Quote:
Originally Posted by Thudlow Boink
I think the confusion revolves around what constitutes a "pattern."
Does something like 0.1010010001000010000010000001... have a pattern? I'd say it does: once you see what's going on, you can keep writing as many digits of that pattern as you want to. It's certainly not random. But it is an irrational number, because it's not just the same digit or sequence of digits repeating over and over.

This is getting on the right track. It's tough to say anything about irrational numbers in general, because the only thing they have in common is that they're not rational. There are larger classes of real numbers than the rationals that are worth talking about; here are the ones I know of:
 Constructible numbers: These are the numbers on the real line that you can get starting with a straightedge, a compass, and the numbers 0 and 1.
 Algebraic numbers: A number a is algebraic if there's some polynomial P(x) with integer coefficients such that P(a) = 0. For instance, 2^{1/3} is algebraic because if you plug it into P(x) = x^{3}  2, you get 0.
 Computable numbers: If you can write a computer program that takes no input and outputs a number r, r is said to be computable.
 Definable numbers: x is a definable number if there's some wellformed formula of first order logic that exactly specifies the value of x.
Each category contains the one above it (and every rational is constructible), but you're adding new numbers at each step. For instance, 2
^{1/2} is constructible but irrational; 2
^{1/3} is algebraic but not constructible;
p is computable but not algebraic; and Chaitin's constant is definable but not computable. It's a little tough to give examples of nondefinable numbers in a finite amount of space, but they're out there. In fact, the set of definable real numbers has Lebesgue measure 0 (as it's a countable set), so in a very real sense almost every real number is not definable.