ( c +id)( a+ib) = ac +ibc + iad +i²bd = (ac-bd) +i( ad+bc) as i² = -1.

Therefore, zw = wz.

(2) If z, w are complex numbers, then Iz+wI² = (z+w)(z+w)’ = (z+w)(z’+w’) = zz’ +ww’ + z’w +zw’ = IzI² + IwI² + z’w +zw’ = IzI² + IwI² + zw’ +(zw’)’ =   IzI² + IwI² + 2Re (zw’). Hence Iz+wI²    IzI² + IwI² + 2IzwI or, Iz+wI²    IzI² + IwI² + 2IzI IwI or, Iz+wI²    (IzI + IwI)² . Now, on taking the non-negative square roots, we get Iz+wI    (IzI + IwI)