DeMoivre's Theorem: Visualization

Move the labeled point to any complex number, , on the polar grid above. The red curve shows all of the powers of that number. To see the first powers, click on the checkbox. You can adjust from 1 to 20. You can also show the segments from 0 to each point to see the relationship between their arguments. Interesting things to explore: 1. What happens if ? 2. What happens if the number is on the real axis ( or )? 3. What if ? 4. What number is always on the red curve, no matter where you drag ?