For example, the square root of 3, we get a repeating sequence of integers (1,2,1,2,…) after the first couple of terms. “e”, the base of the natural log also has a very predictable pattern (2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,…). As far as I know, no known pattern of the continued fraction (at least patterns as obvious) exists for pi.

Actually the digits of Pi are a countable infinity. You can think of this distinction like this: associate the first digit with the whole number 1, the second digit with the whole number 2, and so on. each digit can be associated with one of the whole numbers, thus they can be counted. there’s an infinity of digits, because no matter how high you count, there’s always more.

Having said that, i still don’t know the answer to the question. My guess is that there is no representative sample. There are very likely strings of repeated sequences in the digits of Pi (or any other irrational number), and you could land on one of those strings and convince yourself that the sequence has terminated. I seem to recall that there is something like that fairly early in the sequence of Pi (or is it e?).

Clearly, she knows everything.

-The Mirror Has Two Faces

I will definitely add it to the list of must-read books. I am somewhat familiar with the countable/uncountable dichotomy thanks to a Martin Gardner book I read years ago. So if the digits in Pi are an uncountable infinity, could you get a “representative sample” of Pi using a countably infinite number?

And if so, what good would it do finite creatures like ourselves?

for example, there is a countable infinity of whole numbers (…, -3, -2, -1, 0, 1, 2, 3, …), whereas there is an uncountable infinity of real numbers. there’s even an uncountable infinity of real numbers between say pi/2 and pi, or even between 1.2345 and 1.2346

still, as you say, 280,000/infinity is still zero.

if you’ve never read The Man Who Loved Only Numbers (about Paul Erdos), then put it on your reading list.