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Math Help - Roots of Unity

  1. #1
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    Roots of Unity

    Hi,
    Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
    Here's an example:
    z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}
    z^{6}=e^{\frac{7i\pi}{5}+2\pi in}
    z=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}

    Where n is an integer, i is the complex number and e an exponential.

    Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.

    However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?

    Hope you can help me
    Last edited by imagemania; June 1st 2011 at 04:38 PM.
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  2. #2
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    Quote Originally Posted by imagemania View Post
    Hi,
    Roots of unity is a very easy topic for me, though i am misunderstanding one small part of it which i wish to rectify.
    Here's an example:
    z^{\frac{6}{7}}=e^{\frac{i\pi}{5}}
    z^{6}=e^{\frac{7i\pi}{5}+2\pi in}
    z^{6}=e^{\frac{7i\pi}{30}+\frac{2\pi in}{6}}
    Where n is an integer, i is the complex number and e an exponential.
    Now i undertand the use of 2pi i n, as its roots of unity and therefore will go back to being equal to 1.
    However i don't understand why we pluck in the 2 pi i n in the second step, rather than the first or the last. By raising to the power that induces the 2 pi i n to appear? Why?
    Can you please explain how this question has anything to do with roots of unity/
    It does not make any sense to me.
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  3. #3
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    [Ignore- it double posted for some reason] :\
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  4. #4
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    Well it might not be, but its under the same area in my notes, the section starts of with:
    z=e^{2\pi ik} = 1 where k is an integer.
    z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}
    Then it extrapolates by bringing in n roots

    Oh i also made a mistake there, the last z should not be raised to the 6.
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  5. #5
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    Quote Originally Posted by imagemania View Post
    Well it might not be, but its under the same area in my notes, the section starts of with:
    z=e^{2\pi ik} = 1 where k is an integer.
    z^{\frac{1}{n}}=e^{\frac{2\pi ik}{n} = 1^{\frac{1}{n}}
    Then it extrapolates by bringing in n rootsOh i also made a mistake there, the last z should not be raised to the 6.
    That is still nonsense as far as I can read it.
    Why don't you simply post a clear question?
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