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Math Help - Complex number plane multiplication

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    Complex number plane multiplication

    We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.


    Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.
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    Quote Originally Posted by meggnog View Post
    We know that multiplying the plane of complex numbers by a constant complex number u multiplies all distances by the absolute value of u.


    Assuming that u is not equal to 1, and hence that uz is not equal to z when z is not equal to 0, deduce that multiplication by u is a rotation.
    Let u = \cos{\theta_1} + i\sin{\theta_1}

    and z = \cos{\theta_2} + i\sin{\theta_2}.


    Therefore uz = (\cos{\theta_1} + i\sin{\theta_1})(\cos{\theta_2} + i\sin{\theta_2})

     = \cos{(\theta_1 + \theta_2)} + i\sin{(\theta_1 + \theta_2)}.


    Since the angle has changed, that means you have a rotation.
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