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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?

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We don't know that $\pi$ has no pattern. Your third item isn't a number, but a recipe for generating real numbers, some of which are rational. You may be interested in the concept of "automatic sequences" --- sequences (which could be sequences of decimal digits --- that can be generated by a finite state automaton. –  Gerry Myerson Jan 24 '13 at 5:35
    
@Gerry, $\pi$ has no repeating pattern in that its decimal expansion is not periodic. I think that's all the OP meant. –  Trevor Wilson Jan 24 '13 at 5:59
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$\sum 10^{-j!}$ doesn't have a "repeating pattern." It has a pattern. –  Thomas Andrews Jan 24 '13 at 6:06
    
The third is not a number at all, it is a description of a distribution on $[0,1]$. –  Thomas Andrews Jan 24 '13 at 6:08
    
I think there's supposed to be a factorial in the exponent of the OP at this point of the "recipe": $\sum a_i \cdot 10^{i!}$. Without the factorial it gives rational points, in fact all the points. But with the factorials the decimal has no period, so represents an irrational. –  coffeemath Jan 24 '13 at 6:39

2 Answers 2

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.

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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.

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