Exploring Roots of Unity (and other complex numbers)
Use this applet to explore the roots of unity (roots of one) or the roots of any complex number you can think of. There's a beautiful graphic interpretation of these roots as a consequence of the way complex number multiplication encodes scaling and rotation into one operation. In the applet below, you can choose the "power of the root" ( i.e. 3 would find the 3rd root(s) of the number), and the complex number you want to find the root(s) of. You may choose the complex number either by typing it into the box on the screen, or dragging point Z around the plane. Then, click "Show Roots" to toggle the roots on or off. The BEST part is that you can follow the roots around the circle to their eventual end by sliding the "a" slider from left to right. As you move the slider towards the right, it will animate each of the roots being raised to higher and higher powers (indicated by the power #) until a=1, at which point the roots have been raised to the chosen power, and will all converge on the complex number Z! It's pretty amazing, I hope you enjoy playing around with it : ) Be sure to try vales of Z outside and inside the unit circle.