Complex numbers?

Question

1. Prove that Im(xy) = Re(x) + Im(x)Re(y)
for x = a + bi, y = c + di, i = imaginary

2. If z + 1/z is real, prove that either Im(z) = 0 or |z| = 1
for z = x + yi

Answer 1

1. This should be Im(xy) = Re(x) Im(y) + Im(x) Re(y).

Proof:
LHS = Im ((a + bi) (c + di))
= Im (ac + adi + bci + bdi^2)
= Im ((ac-bd) + (ad + bc) i)
= ad + bc
= Re(x) Im(y) + Im(x) Re(y).

2. z = x + yi, so 1/z = (x - yi) / (x^2 + y^2)
So Im(z + 1/z) = y + (-y)/(x^2 + y^2)
= y (1 - 1/(x^2 + y^2))
= 0 <=> y = 0 or 1/(x^2 + y^2) = 1
<=> y = 0 or x^2 + y^2 = 1
<=> y = 0 or |z| = 1.