Algebraic Numbers

Algebraic numbers

Algebraic numbers –

This is a picture of the algebraic numbers in the complex plane, made by David Moore based on earlier work by Stephen J. Brooks, and available along with other neat stuff at Moore’s site Math and Code.

Algebraic numbers are roots of polynomials with integer coefficients. The integers 0 and 1 are the big dots near the bottom, while $i$ is near the top.

In this picture, the color of a point indicates the degree of the polynomial of which it’s a root:

• red = roots of linear polynomials, i.e. rational numbers,

• green = roots of quadratic polynomials,

• blue = roots of cubic polynomials,

• yellow = roots of quartic polynomials, and so on.

The size of a point decreases exponentially with the ‘complexity’ of the simplest polynomial with integer coefficient of which it’s a root. Here the complexity is the sum of the absolute values of the coefficients of that polynomial.

There are many patterns in this picture that call for explanation! For example, look near the point $i$. Can you describe some of these patterns, formulate some conjectures about them, and prove some theorems? Maybe you can dream up a stronger version of Roth’s theorem, which says roughly that algebraic numbers tend to ‘repel’ rational numbers of low complexity.

David Moore made this image using software created by Stephen J. Brooks on Wikipedia. Moore writes:

I based my code off the author on Wikipedia, Stephen J. Brooks. He had posted his source code here: (which is under a sharealike license) and I just uploaded the project with my local repository to sourceforge here: There aren’t many improvements to the project, but it adds the ability to zoom and pan around with your mouse in real time! It’s quite amazing to look at. (The build process just requires sdl and opengl.)

Visual Insight is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!