Visualising Complex Roots of a Parabola
- maths partner
In 1799, Carl Gauss published his proof stating that every polynomial of order n has n roots. So important was this idea that it became known as The Fundamental Theorem of Algebra. So essentially if you take a quadratic polynomial then this has a highest power (or order) of 2 so according to The Fundamental Theorem of Algebra there must always be two roots (that is where the parabola crosses the x-axis). When the parabola has its turning point (or vertex) below the x-axis, it is clear to see where these roots are (I've coloured them black on the graph). Even when the parabola is sitting on the axis, we can say that the two roots have the same value so they're sitting on top of each other. However, when the parabola is above the x-axis, where are these two roots???? The answer is that we are only looking at a slice of the true parabola - the slice that intersects our REAL dimension!!! In truth the parabola is a much bigger object which can't see because it exists in the IMAGINARY dimension; it's a bit like an iceberg - we only see the part that is above water but this is just a section of a much bigger object that we can't see since it is below the water. So to help us visualise the rest of the parabola that lies in the imaginary dimension, I've drawn the real dimension as a flat plane (surface coloured grey) and represented the imaginary dimension as the space that exists in front and behind this plane. By using this 3D view, we can see that even though the real slice of the parabola is above the axis, the complex part is still intersecting the complex plane twice (which I've coloured in blue). Who knew - we've been working with parabolas for a long time but we've only been looking at a tiny slice of the much bigger object!!