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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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
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
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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We don’t know. However, if it is proved to be a normal number, then yes that is exactly correct.
From an empirical stance, there is the famous [Six nines in pi](https://en.wikipedia.org/wiki/Six_nines_in_pi). There have been longer strings of a repeated single digit that have been discovered since then. You can look at the various sequences of a single digit being repeated here:
* 1 https://oeis.org/A035117
* 2 https://oeis.org/A050281
* 3 https://oeis.org/A050282
* 4 https://oeis.org/A050283
* 5 https://oeis.org/A050284
* 6 https://oeis.org/A050285
* 7 https://oeis.org/A050286
* 8 https://oeis.org/A050287
* 9 https://oeis.org/A048940
* 0 https://oeis.org/A050279
For example, in the sequence for nines, it goes up to 14, meaning that a string of 14 nines in a row is the longest known. For the digit one, it goes up to 13, which begins at position 3,907,688,331,257. Of these, the longest string is of 15 sevens at position 46,970,519,777,308.
Although theoretically, we should be able to check for longer and longer strings as computational power increases, this has an upper bound of our entire physical universe being used as a computer. I don’t know if that’s enough to search for a one million digit string. Already at 15 digits long, you have to search trillions, and so I would imagine that to find a string of a million digits long, it would be necessary to search up to at least the sextillionth digit of pi.
edit: https://newatlas.com/science/pi-world-record-62-8-trillion-digits/
The world record is 62.8 trillion digits of Pi. It took a supercomputer 108 days to calculate it. So a computer **a million times faster** would be able to compute 62.8 quintillion digits in the same amount of time, which is around 6% of the digits needed to calculate my lower bound estimate of 1 sextillion. So a supercomputer **a million times faster** would take several years to calculate 1 sextillion digits, assuming the program used is O(n).
Why do you think we know that?
We absolutely aren’t, it’s just something people believe.
Many assume that pi is a normal number, in which case every sequence would appear. But there is absolutely no guarantee at all that pi is a normal number, and people should stop claiming it is until we have an actual proof.
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