There is at least one occurrence of eight in a row for 0, 1, 2, 3, 4, 5 and 9.
There is at least one occurrence of nine in a row for 6, 7 and 8.
There are none for ten in a row.
“12345678” shows up once as well as “01234567” and “23456789”
My birthday in MMDDYYYY shows up five times.
I was surprised by some of this but I shouldn’t have been because of how large 200 million is.
There is a 99.995% chance of finding any seven digit number. There are a very small number of six digit sequences not found.
If you are a bored, stoned geek and you convert Pi to binary and then move the strings of numbers around enough…you can make a picture of a big eyed alien. As I recall, I also found a cat and an elephant. While I’m still a geek, I’m not bored or stoned enough to do it again.
Further to hajario’s post
I recall reading that the sequence 0123456789 has not been discovered yet. (Probably in David Wells book “curious and interesting numbers”)
How about a poem for pi to 740 decimal places? See here. (It has some quirks, explained on the page there – for example, the digit 0 is encoded with a 10-letter word, and words of more than 10 letters encode two consecutive digits.)
There’s an even longer mnemonic by Mike Keith, the recreational mathematician/linguist I mentioned upthread. His Cadaeic Cadenza encodes the first 3835 digits of pi, and also mimics portions of famous works by Shakespeare, Edgar Allen Poe, Lewis Carroll, T. S. Eliot, and others. The first part, “Near A Raven”, is the most famous; it begins:
The notion that if pi is infinite it therefore contains every combination possible–that somewhere in its infinite length there will be the complete words of Shakespeare and so on-- is flawed.
There is more than one variety of infinity under consideration here and it is wrong to assume that infinity = infinity. The set of every possible combination of numerical characters is a level of infinity that is higher than the level of infinity that the infinite length of characters constituting pi.
Hence, pi can be infinitely long, never repeating itself, without in any way exhausting the entire range of possible numerical character-strings. And hence there are possible numerical character-strings (an infinite number of them, in fact) that do not, in fact, occur anywhere within the infinite string of characters that constitutes pi.
Fair enough! I thought that might be the case, but usually in a SDMB whoosh there is a subtle wit at play. I just don’t get the joke. So if any of you smarter dopers would like to fill me in, I’d be grateful.
If you mean using the actual ZIP compression algorithm specifically, it would probably remain infinite, since it’s not the sort of pattern that that algorithm was designed to compress. If, on the other hand, you’re asking about compression in general, one could write a fairly short program which could keep generating digits of pi indefinitely, and thus achieve an effectively infinite compression ratio.
The notion is flawed, but only because we don’t know that pi is normal. It has nothing to do with different sizes of infinity. The number of finite strings of digits is, in fact, equal to the number of digits in the representation of pi. The number of infinite strings of digits is a larger infinity, but nobody’s trying to find infinite strings of digits.
Hell, I had to google that to find out it means (.)(.)
The underlying algorithm is called Lempel-Zev, in case you’d like to google that for more details. This algorithm takes advantage of any periodicity it can find to reduce the encoding. It also has a bit of overhead. Random noise doesn’t have enough periodicity to make it worthwhile. (That’s why English text and executable programs compress down a lot, but audio files, which have periodicity of a kind that LZ isn’t looking for, don’t compress much.)
For any given fixed portion of pi, the zip encoding is likely to be as big or bigger, if I understand it correctly. Only a little bigger, and sometimes smaller, because a random sequence will show some periodicity.
As Chronos says, the only flaw is that we have to assume it’s normal. That is, we have to assume the digits are random.
Given that assumption, it’s pretty easy to show it’s true. For any given fixed-length string, we can exactly calculate the probability of that string appearing in a random sequence of the same length. Since the probability is a fixed nonzero value, the probability of it occurring somewhere in an infinitely long sequence is exactly 1. See the strong law of large numbers. (I recall it’s the strong law we’re using here, but it might be the weak law. My math-foo is too weak to differentiate the two in a quick look.)
Infinity? No. The number of characters in all the books ever written is finite. The number of distinct ways to fill that huge sequence of characters with arbitrary symbols from all human languages is much much larger … but still finite. If we assume each galaxy in the observable universe has a billion civilizations, each with a library a billion times the size of Earth’s the number will still be finite.
The “flaw” is that, although a normal number will eventually, after only finitely many digits, reach the exact text of Hamlet as well as The Decline and Fall of the U.S.A. (written 2143 by Benjamin Rush XII), you’ll first have to wade through trillions of Hamlet versions in which Hamlet and Ophelia elope in Act I, as well as trillions of faulty versions of Rush’s Rise and Fall in which 9-11 was a CIA plot and Sarah Palin becomes President in 2010.
For what it’s worth, it’s possible that π is not normal but still contains every finite string of digits in its decimal expansion (i.e., that it is “disjunctive”). Not that anyone has any particular reason to suspect this, but it could happen…
