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Math Help - Proving complex numbers

  1. #1
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    Proving 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.
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  2. #2
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    Quote Originally Posted by erickvm123 View Post
    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.
    if you mean by z* the conjugate here is the answer
    let

    z = x+iy , w = u+iv

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

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

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


    z\bar{w} + \bar{z} w = xu+yv+i(yu-vx)+xu+vy -i(uy-vx) = 2(xu+yv)
    and this is always real

    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

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

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