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# FP2: Roots of Unity

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1. if this is true for when Z^n=1 then why, for z^10=1, does k= +/-1,+/-2,+/-3, +/-4 and 5? I have the general solution as cos kpi/5+isin kpi/5. I would be grateful for any explanation😅

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2. (Original post by Mr Pussyfoot)
if this is true for when Z^n=1 then why, for z^10=1, does k= +/-1,+/-2,+/-3, +/-4 and 5? I have general solution as cos kpi/5+isin kpi/5. I would be grateful for any explanation😅

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Those values of k are the principal solutions
3. (Original post by Protoxylic)
Those values of k are the principal solutions
So wouldn't the solutions be all the numbers -1<k<10?

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4. (Original post by Mr Pussyfoot)
So wouldn't the solutions be all the numbers -1<k<10?

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You can either have all the values of k for which the the solutions have argument (0,2pi) or (-pi,pi) these tend to be the principal domains. Technically any domain with period 2pi, but the principal tends to be the above two.
5. (Original post by Mr Pussyfoot)
So wouldn't the solutions be all the numbers -1<k<10?

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They are, remember that the -1 and n-1 solutions are the same as they differ by 2*pi. With -2 and n-2, -3 and n-3 being the same as well etc.
6. By principal do you mean between 0 and 2pi? Sorry I hadn't read the previous posts. Ignore this😂

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7. (Original post by Protoxylic)
You can either have all the values of k for which the the solutions have argument (0,2pi) or (-pi,pi) these tend to be the principal domains.
All right cool I understand now

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8. (Original post by CharlieGEM)
They are, remember that the -1 and n-1 solutions are the same as they differ by 2*pi. With -2 and n-2, -3 and n-3 being the same as well etc.
Thanks for the reminder, I hadn't spotted the relation at first.

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