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Difficult FP1 complex numbers question

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z_1 = \frac{1}{2} (1+i\sqrt3)





z_2 = i







\mathrm{Find all natural numbers}  k \mathrm{such that} z_1^k = z_2^k = 1


    What I've tried:

    - Attempting to repeatedly expand  z_1 = \frac{1}{2} (1+i\sqrt3). When  k = 3 , the result is -1.

    - Realized that  k must be a multiple of 4


    Anything else I can do? I don't have much experience with proofs so I'm not entirely sure how to move on
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    (Original post by ETbuymilkandeggs)
    

z_1 = \frac{1}{2} (1+i\sqrt3)





z_2 = i







\mathrm{Find all natural numbers}  k \mathrm{such that} z_1^k = z_2^k = 1


    What I've tried:

    - Attempting to repeatedly expand  z_1 = \frac{1}{2} (1+i\sqrt3). When  k = 3 , the result is -1.

    - Realized that  k must be a multiple of 4


    Anything else I can do? I don't have much experience with proofs so I'm not entirely sure how to move on
    If you write it in exponential form  z_1=e^{\frac{\pi }{6}i} .
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    (Original post by B_9710)
    If you write it in exponential form  z_1=e^{\frac{\pi }{6}i} .
    Thanks for the reply. I haven't come across that type of maths so far - only FP1's complex numbers. Could you perhaps elaborate on how I could use that fact towards a solution?
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    you can use (a^b)^c = a(b*c) for z_1 and z_2, sub into the constraint equation, and then take logs and solve for k.
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    (Original post by ETbuymilkandeggs)
    

z_1 = \frac{1}{2} (1+i\sqrt3)

z_2 = i

\mathrm{Find all natural numbers}  k \mathrm{such that} z_1^k = z_2^k = 1
    Note that when you multiply 2 complex numbers v, w, you get a new complex number whose:

    1) length is the products of v,w
    2) angle is the sum of the angles of v,w

    Here, |z_1| = |z_2| = 1 \Rightarrow  |z_1|^k = |z_2|^k = 1 so taking integer powers of these numbers doesn't change their lengths.

    Also:

    \text{arg}(z_1) = \arctan(\sqrt{3}) = 60^\circ, \text{arg}(z_2) = 90^\circ

    so

    \text{arg}(z_1^k) =  60 k, \text{arg}(z_2^k) = 90 k

    Now you require that these complex number powers are equal; that means that their lengths must be the same (which they are, so forget this) and that they have the same angle, up to a multiple of 360. So we require that the angles differ by a multiple of 360 i.e. we need:

    90k-60k = 360m for some m \in \{1,2,3, \cdots\}
 
 
 
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