It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? 


Are most numbers transcendental?



What's so special about transcendental numbers?
The set of real numbers is uncountable, but the set of algebraic numbers is countable, so "most" real numbers are transcendental in a very strong sense of "most". This is actually a capsule description of Cantor's proof of the existence of transcendental numbers; just note that an uncountable set cannot be empty. Looking at the interval [0,1], the set of algebraic numbers there have (Lebesgue) measure zero, so a number picked "at random" (uniform distribution) from that interval is transcendental with probability 1. Transcendental numbers are a dime a dozen - but to prove that *particular* real numbers are transcendental is either hard or just *too* hard. It is known that $ e^\pi $ is transcendental, but as to $\pi^e$, nobody knows. Transcendental numbers are studied for their own sake. Important mathematicians found them interesting, so they must be important :) Euler was the first to consider the possibility that there might be real numbers that were not algebraic, Liouville constructed the first ones, Hermite proved e transcendental, and so forth. The closely connected subject of linear forms in logarithms has applications in more mainstream number theory, especially to Diophantine equations.


