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(a) Find all non-zero complex numbers z satisfying conj(z) = i z². (b) If complex numbers z1, z2,..., zn all lie on the unit circle |z| = 1, prove that |z1 + z2 +... + zn| = |1/z1 + 1/z2 +... + 1/zn|.

1. z = i, or z = (sqrt3 - i)/2, or z = (-sqrt3 - i)/2; part (b) proved via conjugates
2. z = -i, or z = (sqrt3 + i)/2, or z = (-sqrt3 + i)/2; part (b) proved
3. z = 1, or z = i, or z = -1; part (b) proved
4. z = i only; part (b) proved

Correct answer: z = i, or z = (sqrt3 - i)/2, or z = (-sqrt3 - i)/2; part (b) proved via conjugates

Solution

Taking modulus, |z| = |z|² so |z| = 1. Writing z = e^(i theta), conj(z) = e^(-i theta) and i z² = e^(i(2theta + pi/2)); solving gives the three roots. For (b), since 1/zk = conj(zk), the second sum is the conjugate of the first and has equal modulus.

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