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Roots of unity

  1. Jun 1, 2014 #1
    Hello everyone.

    How to find the 4th root of -4? I know it's just plugging in the number into the formula but how since n=4, how can we calculate that without calculator? And how to draw it? Here I attached what I have done so far.
     

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  2. jcsd
  3. Jun 1, 2014 #2

    ehild

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    How you draw z=-4? What is the angle of -4? It is not 0 as you wrote.

    ehild
     
  4. Jun 1, 2014 #3
    It would help to express -4 in complex exponential form.
     
  5. Jun 1, 2014 #4

    SteamKing

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    You've made a simple mistake in your calculation of arg(-4).

    z = -4 + i0, or (-4, 0)

    θ = arctan (0/-4) = π

    Even though z = -4, draw it on the complex plane properly.
     
  6. Jun 1, 2014 #5
    Oh yes, that was a careless mistake. Ok, so now I have the values:
    r=4
    n=4
    θ=∏

    But how do I find 4√4 ?
     
  7. Jun 1, 2014 #6

    HallsofIvy

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    You know that [tex]x^4= (x^2)^2[/tex], right? So [tex]\sqrt[4]{4}= \sqrt{\sqrt{4}}[/tex]. What is the square root of 4? What is the square root of that?
     
  8. Jun 1, 2014 #7

    SteamKing

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    Use this information and apply Euler's formula.

    z = r e[itex]^{i θ}[/itex]

    z[itex]^{1/n}[/itex] = r[itex]^{1/n}[/itex] e[itex]^{i kθ / n}[/itex], [itex]0\leq k \lt n[/itex]

    See:

    http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx
     
  9. Jun 1, 2014 #8
    If I compute z = r e[itex]^{i θ}[/itex], wouldn't that bring us back to the start? Because that is -4.
    Could you elaborate please? I don't really get it.
     
  10. Jun 1, 2014 #9
    Thanks! I never thought of that. But what if we were to find the 5th root? I don't think this method can be applied.
     
  11. Jun 1, 2014 #10

    SteamKing

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    That's your problem in a nutshell. You are working with complex numbers and you don't understand what is going on.

    I wrote these two formulas as a reminder of

    1. how to express any number in exponential form, using Euler's formula, and

    2. how to find the n nth roots of said number.

    Your original problem was to find the 4 fourth roots of -4, or in other words, solve the equation

    [itex]z^{4}+4 = 0[/itex] or

    [itex]z^{4}= -4[/itex]

    Let's say the solutions to this equation are the complex numbers

    [itex]ω_{1}, ω_{2}, ω_{3}[/itex], and [itex]ω_{4}[/itex]

    By writing -4 in the form [itex]z = r e^{i θ}[/itex], where z = -4,
    we can use the second formula from the quote,

    z[itex]^{1/n}[/itex] = r[itex]^{1/n}[/itex] e[itex]^{i kθ / n}[/itex]

    to calculate the numerical values of ω

    I really recommend that you study the article linked below very carefully:

    http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx

    For a better visual representation of the cyclic nature of such roots:

    http://mathworld.wolfram.com/RootofUnity.html
     
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