One small point of contention: I think it's not necessarily true that every finite sequence of digits exists somewhere in the digits of pi. It might intuitively appear that way because the digits are infinite and appear to be random. But if they are truly random, then there cannot be a preference for any infinite sequence of digits over any other, in which case all of the infinite sequences which lack our specified finite sequence have as good a chance as all those which contain it.
Looking at it digit by digit, it's clear that as we consider more random digits tending to infinity, the odds of a particular finite sequence not appearing tend towards 0. But that doesn't mean that we can say the odds actually are 0 when the number of digits equals infinity. If anything, I think we should consider the odds as infinitesimal, but not 0. I think we have to allow infinitesimals if we're to talk about infinite random sequences, as otherwise we find that every infinite sequence has a probability of 0... And then since there are infinite finite sequences, it's quite possible some of them might accomplish that infinitesimally unlikely feat of escaping pi.
I bought some round iced ginger cookies to celebrate Pi Day! (I might order a pizza tonight.)
You make a very interesting point. I've pondered that myself in the context of the idea that, if the universe is truly infinite, then exact copies of us must exist elsewhere in that infinitude. The argument, as I understand it, is that there are only so many ways atoms can be arranged, so given infinite occurrences of arrangements, duplicates are unavoidable. More to the point here, every possible arrangement must exist.
But, yeah, does it necessarily? Infinity makes things strange, and some treat it with suspicion because (they believe) nothing in the physical world can be infinite. Abstractions can be, though, so the question with pi remains.
The formula in the post would never return zero odds, but when 'pi-digits' is 'infinity' then the equation returns infinity because no matter how small the term in parentheses gets (it gets smaller the longer the finite string sought), anything times infinity is infinity. So, this formula (which probably only works for a fixed number of 'pi-digits') doesn't make much sense for the infinitude of pi digits.
Instead, perhaps the arrangements argument works. There are only so many ways a given sequence of digits can be arranged. It's 10ⁿ, where n is the sequence length. So, given an infinity of digits, what are the odds a given sequence **doesn't** exist? Perhaps the actual equation governing this is a limit that never drops to exactly zero? I'm not knowledgeable enough to know. But mathematically, a subset of an infinite set is also infinite. 🤷🏼♂️
The tech bros and the mathematicians before them have been trying to square the circle, since forever, and to this date they still have not solved the equation beyond a decimal approximation. π is unsolvable; math lists π as an irrational number.
Ha, yes indeed. Any tech bro trying to square the circle doesn’t know mathematics, because the impossibility of squaring the circle was proved in 1882.
Because of what’s talked about in this post — pi is transcendental!
Happy Pi day!
One small point of contention: I think it's not necessarily true that every finite sequence of digits exists somewhere in the digits of pi. It might intuitively appear that way because the digits are infinite and appear to be random. But if they are truly random, then there cannot be a preference for any infinite sequence of digits over any other, in which case all of the infinite sequences which lack our specified finite sequence have as good a chance as all those which contain it.
Looking at it digit by digit, it's clear that as we consider more random digits tending to infinity, the odds of a particular finite sequence not appearing tend towards 0. But that doesn't mean that we can say the odds actually are 0 when the number of digits equals infinity. If anything, I think we should consider the odds as infinitesimal, but not 0. I think we have to allow infinitesimals if we're to talk about infinite random sequences, as otherwise we find that every infinite sequence has a probability of 0... And then since there are infinite finite sequences, it's quite possible some of them might accomplish that infinitesimally unlikely feat of escaping pi.
I bought some round iced ginger cookies to celebrate Pi Day! (I might order a pizza tonight.)
You make a very interesting point. I've pondered that myself in the context of the idea that, if the universe is truly infinite, then exact copies of us must exist elsewhere in that infinitude. The argument, as I understand it, is that there are only so many ways atoms can be arranged, so given infinite occurrences of arrangements, duplicates are unavoidable. More to the point here, every possible arrangement must exist.
But, yeah, does it necessarily? Infinity makes things strange, and some treat it with suspicion because (they believe) nothing in the physical world can be infinite. Abstractions can be, though, so the question with pi remains.
The formula in the post would never return zero odds, but when 'pi-digits' is 'infinity' then the equation returns infinity because no matter how small the term in parentheses gets (it gets smaller the longer the finite string sought), anything times infinity is infinity. So, this formula (which probably only works for a fixed number of 'pi-digits') doesn't make much sense for the infinitude of pi digits.
Instead, perhaps the arrangements argument works. There are only so many ways a given sequence of digits can be arranged. It's 10ⁿ, where n is the sequence length. So, given an infinity of digits, what are the odds a given sequence **doesn't** exist? Perhaps the actual equation governing this is a limit that never drops to exactly zero? I'm not knowledgeable enough to know. But mathematically, a subset of an infinite set is also infinite. 🤷🏼♂️
The tech bros and the mathematicians before them have been trying to square the circle, since forever, and to this date they still have not solved the equation beyond a decimal approximation. π is unsolvable; math lists π as an irrational number.
The circle remains un broken, un squared.
Ha, yes indeed. Any tech bro trying to square the circle doesn’t know mathematics, because the impossibility of squaring the circle was proved in 1882.
Because of what’s talked about in this post — pi is transcendental!