Solve the equation for complex number.
z = ∣ z ∣ + i ∣ z ∣
Enter real part of sum of all the solutions.
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z z ( 1 − i ) z − i z ⟹ z = ∣ z ∣ + ∣ z ∣ i = ∣ z ∣ ( 1 + i ) = ∣ z ∣ ( 1 + i ) ( 1 − i ) = 2 ∣ z ∣ = 0 Equating the imaginary part on both sides.
Therefore the sum of real part of all the solutions is 0 .
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Let z = ∣ z ∣ e i α . Then ∣ z ∣ e i α = 2 ∣ z ∣ e i 4 π ⟹ ∣ z ∣ = 0