Question #61d86

1 Answer
Aug 27, 2016

For A and B, let #w_1 = x_1+iy_1, w_2 = x_2+iy_2#, where #x_1,x_2,y_1,y_2 in RR#.

A) Proof:
#Re(i*w_1) = Re(i(x_1+iy_1)) = Re(-y_1+ix_1) = -y_1 = -Im(x_1+iy_1)=-Im(w_1)#

B) Proof:
#Im(w_1*i)=Im((x_1+iy_1)i)=Im(-y_1+ix_1)=x_1=Re(x_1+iy_1)=Re(w_1)#

C) Counterexample: Let #w_1 = 1+i, w_2 = 1-i#. Then
#Re(w_1)*Re(w_2) = 1*1 = 1#, but
#Re(w_1*w_2) = Re(2) = 2#

D) Counterexample: Let #w_1 = 1, w_2 = i#. Then
#(Im(w_1))/(Im(w_2)) = 0/1 = 0#, but
#Im(w_1/w_2) = Im(1/i) = Im(i/i^2)=Im(-i)=-1#