# 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?

• 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. Jan 24, 2013 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. Jan 24, 2013 at 5:59
• $\sum 10^{-j!}$ doesn't have a "repeating pattern." It has a pattern. Jan 24, 2013 at 6:06
• The third is not a number at all, it is a description of a distribution on $[0,1]$. Jan 24, 2013 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. Jan 24, 2013 at 6:39

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.
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.