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Math Help - Need some more help with complex numbers, sorry! Bit Urgent as well.

  1. #1
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    Need some more help with complex numbers, sorry! Bit Urgent as well.

    There is this equation: x^5 - i = 0

    Find all the solutions,

    Now im guessing that there are 5 solutions?

    I'm sorry but Im not sure where to go form here; i was searching the net the other night and found something- i think it was on purplemath but I lost it and have been unable to find out how to tackle this problem and ones like it- any help is appreciated! thank you!
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  2. #2
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    Re: Need some more help with complex numbers, sorry! Bit Urgent as well.

    Quote Originally Posted by Rhys101 View Post
    There is this equation: x^5 - i = 0

    Find all the solutions,

    Now im guessing that there are 5 solutions?

    I'm sorry but Im not sure where to go form here; i was searching the net the other night and found something- i think it was on purplemath but I lost it and have been unable to find out how to tackle this problem and ones like it- any help is appreciated! thank you!
    You are correct, there are five solutions. All will be evenly spaced around a circle, and so have the same modulus and are separated by an angle of \displaystyle \begin{align*} \frac{2\pi}{5} \end{align*}.

    Write \displaystyle \begin{align*} x = r\,e^{i\theta} \end{align*}, then you have

    \displaystyle \begin{align*} \left(r\,e^{i\theta}\right)^5 - 1e^{\frac{\pi}{2}i} &= 0 \\ \left(r\,e^{i\theta}\right)^5 &= 1e^{\frac{\pi}{2}i} \\ r^5e^{5i\theta} &= 1e^{\frac{\pi}{2}i} \\ r^5 = 1 \textrm{ and } 5\theta &= \frac{\pi}{2} \\ r = 1 \textrm{ and } \theta &= \frac{\pi}{10}  \end{align*}

    So the first of these roots is \displaystyle \begin{align*} 1e^{\frac{\pi}{10}i} \end{align*}. Now keep adding and subtracting \displaystyle \begin{align*} \frac{2\pi}{5} \end{align*} to the argument in order to get all five roots in \displaystyle \begin{align*} -\pi < \theta \leq \pi \end{align*}.
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    Re: Need some more help with complex numbers, sorry! Bit Urgent as well.

    Thank you! I think I am going to need to do some more study before I understand it properly, but this is where I need to be, do thank you!
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    Re: Need some more help with complex numbers, sorry! Bit Urgent as well.

    Yes, x^5 = i has five "roots of unity." Just like Prove It said, we let x = re^{i \theta}, then


    (re^{i \theta})^5 = e^{i \frac{\pi}{2}} = i


    You know that the norm, or magnitude, of i is 1. Hence, find all values of \theta that satisfy.
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