I need help with this problem, I'm very confused

Prove that for all complex numbers z and w:

1. zw* + z*w is always real

2. zw* - z*w is purely imaginary or zero.

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- Jan 9th 2010, 09:39 AMerickvm123Proving complex numbers
I need help with this problem, I'm very confused

Prove that for all complex numbers z and w:

1. zw* + z*w is always real

2. zw* - z*w is purely imaginary or zero. - Jan 9th 2010, 10:55 AMAmer
if you mean by z* the conjugate here is the answer

let

$\displaystyle z = x+iy , w = u+iv $

$\displaystyle \bar{z} = x -iy , \bar{w} = u-iv $

$\displaystyle z\bar{w} = xu+yv+i(yu-vx) $

$\displaystyle \bar{z} w = xu+vy +i(vx-uy) = xu+vy -i(uy-vx)$

$\displaystyle z\bar{w} + \bar{z} w = xu+yv+i(yu-vx)+xu+vy -i(uy-vx) = 2(xu+yv) $

and this is always real

$\displaystyle z\bar{w} - \bar{z} w = xu+yv+i(yu-vx) - (xu+vy -i(uy-vx)) = 2i(yu-vx) $

and this maybe 0 if yu=vx or imaginary if yu dose not equal vx

and it is known that

$\displaystyle z + \bar{z} = 2Re(z) $ Re is the real part of z

$\displaystyle z - \bar{z} = 2Im(z) $ Im is the imaginary part of z