Complex Numbers Squared
Power function in the real number system can be generalized as y = x n . A power function in the complex plane is represented as w = z^n where z is any complex number and n is an integer. Using Geometer’s Sketchpad, we are able to make a custom transformation representing the operation z. In this section, we observe the geometry of z^2 as a transformation acting on different pre-images.
A line segment in the complex plane can act like a vector if one of the points is at the origin but once the line is taken off of the origin it begins to warp and bend which shows that complex numbers can be represented in a different way.
- When the pre-image of a line segment is horizontal the image of the line segment has a parabolic shape on a closed interval that is opened to the right and is horizontal.
- Definition 4.1. A parabolic shape is when a figure is curved and both ends of the figure are in the same direction to create a figure similar to that of a U.
- When it is vertical it shows an after image of a parabolic shape on a closed interval opened to the left and horizontal.
- A line segment that has a positive slope will have a after image of a positive parabola opened up on a closed interval. A negative slope will have a negative parabola opening at the bottom on a closed interval.