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Math Help - Complex Plane

  1. #1
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    Complex Plane

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

    Explain why any u with absolute value =1 can be written in the form cos(theta) + isin(theta) for some angle theta, and conclude that multiplication by u rotates the point 1 (hence the whole plane) through angle theta.

    Thanks in advance!
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  2. #2
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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.

    Explain why any u with absolute value =1 can be written in the form cos(theta) + isin(theta) for some angle theta, and conclude that multiplication by u rotates the point 1 (hence the whole plane) through angle theta.

    Thanks in advance!
    If u is a complex number, it can be written as

    u = x + iy.


    But x can be written as r\cos{\theta}, and y can be written as r\sin{\theta}, where r = |u|.

    So that means

    u = r\cos{\theta} + ir\sin{\theta}

     = r(\cos{\theta} + i\sin{\theta}).

    But you are told |u| = r = 1.


    So therefore u = \cos{\theta} + i\sin{\theta}.


    Now if you were to multiply u by 1, you get back u, so you end up rotating by an angle of \theta through the complex plane.
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