A picture of the algebraic numbers in the complex plane.

The algebraic numbers

This is a picture of the algebraic numbers in the complex plane. 

What does that mean?  Well, algebraic numbers are roots of polynomials with integer coefficients. In the picture, the integers 0 and 1 are the big dots near the bottom, while the imaginary number i (the square root of –1) is near the top.

The colour 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.