An algebraic number is a number that is a solution of a non-zero polynomial equation with integer coefficients. For example, √2 is an algebraic number because it is a solution to the equation x2-2=0. A number that is not algebraic is called transcendental (Maor, 1994).
Figure: Charles Hermite, 1822-1901
The first number proven to be transcendental was in 1844 by French mathematician Joseph Liouville. Nearly 30 years later, another French mathematician Charles Hermite proved that the number e was also transcendental. Since Hermite, several other numbers have been proven to be transcendental including Π, eΠ, and ab where a is any algebraic number excluding 0 and 1 and b is any irrational algebraic number (e.g. 2√2) (Maor, 1994).
It is rather difficult to prove a number is transcendental because one must show there does not exist a non-zero polynomial with integer coefficients for which the number is a root. However, in 1874 German mathematician Georg Cantor proved that irrational numbers actually outnumber rational numbers and transcendental numbers outnumber algebraic numbers (Maor, 1994). Think about it, that's crazy!
The numbers ΠΠ, Πe, and ee have still not been proven to be transcendental (Maor, 1994).
