[[File:Algebraicszoom.png|thumb|alt=refer to caption|Algebraic numbers on the [[complex plane]] colored by polynomial degree. (red = 1, green = 2, blue = 3, yellow = 4). Points become smaller as the integer polynomial coefficients become larger.]]
To proveprovee that the set of real algebraic numbers is countable, define the ''height'' of a [[polynomial]] of [[degree of a polynomial|degree]] ''n'' with integer [[coefficient]]s as: ''n'' − 1 + |''a''<sub>0</sub>| + |''a''<sub>1</sub>| + ... + |''a''<sub>''n''</sub>|, where ''a''<sub>0</sub>, ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> are the coefficients of the polynomial. Order the polynomials by their height, and order the real [[Root of a polynomial|root]]s of polynomials of the same height by numeric order. Since there are only a finite number of roots of polynomials of a given height, these orderings put the real algebraic numbers into a sequence. Cantor went a step further and produced a sequence in which each real algebraic number appears just once. He did this by only using polynomials that are [[irreducible polynomial|irreducible]] over the integers. The following table contains the beginning of Cantor's enumeration.<ref>{{harvnb|Cantor|1874|pp=259–260}}. English translation: {{harvnb|Ewald|1996|p=841}}.</ref>
