Powers of Complex Numbers

This environment allows you to explore what happens when you raise a complex number to a power. Drag the black dot to position the complex number Z = a + bi. Show or hide the complex conjugate of Z, i.e. Z* = a - bi Challenge - Why are there spirals? How are the spirals of Z and Z* related? Why? What is the circle and why is it there? Can you solve for z in the equation z + iz = i ?