Difference between irrational numbers with and without a pattern. I'm not sure how to talk about what I want to talk about, so I'll give some examples.
The number $\pi$ is irrational and has no repeating pattern, but is computable by an easy rule; divide the circumference of a circle by its diameter.
Now consider the number $\sum_{i=1}^{\infty}10^{-(i!)}.$ This has a pattern, and by definition generated by a defined rule. But the number is still irrational.
My question is, is there a mathematical concept similar to, but more general than, rationality that differentiates between these different types of numbers?
 A: The first two numbers are examples of computable numbers.  A  computable number is defined, more or less, as a number $x$ such that there is a (deterministic) computer program that spits out the digits of $x$ in sequence.  For example, there is a computer program that outputs "3", then "1", then "4", and so on for all the decimal digits of $\pi$ in sequence.  Although there are uncountably many real numbers, there are only countably many computable numbers because there are only countably many computer programs, so in a sense "most" numbers are not computable.
The third "number" would be called a random variable.  In this example, it is computable with probability zero.
A: You might be thinking about constructable numbers, I'm not sure. 
Regardless, the third "number" you consider is not really a number. You describe there a random variable. And the claim that it is an irrational number is incorrect. Firstly, it's not a number at all, but even using a more liberal interpretation it is certainly possible that all the digits are chosen to be, say, $9$, in which case the number chosen is precisely equal to $1$, a perfectly rational number. 
You might be trying to distinguish between rational numbers whose decimal expansions exhibit some pattern that can be finitely described vs. those that do not. So you will have to be more precise about the patterns you are considering. 
