# how are we sure that every arrangement of number appears somewhere in pi? How do we know that a string of a million 1s appears somewhere in pi?

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how are we sure that every arrangement of number appears somewhere in pi? How do we know that a string of a million 1s appears somewhere in pi?

In: 1229

well that’s the thing with irrational numbers.

they don’t end after the decimal point

and somewhere within that infinite string of numbers is everything we ever knew and will know

just as well as the thing with the apes on typewriters. they’ll eventually write all of Shakespeare’s works by just mashing the keys. it’s bound to happen SOMEtime

We have calculated Pi to a very large length and we have observed the occurrence of each digit at least once, which implies that every digit has a probability of occurring, no matter how small. So in the infinitely long sequence of Pi, there is a chance that every possible combination will occur at some point because of the presence of a probability

if the decimal expansion of pi is infinite, then you can do the following: split the decimal into buckets of 10 digits groups. But because it’s infinitely long, there must be an unlimited number of these buckets.

However, a 10 digit group can have a maximum of 3628800 permutation of digits – that is, only this many unique permutations. But there’s infinitely many buckets in the decimal of pi, so it must imply that after at most 3628800 buckets, the group of digits must repeat.

There’s nothing special about 10 digit groups above – any digit groups work (just a larger permutation number).

The actual answer is: we aren’t.

The property you are talking about “that every arrangement of number appears” is called *normality.* And we have absolutely no proof that pi is normal. So far it appears to be normal, but we have nothing that proves that it will continue to be normal. It is perfectly possible, for example, that the number 9 stops appearing at some point.

In fact, other than specific numbers constructed to be normal or not normal, we have no general test for normality *at all.*

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