Results 1 to 6 of 6

Math Help - complex numbers

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    239

    complex numbers

    Express the comlex number (√3+i) in trigonometric form. Hence find the smallest positive integer n that (√3+i)^n is a real number.

    This is how I have started it however im really not sure if this is right hoping someone could help.

    ArgZ = tan^-1 (1/√3) = 30

    |Z| = √3+1 = 2

    Z = 2(cos30+isin30)

    Is this correct??

    Also how do i complete the question??

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,386
    Thanks
    1476
    Awards
    1
    To finish this problem it is best to use number notation.
    \left( {\sqrt 3 + i} \right) = 2\left( {\cos \left( {\frac{\pi }<br />
{6}} \right) + i\sin \left( {\frac{\pi }{6}} \right)} \right).
    Now real numbers have arguments that are integral multiples of pi
    So the second answer is n=6.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member Henderson's Avatar
    Joined
    Dec 2007
    Posts
    124
    Thanks
    2
    Using DeMoivre,


    So you're finding n so that 2^n(cos (nx) + i sin (nx)) is real.

    The only way to do that is if your imaginary part is zero, as in sin (nx) = 0.

    Since x is 30 degrees, your first option is n=6, as  sin (6*30) = sin (180) = 0.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2008
    Posts
    239
    Quote Originally Posted by Plato View Post
    Now real numbers have arguments that are integral multiples of pi
    Could you please explain this sentence lol, unsure where the 6 came from :/ thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Sep 2008
    Posts
    239
    still on the topic of complex numbers, could anybody tell me if this is correct?

    complex number Z and conjugate Zbar satisfy equation

    3Z + Zbar = (11+7i)/(1+i)

    find Z in form x + iy


    This is how i attempt to answer it but unsure if its correct

    let Z = x + iy
    Zbar = x - iy

    3(x+iy) + (x-iy) = (11+7i)/(1+i)

    3 (x+iy) + (x-iy) = 10 + 6i

    2(x + iy) = 10 + 6i

    (x+iy) = (10 + 6i))/2

    x + iy = 5 + 3i


    is this correct anybody please???
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,386
    Thanks
    1476
    Awards
    1
    Where are you getting your calculations?
    \frac{{11 + 7i}}{{1 + i}} = \frac{{\left( {11 + 7i} \right)\left( {1 - i} \right)}}<br />
{2} = 9 - 4i.

    \begin{gathered} 3z + \overline z = 9 - 2i \hfill \\ 3(x + yi) + (x - yi) = 9 - 2i \hfill \\ 4x + 2yi = 9 - 2i \Rightarrow \quad x = \frac{9}{4}\,\& \,y = - 1 \hfill \\ <br />
\end{gathered}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. raising complex numbers to complex exponents?
    Posted in the Advanced Math Topics Forum
    Replies: 10
    Last Post: March 25th 2011, 10:02 PM
  2. Replies: 1
    Last Post: September 27th 2010, 03:14 PM
  3. Replies: 2
    Last Post: February 7th 2009, 06:12 PM
  4. Replies: 1
    Last Post: May 24th 2007, 03:49 AM
  5. Complex Numbers- Imaginary numbers
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 24th 2007, 12:34 AM

Search Tags


/mathhelpforum @mathhelpforum