What “normal” would mean for pi
A phone number is a finite string of digits, so the question is whether that string appears somewhere in pi’s decimal expansion—not whether pi contains phone numbers in any special sense. If pi is normal in base 10, every possible block of digits of length k would occur with limiting frequency 1 in 10^k. In plain terms, each one-digit pattern would appear about one-tenth of the time, each two-digit pattern about one-hundredth, and so on, across the expansion.
That condition would guarantee that every finite digit string appears, including any particular phone number. It would also mean the string appears again and again, though the gaps between appearances need not be small or predictable.
Why endless digits are not enough
Pi is irrational, which means its decimal expansion never ends or settles into a repeating cycle. But that alone says very little about which patterns occur. An infinite sequence can avoid a particular block forever: repeating 123 endlessly produces an infinite decimal-like sequence with no 456 anywhere in it.
Normality is a much stronger claim about the long-run distribution of digits. Finding a block in a long stretch of pi’s digits—or seeing digits that look evenly spread—cannot prove that the pattern will appear everywhere it could, or that the frequencies will settle as required. For pi, normality remains unproved. So a phone number may be somewhere in its digits, but the known facts about pi do not guarantee it.
