Does Pi necessarily contain any sequence of numbers?
At work today, concerned about whether our new product release will actually work, we discussed the above question (totally unrelated to our product).
More formally, is there any property of Pi (assuming an infinite level of precision) that means it must contain any (possibly infinite) sequence of digits. For example, does it (by it's nature) contain the sequence 123.
Pi in fact DOES contain the sequence 123 (see 10000 Digits of Pi) but that's not the point! Can we prove that a non-random number must contain it.
After much arguing about the interaction between probability theory and infinity, trancendental numbers entropy and other related concepts, we seem to have agreed (between a Maths major, a Computer Science major and an Archaeoloy major ;) that it can't be concluded from the obvious properties of Pi, but we didn't rule out that a rigerous attack of the various Pi algorithms might in fact be able to prove otherwise.
Google and Wikipedia haven't shed any further light on the subject - can anyone comment further? (Note to self: get Dad to read this blog entry).
Clarification: We are of course dealing with Pi in decimal form. And yes, it is a stupid question that is of interest purely because of our human interest in patterns and order!
Image courtesy of xkcd A webcomic of romance, sarcasm, math, and language.
03:37 PM, 28 Jun 2005 by Mark Aufflick Permalink
Not sure, but...
I suspect you could use a variant of the triangular proof to show that it doesn't. The number of finite sequences within the decimal representation of pi is, surely ℵ-null (??), i.e. an enumerable sequence of them exists. Using the triangular proof you might create a sequence at any given size that is a counter example to the proposition.
I've been wrong before and I'm already doubting my suggestion. I do believe, however, that the result depends on the cardinality of the set of finite digit sequences. But then you also said "possibly infinite"; that takes me even further out of my depth than I was before, from ℵ-null metres to ℝ metres in the drink.
Semi from TramTown
by Unregistered Visitor on 07/02/05
Base Pi
That was along the lines of my thinking (although vastly more mathematically informed!), that the sequence of Pi can be enumerate, given sufficient time, and is therefore not random. Has anyone proven that Pi (in decimal format) has infinate precision, or might it be possible to enumerate the whole thing? As stated previously, this is all based on a decimal representation of Pi in base 10. If we were calculating in base Pi, the answer would be quite different ;)
by Mark Aufflick on 07/03/05
Pi contains everything
If you're able to convert all atomic data of yourself into digits then those digits should be located in pi somewhere. So that includes all of your memories, feelings, everything that makes you who you are. Not only are you located in pi somewhere, you are located in pi in an infinite number of places. Expanding further if you convert all data of the universe into digits that also should be located in pi somewhere in an infinite number of places. The same holds true for every possible alternate universe and every possible person that doesn't have to exist. Food for thought :-)
by Unregistered Visitor on 01/07/08
Infinitives
Slightly rambling but sometimes on point discussion of Pi and the nature of 'countable' infinitive numbers:
We are in Digits of Pi and Live Forever
by Mark Aufflick on 03/05/08
Short proof that transcendental numbers don't necessarily contain all sequences
Fact: pi is transcendental, meaning that it is infinite and nonrepeating. Fact: We cannot know all of pi, because it is infinite. An infinite, nonrepeating sequence does not necessarily contain all possible sub-sequences. Imagine, if you will, that somewhere beyond the number of digits we know for pi, the number 9 unexpectedly stops occurring. Instead of continuing as an infinite, nonrepeating sequence of the digits 0 through 9, it becomes an infinite, nonrepeating sequence of the digits 0 through 8. Such a sequence is obviously possible, because we can express pi in base 9, and it's still transcendental. After the disappearance of 9, then, any sequence containing 9 that didn't appear before its disappearance WILL NEVER APPEAR. Now, note that this is a proof that pi doesn't NECESSARILY contain all sequences, not a proof that there is some sequence that it absolutely DOESN'T contain. Because pi is infinite, we can never know the actual truth, but it is provably possible that it might not.
by Unregistered Visitor on 03/15/08
Pi contains all finite sequences
I read somewhere long ago that Pi contains all finite sequences. So the probability of finding your own phone number in pi is 1. pi is sufficiently random and non-repeating that all finite sequences occur there. I'm pretty confident that there wasn't a proof in the book I saw this in :-) Anonymous Stuart- stuart.cooper@gmail.com
by Unregistered Visitor on 07/01/05