Q: Since pi is infinite, do its digits contain all finite sequences of numbers?
Mathematician: As it turns out, mathematicians do not yet know whether the digits of pi contains every single finite sequence of numbers. That being said, many mathematicians suspect that this is the case, which would imply not only that the digits of pi contain any number that you can think of, but also that they contains a binary representation of britney spears’ DNA, as well as a jpeg encoded image of you making out with a polar bear. Unfortunately, to this day it has not even been proven whether every single digit from 0 to 9 occurs an unlimited number of times in pi’s decimal representation (so, after some point, pi might only contain the digits 0 and 1, for example). On the other hand, since pi is an irrational number, we do know that its digits never terminate, and it does not contain an infinitely repeating sequence (like 12341234123412341234…).
One thing to note is that when mathematicians study the first trillion or so digits of pi on a computer, they find that the digits appear to be statistically random in the sense that the probability of each digit occurring appears to be independent of what digits came just before it. Furthermore, each digit (0 through 9) appears to occur essentially one tenth of the time, as would be expected if the digits had been generated uniformly at random.
While tests performed on samples can never unequivocally prove that a sequence is random (in fact, we know the digits of pi are not random, since we know formulas to generate them) the apparent randomness in pi is consistent with the idea that it contains all finite sequences (or, at least, all fairly short ones). In particular, if we generate a number from an infinite stream of digits selected uniformly at random, then there is a probability of 100% that such a number contains each and every finite sequences of digits, and pi has the appearance of being statistically random.
The following rather remarkable website allows you to search the digits of pi for specific integer sequences:
http://www.angio.net/pi/piquery