Fundamental theorem of algebra
A classic proof for the fundamental theorem of algebra (that is, each non-constant polynomial has a root among the complex numbers) uses topology.
It is to prove that has a root in the complex plane, that is, a is required (on the left) for which (on the right) goes through the origin (X).
The circles on the left are always transformed to a closed curve on the right which wind around the origin, if is sufficiently large. On the other hand, if , the point on the left is transformed to a point on the right. Between these two extremal situations the right curve continuously changes. This results in a position when the right curve obviously crosses the origin.
You can try some activities out:
- Change the radius of the circle to find an image of the circle which crosses the origin on the right.
- Trace the circles to remember which radiuses were already checked.
- Show the image of the circle to immediately know that a radius will be fine or not.
- Trace the images of the circles to have a general view about the map .
- Change to different polynomials. Learn how the higher order polynomials behave.
- You can also try non-polynomial functions. For example, is an interesting option.