Changes

'''Algebraic numbers''', are real numbers which are roots of some finite degree polynomial with integer coefficients.  Examples of Algebraic Numbers are 2, 1/3, and <math>\sqrt{2}</math>.

'''Algebraic numbers''', are real numbers which are roots of some finite degree [[polynomial]] with integer coefficients.  Examples of algebraic numbers are 2, 1/3, and <math>\sqrt{2}</math>.

In contrast, [[transcendental numbers]] are real numbers which are not roots of any finite degree polynomial with integer coefficients.  The most well-known transcendental numbers are [[e]] and <math>\pi</math> ([[pi]]).

In contrast, [[transcendental numbers]] are real numbers which are not roots of any finite degree polynomial with integer coefficients.  The most well-known transcendental numbers are [[e]] and <math>\pi</math> ([[pi]]), though this must be proven using advanced [[calculus]] or [[number theory]].

The set of Algebraic Numbers is countably [[infinite]].  This follows from the fact that the set of finite degree polynomials with integer coefficients is countably [[infinite]].

The set of algebraic numbers is countably [[infinite]].  This follows from the fact that the set of finite degree polynomials with integer coefficients is countably [[infinite]]. Conversely, the set of transcendental numbers is [[uncountable]] because for any finite degree polynomial with integer coefficients there are uncountably many numbers which are not solutions to it.

 

 

 

A [[random]] real number is probably transcendental, though the exact value of this [[probability]] has baffled [[statistician]]s for almost 100 years.

[[Category:Mathematics]]

[[Category:Mathematics]]