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Thread: [SOLVED] Is this answer and steps correct?

  1. #1
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    [SOLVED] Is this answer and steps correct?

    If not, please do correct me.

    The question is,

    The complex number z is given by z = 1 - i. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$.

    I don't know, I just feel that something is not right here even though I've checked many times. Here's my answer:

    $\displaystyle -1/2 - (5/2)i$

    Ok. I've figured out how to use LaTex.

    The complex number z is given by $\displaystyle z = 1 - i$. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$

    Here is how I expressed it in the form p + qi:

    =$\displaystyle z^2 - \frac{1}{z}$

    =$\displaystyle (1 - i)^2 - \frac{1}{1 - i}$

    =$\displaystyle (1 - 2i - 1) - \frac{1}{1 - i}$

    =$\displaystyle \frac{(1 - 2i - 1)(1 - i) - 1}{1 - i}$

    =$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}$

    =$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}\;\text{x}\,\frac{1 + i}{1 + i}$

    =$\displaystyle \frac{(-3 - 2i)(1 + i)}{2}$

    =$\displaystyle \frac{-3 - 5i + 2}{2}$

    =$\displaystyle \frac{-1 - 5i}{2}$

    =$\displaystyle \text{-}\,\frac{1}{2}\;\text{-}\,\frac{5}{2}\;\text{i}$

    =Correct or Wrong? Please help to evaluate! Thanks. If this is correct then that means my modulus and argument are also correct.
    Last edited by mark1950; Jun 3rd 2009 at 06:17 PM.
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  2. #2
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    try again
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  3. #3
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    Quote Originally Posted by mark1950 View Post
    If not, please do correct me.
    The question is,
    The complex number z is given by z = 1 - i. Express $\displaystyle z^2 - 1/z$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - 1/z$.
    First, notice that your exponents did not show up in your OP.
    I added the correct LaTeX codes.
    Now it is not clear what you are asking.
    Is it $\displaystyle z^2 - \frac{1}
    {z}\;\text{ or } \,\frac{{z^2 - 1}}
    {z}$
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  4. #4
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    Ok. I've figured out how to use LaTex.

    The complex number z is given by $\displaystyle z = 1 - i$. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$

    Here is how I expressed it in the form p + qi:

    =$\displaystyle z^2 - \frac{1}{z}$

    =$\displaystyle (1 - i)^2 - \frac{1}{1 - i}$

    =$\displaystyle (1 - 2i - 1) - \frac{1}{1 - i}$

    =$\displaystyle \frac{(1 - 2i - 1)(1 - i) - 1}{1 - i}$

    =$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}$

    =$\displaystyle \frac{(-2i - 2 - 1)}{1 - i}\;\text{x}\,\frac{1 + i}{1 + i}$

    =$\displaystyle \frac{(-3 - 2i)(1 + i)}{2}$

    =$\displaystyle \frac{-3 - 5i + 2}{2}$

    =$\displaystyle \frac{-1 - 5i}{2}$

    =$\displaystyle \text{-}\,\frac{1}{2}\;\text{-}\,\frac{5}{2}\;\text{i}$

    =Correct or Wrong? Please help to evaluate! Thanks. If this is correct then that means my modulus and argument are also correct.
    Last edited by mark1950; Jun 3rd 2009 at 06:11 PM.
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  5. #5
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    Quote Originally Posted by mark1950 View Post
    Ok. I've figured out how to use LaTex.

    The complex number z is given by $\displaystyle z = 1 - i$. Express $\displaystyle z^2 - \frac{1}{z}$ in the form p+qi where p and q are real and find the modulus and argument of $\displaystyle z^2 - \frac{1}{z}$

    Here is how I expressed it in the form p + qi:

    =$\displaystyle z^2 - \frac{1}{z}$

    =$\displaystyle (1 - i)^2 - \frac{1}{1 - i}$

    =$\displaystyle (1 - 2i - 1) - \frac{1}{1 - i}$
    At this point I would combine like terms to simplify things: (1 - 2i - 1) = -2i:

    = $\displaystyle -2i - \frac{1}{1 - i}$

    Now combine:

    = $\displaystyle \frac{-2i(1 - i) - 1}{1 - i}$

    = $\displaystyle \frac{-2i - 2 - 1}{1 - i}$

    = $\displaystyle \frac{-2i - 3}{1 - i}$

    Now multiply the numerator & denominator by the denominator's conjugate:

    = $\displaystyle \frac{-2i - 3}{1 - i}$

    = $\displaystyle \frac{-2i - 3}{1 - i} \times \frac{1 + i}{1 + i}$

    = $\displaystyle \frac{-2i + 2 - 3 -3i}{1 + 1}$

    = $\displaystyle \frac{-1 -5i}{2}$

    = $\displaystyle -\frac{1}{2} -\frac{5}{2}i$

    We get the same answer nevertheless.

    The modulus of this complex number would be

    $\displaystyle |z| = \sqrt{\left( -\frac{1}{2} \right)^2 + \left( -\frac{5}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{25}{4}} = \frac{\sqrt{26}}{2}$

    The argument would be found by solving

    $\displaystyle tan(\theta) = \frac{-5/2}{-1/2} = 5$

    $\displaystyle \theta = tan^{-1}(5) + 180\ deg$ (because the point would be in Quadrant III)

    $\displaystyle \theta = 258.69\ deg$


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    Last edited by yeongil; Jun 3rd 2009 at 09:06 PM.
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  6. #6
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    Oh, ok thanks!
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