1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Imaginary number solution?

  1. Dec 13, 2009 #1
    I'm doing some practice problems for my mechanics exam tomorrow (good ol' SHM) and I can't solve this for the life of me:

    Determine: (-1+i)^(1/3)

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Dec 13, 2009 #2

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    Write (-1 + i) in polar form (rexp(itheta)). Then multiply the angle by 1/3, and take the cube root of the radius.
     
  4. Dec 13, 2009 #3
    Thanks very much for the answer. IT WORKS!
    However, I don't understnad why/how. Whats is the reasoning behind multiplying then angle and then cube rooting the radius?
     
  5. Dec 13, 2009 #4
    Think about the laws of exponents. If [tex]-1 + i = r e^{ i \theta }[/tex], then [tex](-1 + i)^{ \frac{ 1 }{ 3 } } = ( r e^{ i \theta } )^{ \frac{ 1 }{ 3 } } = \sqrt[3]{r} ( e^{ i \theta } )^{ \frac{ 1 }{ 3 } } = \sqrt[3]{r} e^{ i \frac{ \theta }{ 3 } }[/tex].
     
  6. Dec 13, 2009 #5

    nicksauce

    User Avatar
    Science Advisor
    Homework Helper

    Taking the cube root of the radius should be obvious. The radius is a real number, and the cube root satisfies (AB)^1/3 = A^1/3 * B^1/3. Diving the angle by 3 is an application of De Moivre's formula:

    http://en.wikipedia.org/wiki/De_Moivre's_formula
     
  7. Dec 14, 2009 #6
    To multiply two complex numbers in polar form [tex]a \angle b^\circ [/tex] and [tex]c \angle d^\circ[/tex], you multiply the radii (a x c) and add the angles (b + d), so the product is [tex]ac \angle (b+d)^\circ[/tex]

    Cubing a complex number [tex]a \angle b^\circ[/tex] is then [tex](aaa) \angle(b+b+b)^\circ = a^3 \angle(3b)^\circ[/tex]

    So, to find the cube root of a complex number, you can see that you take the cube root of the radius and divide the angle by 3.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Imaginary number solution?
  1. Imaginary Numbers (Replies: 4)

  2. Imaginary Numbers (Replies: 14)

Loading...