This is true for normal numbers [1], but is definitely not true for all non-repeating (irrational) numbers. Pi has not been proven to be normal. There are many non-repeating numbers that are not normal, for example 0.101001000100001...
Storing the index into pi for a file would usually take something like as much space as just storing the file, and storing or calculating enough digits to use that index would be impossible with the technology of today (or even probably the next century).
The short version is that the size of the reals is a "bigger infinity" than the size of the rationals, so they effectively have 'zero weight'.
Reference (very technical): https://math.stackexchange.com/questions/508217/showing-that...
The joke lies in the fact that saying "100% of real numbers" isn't *technically* the same thing as saying "all real numbers", because there's not really a good way to define a meaning for "100%" that lets you exclude rational numbers (or any other countable subset of the reals) and get something other than 100%.
Right. I'm pretty sure actually that it was a joke...
may I interest you in the difference between *irrational* numbers and *normal* numbers?
look at https://en.wikipedia.org/wiki/Liouville_number - no repeats, but minuscule "contained information"
It is somewhat shocking that again and again this logical fallacy comes up. Why do people think that this is true? It doesn't even sound true.
It's sort of like the idea that if the universe is infinitely big and mass and energy are randomly distributed throughout the universe, then an exact copy of you on an exact copy of Earth is out there somewhere.
This property of infinity has always fascinated me, so I'm very curious for where the logical fallacy might be.
A number that contains all other numbers infinitely many times (uniformly) would be called normal, but no one has managed to prove this for pi yet. In fact, no one even managed to prove that pi doesn't contain only 0s and 1s like the above after the X-th digit.
No. Example: 0.1011011101111011111... does never repeat, yet there is no 2 in there, neither is there 00 in there.
The question is really 'Does every series of numbers of arbitrary finite length appear in pi?' I can't answer that because I'm not a mathematician, but I also can't dismiss it, because I'm not a mathematician. It sounds like a fair question to me.
So what? Mathematicians can't answer it either. It is an open question and because it is an open question claiming it is or isn't true makes no sense.
>The fact you can't encode arbitrary data in a structured-but-irrational number doesn't mean you can't encode data in a 'random' irrational number.
You can not encode data in a random number. If it is random you can not encode data in it, because it is random. I am not sure what you are saying here.
I demonstrated that numbers where the digits go on forever and never repeat exist, which don't contain every single possible substring of digits. Therefore we know that pi can either be such or a number or it is not, the answer to that is not known. Definitely it is not a property of pi being infinitely long and never repeating.
That's why I put random in quotes. Pi is not a random number. You can encode data in it eg find a place that matches your data and give people the offset. That's not very helpful for most things though.
That obviously applies to 0.00... = 0 as well, it contains 0, then 00, then 000 and so on. So every number and therefore every piece of information is contained in 0 as well, given the right encoding. Obviously if you can choose the encoding after choosing the number all number "contain" all information. That is very uninteresting though and totally misses the point.
Put another way, the program which searches those works of art in the digits of pi will never finish (for a sufficiently complex work of art). And if it never finishes, does it actually exist?
Citation needed.
Believing in real numbers requires you to believe in far more than infinity. How many physicists reject real numbers?
To answer that question, you would have to dismiss with experimental evidence all models people can come up with that try to explain the universe without "infinities". It's neither completely clear what that would mean, nor whether it's even in principle possible to determine experimentally (it's also most likely completely irrelevant to any practical purpose).
feel free to prove me wrong. I never said it's efficient, the point is just that the information is out there. If pi has the following subnumbers 00, 01, 10, 11 in there, we can construct every perceivable data we can encode as binary. Even with 0 and 1. So we can construct a file by pointers to these four numbers. The bigger substrings we can match, the bigger the compression ratio. The set of pointers might even be way bigger than the file itself. It's nowhere near efficient or clever, but just entertaining
I don't think you can argue against IP because the way you arrange the pointers is IP itself, but still a funny thought experiment anyway
I'm not saying, that every piece of information is in there end to end, but that there are parts in there which can be used to construct it. I think I should've made the "encoded" part a bit more transparent haha. But I love the discussion that I kicked off!
you might find this to be pretty cool. It's similar to what you're describing. Whoever made it has an algorithm where you can look up "real" strings of text and it'll show you where in the library it exists. you can also just browse at random, but that doesn't really show you anything interesting (as you would expect given it's all random).
...and can't because there is no original corpus that the locality hashing algorithm can use as a basis
Borges wrote a famous short story, “The Library of Babel,” about a library where:
“... each book contains four hundred ten pages; each page, forty lines; each line, approximately eighty black letters. There are also letters on the front cover of each book; these letters neither indicate nor prefigure what the pages inside will say.
“There are twenty-five orthographic symbols. That discovery enabled mankind, three hundred years ago, to formulate a general theory of the Library and thereby satisfactorily resolve the riddle that no conjecture had been able to divine—the formless and chaotic nature of virtually all books. . .
“Some five hundred years ago, the chief of one of the upper hexagons came across a book as jumbled as all the others, but containing almost two pages of homogeneous lines. He showed his find to a traveling decipherer, who told him the lines were written in Portuguese; others said it was Yiddish. Within the century experts had determined what the language actually was: a Samoyed-Lithuanian dialect of Guaraní, with inflections from classical Arabic. The content was also determined: the rudiments of combinatory analysis, illustrated with examples of endlessly repeating variations. These examples allowed a librarian of genius to discover the fundamental law of the Library. This philosopher observed that all books, however different from one another they might be, consist of identical elements: the space, the period, the comma, and the twenty-two letters of the alphabet. He also posited a fact which all travelers have since confirmed: In all the Library, there are no two identical books. From those incontrovertible premises, the librarian deduced that the Library is “total”—perfect, complete, and whole—and that its bookshelves contain all possible combinations of the twenty-two orthographic symbols (a number which, though unimaginably vast, is not infinite)—that is, all that is able to be expressed, in every language.”
I've done the (simple) math on this -- in fact I'm writing a short book on the philosophy of mathematics where it's of passing importance -- and the library contains some 26^1312000 books, which makes 202T look like a very small number.
So though everything you describe is encoded in Pi (assuming Pi is infinite and normal) we're a long, long way away from having useful things encoded therein...
Also, an infinite and normal Pi absolutely repeats itself, and in fact repeats itself infinitely many times.
- it asked for my birthday (e.g. 25th Feb 1986) using a day / month / year form
- then converted to the m/dd/yy form (i.e. a string 22586),
- found that string in Pi,
- forgot my birthday and messed up displaying that somehow when converting back - saying that it found my birthday of 22 / 5 / 86
I just submitted a sub-page of that site, which has some discussion that touches more on the layout of the library as described by Borges: https://news.ycombinator.com/item?id=40970841
But it's also true that pi may not contain every _possible_ sequence of decimals, no matter what base you pick. Like the Riemann hypothesis, it seems very likely and people have checked a lot of statistics, but nobody has proven it beyond a (mathematical) shadow of doubt.
I don’t know if there would be any logical issue with this approach. The only logistical difficulty I can figure out is computing enough decimals and search the pattern in it, but I guess that such a voluminous pre-computed approximation can greatly help.
Actually any resources related to that point could be fun to explore
https://en.wikipedia.org/wiki/Pigeonhole_principle#Uses_and_...
So that means that if we give a roomful of infinite monkeys an infinite number of hand-cranked calculators and an infinite amount of time, they will, as they calculate an infinite number of digits of pi, also reproduce the complete works of Shakespeare et al.
Wouldn't the encoded information have to have a finite length? For example, pi doesn't contain e, does it?
Assuming we are only interested in base 10 and that pi contains e means that at some point in the sequence of decimal digits of pi (3, 1, 4, 1, 5, 9, 2, ...) there is the sequence of decimal digits of e (2, 7, 1, 8, 2, 8, ...), then I believe that question is currently unanswered.
Pi would contain e if and only if there are positive integers n and m such that 10^n pi - m = e, or equivalently 10^n pi - e = m.
We generally don't know if combinations of e and pi of the form a pi + b e where a and b are algebraic are rational or not.
Even the simple pi + e is beyond current mathematics. All we've got there is that at least one of pi + e and pi e must be irrational. We know that because both pi and e are zeros of the polynomial (x-pi)(x-e) = x^2 - (pi+e)x + pi e. If both pi+e and pi e were rational then that polynomial would have rational coefficients, and the roots of a non-zero polynomial with rational coefficients are algebraic (that is in fact the definition of an algebraic number) and both pi and e are known to not be algebraic.
You reminded me of this Person of Interest clip: https://www.youtube.com/watch?v=fXTRcsxG7IQ