https://github.com/philipl/pifs
πfs is a revolutionary new file system that, instead of wasting space storing your data on your hard drive, stores your data in π! You'll never run out of space again - π holds every file that could possibly exist! They said 100% compression was impossible? You're looking at it!
It's almost sure to be the case, but nobody has managed to prove it yet.
Simply being infinite and non-repeating doesn't guarantee that all finite sequences will appear. For example, you could have an infinite non-repeating number that doesn't have any 9s in it. But, as far as numbers go, exceptions like that are very rare, and in almost all (infinite, non-repeating) numbers you'll have all finite sequences appearing.
Exceptions are infinite. Is that rare?
Very rare in the sense that they have a probability of 0.
There are lot that fit that pattern. However, most/all naturally used irrational numbers seem to be normal. Maths has, however had enough things that seemed 'obvious' which turned out to be false later. Just because it's obvious doesn't mean it's mathematically true.
A number for which that is true is called a normal number. It’s proven that almost all real numbers are normal, but it’s very difficult to prove that any particular number is normal. It hasn’t yet been proved that π is normal, though it’s generally assumed to be.
0.101001000100001000001 . . .
I'm infinite and non-repeating. Can you find a 2 in me?
You can't prove that there isn't one somewhere
You can, it's literally the way the number is defined.
Defined where ?
It's implicitly defined here by its decimal form:
0.101001000100001000001 . . .
The definition of this number is that the number of 0s after each 1 is given by the total previous number of 1s in the sequence. That's why it can't contain 2 despite being infinite and non-repeating.
Why couldn’t you?
Because you'd need to search through an infinite number of digits (unless you have access to the original formula)
The jury is out on whether every finite sequence of digits is contained in pi.
However, there are a multitude of real numbers that contain every finite sequence of digits when written in base 10. Here's one, which is defined by concatenating the digits of every non-negative integer in increasing order. It looks like this:
0 . 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
fun fact, "most" real numbers have this property. If you were to mark each one on a number line, you'd fill the whole line out. Numbers that don't have this property are vanishingly rare.
Not just any all finite number sequence appear in pi
The term for what you're describing is a "normal number". As @lily33@lemm.ee correctly pointed out it is still an open question whether pi is normal. This is a fun, simple-language exploration of the question in iambic pentameter, and is only 3 minutes and 45 seconds long.
Merry Christmas!
Yes
Can you prove this? Or link a proof?
I don't know of one but the proof is simple. Let me try (badly) to make one up:
If it doesn't go into a loop of some kind, then it necessarily must include all finite strings (that's a theoretical compsci term).
Basically, take a string of any finite length, and then view pi in inrements of this length. Calculate it out to double the amount of substrings of length of your target string's interval you have [or intervals]. Check if your string one of those intervals. If not, do it again until it is, doubling how long you calculate each time.
Because pi is non-repeating, each doubling in intervals must necessarily include at least one new interval from all other previous ones. And because your target string length is finite, you have a finite upper limit to how many of these doublings you have to search. I think it's n in the length of your target string.
Someone please check my work I'm bad at these things, but that's the general idea. It's also wildly inefficient This doesn't work with Infinite strings because of diagnonalization.
No this does not work. Counter example can be found in the comments here of a non-repeating number that definitely does not contain all finite strings.
Edit: I think the confusion is about the word non-repeating. Non repeating does not mean a subsequence cannot repeat but that you cannot write the number as a rational or with a finite decimal representation. I.e. it's not 3.ba repeating. Where a is a finite sequence that repeats infinitely and b is a finite sequence.
Edit edit: another assumption you make is that pi does not go into a loop of some kind. You would need to prove that.
Are you talking about a different base/character set? I think every single person understands that.
See my other comment
Yes.
And if you're thinking of a compression algorithm, nope, pigeonhole principle.
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