Show that the points on an Argand diagram that represent the roots of ((z+1)/z)^6 = 1 lie on a straight line.

We want to simplify this equation to one that we know how to solve. If we let ((z+1)/z) = w, then we need to solve w^6 = 1, which is more familiar. Now we try to find the modulus and argument of w. w = re^(iθ) so by De Moivre's Theorem we have (r^6)(e^(i6θ)) = 1. If two complex numbers are equal then their moduli must be equal so r^6 = 1. Then r = 1 since r is greater than or equal to zero for any complex number; i.e. r cannot equal -1. The argument of a complex number is not unique so 1 = e^(i(0)) = e^(i(2kπ)) for any integer k. This is easiest to understand graphically. The argument of the product of two complex numbers is the sum of their individual arguments so multiplying by e^i(2π) effectively rotates the complex number by 2π radians, so it is unchanged. So w^6 = e^(i(2kπ)) so w = e^i((2kπ)/6) so (z+1/z) = e^i((kπ)/3). Rearranging, z = 1/(e^((ikπ)/3)-1) = 1/(cos(kπ/3)-1+isin(kπ/3)). Now we need to substitute values of k to find as many unique values of z as possible. Substituting six consecutive values of k is sufficient since the seventh will give the same value of z as the first. If we do this (using a calculator) for k = 0, 1, ..., 5 we get five unique values of z whose real part is -1/2. So the roots of ((z+1)/z)^6 = 1 lie on the straight line Re(z) = -1/2.