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Math Help - nth roots of unity

  1. #1
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    nth roots of unity

    Hello everybody,

    I'm having some difficulties with some problems about complex numbers and I would appreciate it if you could help me. It's mainly about the nth roots of unity.

    1. I know that the four 4th roots of unity are 1, i, -1 and -i. I need to determine if it is possible to write them as 1, w, w and w where w is defined as cis pi/2.

    And then I need to show that 1+w+w+w = 0.


    2. The second problem is also about this topic. I need to find the 5th roots of unity and then I have to display them on an Argand diagram. The second part of this is assuming that w is the root with the smallest positive argument, I need to show that the roots are 1, w, w, w and w. This is very similar to the first problem, so hopefully if you help me with one of them I'm able to do the second one alone.

    3. Third: w=x+yi and P(x,y) move in the complex plane, I need the cartesian equation for |w-i| = |w+1+i| [note: |w-i| means modulus of w-i]

    4. Next: z1= cos(pi/6) + isin(pi/6) and z2 = cos(pi/4) + isin(pi/4). What is the expression for (z1/z2) in the form of z=a+bi.

    5. And the last: What is the fifth root of i ?


    Sorry if it seems too easy for you, but unfortunately for me it's not. Thank you for your time and effort.
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  2. #2
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    Hello, Instigator!

    Here are the last two . . .


    4. z_1\:=\:\cos\frac{\pi}{6} + i\sin\frac{\pi}{6} and z_2\:=\:\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}

    What is the expression for: . \frac{z_1}{z_2} in the form of z\:=\:a+bi

    We have: . \frac{z_1}{z_2} \;=\;\cos\left(\frac{\pi}{6}-\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{6} - \frac{\pi}{4}\right)\;=\;\cos\left(\text{-}\frac{\pi}{12}\right) + i\sin\left(\text{-}\frac{\pi}{12}\right)

    . . Therefore: . \frac{z_1}{z_1} \;=\;\cos\frac{\pi}{12} - i\sin\frac{\pi}{12}



    5. What is the fifth root of i ?
    Since i \:=\:\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}

    . . then one of the fifth roots is: . i^{\frac{1}{5}} \;=\;\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)^{\frac{1}{5}}\;=\;\cos\f  rac{\pi}{10} + \sin\frac{\pi}{10}

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  3. #3
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    Quote Originally Posted by Instigator View Post
    1. I know that the four 4th roots of unity are 1, i, -1 and -i. I need to determine if it is possible to write them as 1, w, w and w where w is defined as cis pi/2.

    And then I need to show that 1+w+w+w = 0.
    Let \omega = e^{2\pi i/n} now use geometric series.
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