The fundamental theorem of algebra: a visual approach (Q297479)

In this article the author gives a visual proof of the Fundamental Theorem of Algebra, according to which every polynomial with real or complex coefficients has at least one root in the complex numbers. The basic idea is to encode every complex number by letting its absolute value determine brightness and the argument a certain color. Then the values of a polynomial in the complex plane may be visualized by giving the color \(f(z)\) to the point \(z\). Roots of the polynomial are black, and points far away from the origin become brighter and brighter. Argand's inequality then becomes the fact that if a point is not black, there is a nearby point that is darker. This is then exploited in the usual way: \(|f|\) attains a minimum in the interior of a compact set (corresponding to the darkest point in this set), and if this minimum is not \(0\), there is an even darker point nearby.