find values of z such that (1-z)^n=z^n ;; n not equal to 1

Question

again, values of z that solve (1-z)ⁿ=zⁿ n not equal to 1

Answer 1

Yefim has generated a correct solution, and the only real solution. For n >= 3, we have complex solutions as well. The real component of the complex solutions will be 1/2. This, along with de moivre's equation, will help us identify our complex solutions (the n'th power will be real, and z and 1-z are complex conjugates, so the n'th powers will be equal). For n odd, our solutions are1/2+1/2 i tan 180°x/n,1/2- 1/2 i tan 180°x/n, x = 1, ... (n-1)/2; for n even, our solutions are1/2+1/2 i tan 180°x/n,1/2-1/2 i tan 180°x/n x = 1, ... ,n/2 - 1

(to see how we derived the solution, note that r(cos θ + i sin θ) = r cos θ(1 + i tan θ); we just let r cos θ = 1/2 to have 1/2 as the real component.)

Answer 2

(1 - z)ⁿ = zⁿ, 1 - z = z; 2z = 1; z = 1/2