An irrational number is a real number that cannot be written in the form p / q, where p and q are integers, q <> 0, and p and q have no common factors other than 1. You encounter irrational numbers all the time - sqrt(2) and pi, to name two.
A transcendental number is an irrational number that is not the solution of any finite polynomial with rational coefficients.
For example, an irrational like sqrt(2) is the solution of any number of finite polynoimials, like x<sup>2</sup> - 2 = 0, or
x<sup>4</sup> + 4 = 4x<sup>2</sup>.
For a transcendental number, like pi or e, no such polynomials exist.
A transcendental number is an irrational number that is not the solution of any finite polynomial with rational coefficients.
For example, an irrational like sqrt(2) is the solution of any number of finite polynoimials, like x<sup>2</sup> - 2 = 0, or
x<sup>4</sup> + 4 = 4x<sup>2</sup>.
For a transcendental number, like pi or e, no such polynomials exist.