Finding Roots of Complex Numbers

Finding Roots of Real Numbers

Consider the equation . Clearly there is only one solution here! Now consider which we can express as . We know there are two solutions: and Now consider which again we can express as . From The Fundamental Theorem of Algebra we know that there must be three solutions but we can only think of 1? Surprise, surprise the other two solutions lie in the complex plane. Experiment with the activity below. You should notice some nice patterns that emerge:
  • Firstly, roots occur at equal rotations to each other - think spokes on a bicycle where the number of "spokes" is equal to the number of roots
  • All roots occur within a full rotation of a circle so the spacing between each root should be or
  • For values other than 1 (e.g. or ) the radius of the circles decrease (or increase if between -1 and 1) by a factor of the th root (look at the green circles - each time you raise n you are effectively taking another root).*
*This makes sense... Think about clearly there will just be one root that lies on the circle with radius 2. Now think about which will give two roots but each one will lie on the circle with radius since Now think about which will give three roots but each one will lie on the circle with radius since And the same thing for and etc. So the size of the radius is determined by the amount of roots you've taken.

Complex Roots

Notice that the same is true for complex numbers...