The Fundamental Theorem of Algebra: A Visual Approach

A former student forwarded to me the following article concerning a visual way of understanding the Fundamental Theorem of Algebra, which dictates that every nonconstant polynomial has at least one complex root: http://www.cs.amherst.edu/~djv/FTAp.pdf. The paper uses a very clever idea, from the opening paragraphs:

[I]f we want to use pictures to display the behavior of polynomials defined on the complex numbers, we are immediately faced with a difficulty: the complex numbers are two-dimensional, so it appears that a graph of a complex-valued function on the complex numbers will require four dimensions. Our solution to this problem will be to use color to represent some dimensions. We begin by assigning a color to every number in the complex plane… so a complex number can be uniquely specified by giving its color.

We can now use this color scheme to draw a picture of a function f : \mathbb{C} \to \mathbb{C} as follows: we simply color each point z in the complex plane with the color corresponding to the value of f(z). From such a picture, we can read off the value of f(z)… by determining the color of the point z in the picture…

The article is engagingly written; I recommend it highly.

 

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1 Comment

  1. Oh, that’s awfully good and I’d never thought to use color in that way to plot complex-valued functions. That’s a great notion.

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