Plotting Complex Roots of Parabola
- Author:
- maths partner
In the last section we realised that we've only been seeing the tip of the iceberg when it comes to parabolas since we only see the part that slices through the real dimension. In truth the majority of the parabola exists in the imaginary dimension which we represented as the space in front and behind the real x-y plane.
This is a useful visualisation and one that we should have in our minds as we continue through the next sections, however it can get kind of messy drawing things in 3D all the time (especially on a 2D computer screen!) Instead let's only plot the roots of this parabola since these will either be on the x-axis or imaginary axis. That way, we only need two axes - much nicer to work with (see below)!
You can clearly see now that when the parabola is below the x-axis, the roots are real (black) and when the parabola is above the x-axis, the roots are imaginary (blue).
In fact, we can extend this visualisation technique so that we collapse all real axes into one horizontal "real"-axis and have the vertical axis as the "imaginary"-axis. This view is called the Complex Plane* or an Argand Diagram.
*called "complex" since it involves more than one component - namely real and imaginary components