Results 1 to 5 of 5

Math Help - Complex Roots of Unity

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    49

    Complex Roots of Unity

    Hello, I am having trouble drawing the unit circle in complex roots of unity.

    For exaple the question: z^4 = -16

    I have worked out that r = 2, and that each angle is pi/2. How do i Solve the question? I thought that z1 would equal -1, but it does not. Can someone please explain how to do it?
    EDIT: it was actually z^4 = -16
    Last edited by noobonastick; March 31st 2009 at 11:37 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by noobonastick View Post
    Hello, I am having trouble drawing the unit circle in complex roots of unity.

    For exaple the question: z^4 = 16

    I have worked out that r = 2, and that each angle is pi/2. How do i Solve the question? I thought that z1 would equal -1, but it does not. Can someone please explain how to do it?
    Hi

    Let z = r\:e^{i\theta} then z^4 = r^4\:e^{4i\theta}

    r^4\:e^{4i\theta} = 16 implies
    r^4 = 16 and 4\:\theta = 2k\pi

    which means r = 2 and \theta = k\frac{\pi}{2}

    k=0 => z=2
    k=1 => z=2i
    k=2 => z=-2
    k=3 => z=-2i
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    Posts
    49
    Sorry we use the form cos(theta) +isin(theta). Could you explain it in those terms.
    (i also edited my original post)
    EDIT: dont worry i have figured it out. Thanks
    Last edited by noobonastick; March 31st 2009 at 12:03 PM.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,110
    Thanks
    2
    Quote Originally Posted by noobonastick View Post
    Sorry we use the form cos(theta) +isin(theta). Could you explain it in those terms.
    (i also edited my original post)
    EDIT: dont worry i have figured it out. Thanks
    What are you not seeing?

    You have \theta = k \cdot \frac{\pi}{2}.

    Substitute k = 0, 1, 2, and 3

    It matters not what notation you use.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,603
    Thanks
    1421
    -16 is at \theta= \pi as measured from the positive x-axis. The first root (the principle fourth root of unity) will be at \pi/4 and the other three will be \pi/4+ \pi/2= 3\pi/4, 3\pi/4+ \pi/2= 5\pi/4, and [tex]5\pi/4+ \pi/2= 7\pi/4[/itex] on the circle of radius 2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Roots of Unity
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: June 1st 2011, 04:39 PM
  2. Sum over roots of unity
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 4th 2011, 10:31 AM
  3. Complex - six roots of unity :S
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: June 2nd 2010, 02:31 AM
  4. nth roots of unity?
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 25th 2009, 10:55 AM
  5. Replies: 5
    Last Post: January 13th 2009, 03:37 AM

Search Tags


/mathhelpforum @mathhelpforum