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Thread: 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 $\displaystyle u$ is a complex number, it can be written as

    $\displaystyle u = x + iy$.


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

    So that means

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

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

    But you are told $\displaystyle |u| = r = 1$.


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


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