Critic Ivor Smallbrain is discussing the film Sets, Lines and Videotape with his two chief editors, Sir Giles Tantrum and Lord Overthetop. They are sitting at a circular table of radius 1 . Ivor is bored and notices in a daydream that he can draw real and imaginary axes, with origin at the center of the table, in such a way that Tantrum is represented by a certain complex number \(z\) and Overthetop is represented by the complex number \(z+1\). Breaking out of his daydream, Ivor suddenly exclaims, "You are both sixth roots of 1 !"

Prove that Ivor is correct, despite the incredulous editorial glares.