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Thread: complex numbers

  1. #1
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    complex numbers

    z is a complex number and the argument of z^4 is equal to the argument of z. find two possible values between -180 and 180 degrees of the argument of z.

    i am not really sure how to do this
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    Re: complex numbers

    Quote Originally Posted by edwardkiely View Post
    z is a complex number and the argument of z^4 is equal to the argument of z. find two possible values between -180 and 180 degrees of the argument of z.

    i am not really sure how to do this
    Hint: Given the absolute value r and argument t, z can be written as z = r exp(i t) where i^2=-1. Now find z^4 and its argument.
    If you still find it difficult, I will post all the steps of solution.

    .
    ..
    ...
    Be patient, TEX will load, eventually...
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  3. #3
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    Re: complex numbers

    Quote Originally Posted by edwardkiely View Post
    z is a complex number and the argument of z^4 is equal to the argument of z. find two possible values between -180 and 180 degrees of the argument of z.
    I that that the word equal above is used in the sense of equivalence classes.
    That is: $z=2\exp \left( {\frac{{2\pi \bf{i}}}{3}} \right)$ and $w=8\exp \left( {\frac{{8\pi\bf{i}}}{3}} \right)$ are different complex numbers they both have the same argument.
    (Sorry, but it is against my religion to use degrees.)

    So in this question, suppose that $z=r\exp \left(\theta\pi\bf{i} \right)$ so that $z^4=r^4\exp \left(4\theta\pi\bf{i} \right)$.

    Solve: $4\theta=\theta+2\pi,~-\pi<\theta\le\pi,~\&~4\theta=\theta+4\pi$ Change to the Devil's degrees.
    Thanks from edwardkiely
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    Re: complex numbers

    i am confused with you notation. usually a complex number for me is given by z=r(cosθ +isinθ) so z^4 =r^4(cos4θ+isin4θ) . i am not sure really where to go from here?
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  5. #5
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    Re: complex numbers

    For the arguments to be the same, you need $4\theta = \theta+2\pi n$ for some integer value of $n$.

    This gives $\theta = \dfrac{2\pi n}{3}$. In the interval $-\pi = -180^\circ \le \theta \le 180^\circ = \pi$

    So, $-1 \le n \le 1$ to keep $\theta$ in the appropriate range. This gives you three values. $\theta = -\dfrac{2\pi}{3}, \theta = 0, \theta = \dfrac{2\pi}{3}$
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    Re: complex numbers

    Quote Originally Posted by edwardkiely View Post
    i am confused with you notation. usually a complex number for me is given by z=r(cosθ +isinθ) so z^4 =r^4(cos4θ+isin4θ) . i am not sure really where to go from here?
    Using the standard mathematical notation: $r(\cos(\theta)+\bf{i}\sin(\theta))=r\exp(\bf{i} \pi \theta)$.
    It is a much more compact notation.
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  7. #7
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    Re: complex numbers

    i understand that 4θ=θ so you are equaling their angles but i am not sure where 360(n) comes from on the right hand side of the equation.
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  8. #8
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    Re: complex numbers

    Quote Originally Posted by edwardkiely View Post
    i understand that 4θ=θ so you are equaling their angles but i am not sure where 360(n) comes from on the right hand side of the equation.
    When you turn 360 degrees, you are facing the exact same direction. The only way $4\theta = \theta$ is if $\theta=0$. But, if you turn a multiple of 360 degrees, you are facing the same direction.
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