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Math Help - What is the third root of i?

  1. #1
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    What is the third root of i?

    What is the third root of i?
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  2. #2
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    Quote Originally Posted by Manizzle View Post
    What is the third root of i?
    z^3 = i

    \Rightarrow r^3 \text{cis} (3 \theta) = \text{cis} \left( \frac{\pi}{2} + 2n \pi\right) where n is an integer.

    Therefore r = 1 and \theta = \, ..... (get three values of \theta that give three distinct solutions for z).

    This gives the three cube roots of i in polar form. Convert to Cartesian form as required.
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    Quote Originally Posted by Manizzle View Post
    What is the third root of i?
    .

    The general formula that give the n, nth roots of a complex No z is:


    .... |z|^\frac{\ 1}{n}[cos\frac{\ t + 2\pi j}{n} +isin\frac{\ t + 2\pi j}{n}] = |z|^\frac{\ 1}{n}  e^\frac{\ i(t+2\pi j)}{n}...........,where 0\leq j<n

    And in our case n=3 ,hence 0\leq j<3, t=π/2 ,|z|=1 and for:

    .........j=0,j=1,j=2.....we get the 3 roots of i.............................
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  4. #4
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    Quote Originally Posted by archidi View Post
    .

    The general formula that give the n, nth roots of a complex No z is:


    .... |z|^\frac{\ 1}{n}[cos\frac{\ t + 2\pi j}{n} +isin\frac{\ t + 2\pi j}{n}] = |z|^\frac{\ 1}{n}  e^\frac{\ i(t+2\pi j)}{n}...........,where 0\leq j<n

    And in our case n=3 ,hence 0\leq j<3, t=π/2 ,|z|=1 and for:

    .........j=0,j=1,j=2.....we get the 3 roots of i.............................
    1. j is used in some circumstances to represent \sqrt{-1} so it's not the ideal pronumeral to use here.

    2. j could be confused with J which is a common symbol for rational numbers. So again, not the ideal choice of pronumeral.

    3. It needs to be made very clear that the j represents the set of integers.

    4. The values of these integers is not restricted to 0, 1, ... n. In fact, the required values can be any n consecutive integers, including negative values.
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