r/askmath • u/Loose-Balance3225 • May 06 '26

Discrete Math Can infinity contain infinity

If pi has no end it has to have every combination of numbers but could it hold an infinite combination? Like 1 2 3 4... To infinity

Please help this is missing my brain up

• Upvotes

66% Upvoted

60 comments sorted by

View all comments

•

u/2ndcountable May 06 '26

If pi is a normal number, as we suspect, it 'must' contain any finite sequence of digits. That is, given any finite sequence of digits, like '3333' or '123456789', it must exist within pi. However, even if pi is normal, it does not have to contain a given infinite sequence of digits. For example, we know for sure that pi does not contain the infinite sequence '333333333...', since that would mean its decimal expansion is eventually periodic and hence that pi is rational, which we know to be false.

•

u/mousicle May 06 '26

As far as normal numbers, right now we have no test as to whether a number is normal or not. The only numbers we know for sure are normal are ones specifically constructed to be normal like 0.123456789101112 ... We do know that almost all numbers are normal though.

•

u/Bills_afterMATH May 06 '26

What do you mean by test? For example, it’s trivial to show that x is normal in base b iff the sequence (x*b^n) is uniformly distributed mod 1. The set of numbers normal in base b was proven to be Pi_3^0 complete by Ki and Linton in 1994. So any condition that can characterize normality has to be a Pi_3^0 condition. Are you talking about some condition that checks (but can sometimes not determine) if a number is normal based on a finite amount of information?