Editing Definable real number (section)

==Real algebraic numbers ==
[[File:Algebraicszoom.png|thumb|Algebraic numbers on the [[complex plane]] colored by degree (red=1, green=2, blue=3, yellow=4)]]

A real number <math>r</math> is called a real [[algebraic number]] if there is a [[polynomial]] <math>p(x)</math>, with only integer coefficients, so that <math>r</math> is a root of <math>p</math>, that is, <math>p(r)=0</math>.  
Each real algebraic number can be defined individually using the order relation on the reals. For example, if a polynomial <math>q(x)</math> has 5 real roots, the third one can be defined as the unique <math>r</math> such that <math>q(r)=0</math> and such that there are two distinct numbers less than <math>r</math> at which <math>q</math> is zero.

All rational numbers are constructible, and all constructible numbers are algebraic.  There are numbers such as the cube root of 2 which are algebraic but not constructible.

The real algebraic numbers form a [[field extension|subfield]] of the real numbers. This means that 0 and 1 are algebraic numbers and, moreover, if <math>a</math> and <math>b</math> are algebraic numbers, then so are <math>a+b</math>, <math>a-b</math>, <math>ab</math> and, if <math>b</math> is nonzero, <math>a/b</math>.

The real algebraic numbers also have the property, which goes beyond being a subfield of the reals, that for each positive integer <math>n</math> and each real algebraic number <math>a</math>, all of the <math>n</math>th roots of <math>a</math> that are real numbers are also algebraic.

There are only [[Countable set|countably many]] algebraic numbers, but there are uncountably many real numbers, so in the sense of [[cardinality]] most real numbers are not algebraic.  This [[nonconstructive proof]] that not all real numbers are algebraic was first published by
Georg Cantor in his 1874 paper "[[Georg Cantor's first set theory article|On a Property of the Collection of All Real Algebraic Numbers]]".

Non-algebraic numbers are called [[transcendental numbers]]. The best known transcendental numbers are [[Pi|{{pi}}]] and {{mvar|[[e (mathematical constant)|e]]}}.