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Math Help - representing this complex fracture into exponent

  1. #1
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    representing this complex fracture into exponent

    \frac{2i}{2+i}

    i don know how to separate the complex and imaginary part of this fracture?
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    Quote Originally Posted by transgalactic View Post
    \frac{2i}{2+i}

    i don know how to separate the complex and imaginary part of this fracture?

    Twice you wrote "fracture": perhaps you meant "fraction"? Anyway, multiplying by the denominator's conjugate:

    \frac{1}{2+i}=\frac{2-i}{5}= \frac{2}{5}-\frac{1}{5}\,i , and now just multiply this by 2i ...

    Tonio
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    Quote Originally Posted by transgalactic View Post
    \frac{2i}{2+i}

    i don know how to separate the complex and imaginary part of this fracture?
    \frac{2i}{2 + i} = \frac{2i(2 - i)}{(2 + i)(2 - i)}

     = \frac{4i - 2i^2}{4 - i^2}

     = \frac{2 + 4i}{5}

     = \frac{2}{5} + \frac{4}{5}i.


    Now putting into polar form...

    |z| = \sqrt{\left(\frac{2}{5}\right)^2 + \left(\frac{4}{5}\right)^2}

     = \sqrt{\frac{4}{25} + \frac{16}{25}}

     = \sqrt{\frac{20}{25}}

     = \frac{2\sqrt{5}}{5}.


    \theta = \arctan{\frac{\frac{4}{5}}{\frac{2}{5}}}

     = \arctan{2}.


    So z = \frac{2\sqrt{5}}{5}\,\textrm{cis}\,\arctan{2}

     = \frac{2\sqrt{5}}{5}e^{i\arctan{2}}.
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    thanks
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