Question

Prove conjugate (z1*z2)=conjugate z1 * conjugate z2

Answer 1

If
`z_1 = a+bi` and
`z_2=c+di`, then

`z_1 z_2 = (a+bi)(c+di) = ac+adi+bci+bdi^2 = (ac-bd)+(ad+bc)i`

So
`\overline{z_1 z_2} = (ac-bd) - (ad+bc)i`.

Next, compute
`\overline{z_1} \cdot \overline{z_2}`.

`\overline{z_1} \cdot \overline{z_2} = (a-bi)(c-di)`

`= ac-adi-bci+bdi^2 = (ac-bd) -(ad+bc)i`