Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

Harry Wiggins,Johann Engelbrecht,Ansie Harding

doi:10.5539/jmr.v10n6p91

Abstract

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called &quot;imaginary&quot; roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.&nbsp; We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

Highlights

For centuries mathematicians have spent time and energy on investigating two issues concerning polynomials how to visualize complex zeroes how to visualize polynomials with complex coefficients.We address the first of these issues for a range of well-known functions in an elegant way
One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called ”imaginary” roots of a polynomial
The approach outlined here gives a fresh view to graphical representation of complex roots of functions and a general understanding of the functions that we deal with every day

Summary

Introduction

How to visualize polynomials with complex coefficients. We address the first of these issues for a range of well-known functions in an elegant way. We came to have two planes, the Cartesian plane that can be used for showing the graph of a function and where its real roots lie and the Argand plane that gives a visual representation of complex numbers. This approach was further developed in Harding and Engelbrecht (2007b) which led to the idea of sibling curves, which turned out to be a rich and useful way of visualizing zeroes of polynomials and other well-known functions as well as visualizing complex functions in three dimensions. The projection of these sibling curves on the complex domain is a hyperbola (Figure 6) and not two perpendicular straight lines as was the case with real quadratic polynomials.

Hyperbolic Paraboloid

Conclusion