Question

Question #61d86

Answer 1

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`