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Thread: Complex numbers help

  1. #1
    Mar 2009

    Complex numbers help

    Hey I really need some help with some questions!:

    a) Express

    in exponential form

    - I got 8e^(-3pi/4) is this correct?

    b) Find three cube roots of

    , giving answers in simplified exponential form
    - I have no idea how to do this, please help?

    c) Then approximate the answers in Cartesian form, a + bi, where a and b are given to two decimal places

    thanks guys
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  2. #2
    MHF Contributor

    Apr 2005
    Any complex number, x+ iy, can be written in exponential form, [tex]re^{i\theta}[tex] where $\displaystyle r= \sqrt{x^2+ y^2}$ and $\displaystyle \theta= tan^{-1}(\frac{y}{x})$.
    Yes, $\displaystyle 8e^{-3\pi/4}$ is correct. That same answer could be written [tex]8e^{5\pi/4}[tex] because $\displaystyle \frac{-3\pi}{4}+ 2\pi= \frac{5\pi}{4}$.

    DeMoivre's theorem: if $\displaystyle z= re^{i\theta}$, then $\displaystyle z^n= r^ne^{in\theta}$ which also works if n is a fraction: $\displaystyle z^{1/n}= r^{1/n}e^{\frac{i\theta}{n}}$ and you can get all n roots by repeatedly adding $\displaystyle 2\pi$ to $\displaystyle \theta$ before dividing by n.
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  3. #3
    MHF Contributor
    Nov 2008
    Quote Originally Posted by omghelpman View Post
    I got 8e^(-3pi/4) is this correct?
    It is actually $\displaystyle 8\:e^{-3i\frac{\pi}{4}}$
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