Originally Posted by
romsek mmk
the short way of putting it I guess is that algebraic numbers are the solutions to some polynomial equation with integer coefficients. Rational roots of rational numbers fall into this category as well as rationals, integers, etc.
A transcendental real number is a real number that isn't algebraic.
A transcendental function is one that can produce a transcendental real number. This is most of the functions you know. All the trig functions, the exponential and log function are transcendental.