(Sorry if this has been asked before - Pi is too short of a search term)
If Pi’s decimal representation never ends, it never settles into a permanently repeating pattern and it is infinite, does that mean that every possible combination of numbers would eventually come up?
Like for example nine 9’s in a row. Or would it mathematical properties preclude this?
That’s thought to be true, but has never been formally proven.
The digits of π have no apparent pattern and have passed tests for statistical randomness, including tests for normality; a number of infinite length is called normal when all possible sequences of digits (of any given length) appear equally often. The conjecture that π is normal has not been proven or disproven.
It’s not guaranteed that a non repeating number sequence contains every possible sequence,
for example 123112233111222333111122223333… Is non repeating, but never contains the sequence 789
For pi, I belueve its unknown whether it does or not
It certainly doesn’t follow directly from the “never ends, never repeats” property. The number whose decimal representation is “0.101001000100001000001…” also never ends and never repeats, but doesn’t contain any sequence containing “2” (or even just “11”).
The property you are looking for is “Normality”, which I see I’ve been ninjaed on.
For a normal number, the key point is that every possible finite sequence of numbers will be contained within it. So if Pi is normal, it can’t contain other irrational, non-ending numbers like e, but it does contain all the works of Shakespeare, as well as everything humanity has written, or will ever write in its existence (assuming the words can be represented as ASCII characters which are converted to their Base-10 representation). Heck, throw in there every piece of digital data humanity has or will ever create.
If pi is normal, and (in principle) can be proven so, then pi contains every possible proof of its own normality, infinitely many times.
Some people seem to think that this is some special property of the number pi. Actually, this is true of most* numbers.
*(It’s true of every irrational number, and most of the real numbers are irrational, in a specific, provable mathematical sense.)
And yes, this has come up before. Since the question has been answered, it’s probably not worth searching out earlier answers; but here’s an old thread on a related question that some of us might find interesting:
Sorry, I could only find eight 9s in a row in this page which calls itself the Pi-Search Page. They only seem to use the first 200 Million digits of Pi, but we know several billion by now, and counting. The explaining text reads:
Results
The string 99999999 occurs at position 66780105. This string occurs 1 times in the first 200M digits of Pi.
counting from the first digit after the decimal point. The 3. is not counted.The string and surrounding digits:
141994430187282261379999999931798954323109584210
You can look up any strings you fancy, if they are shorter than 7 numbers in my experience they usually show up. People seem to try out their birthday often, so did I. Now, how do I prove that the page tells the truth and is not making it all up on the fly?
Ask it again. Unless it saves all the requests it won’t know it’s been asked before so it will make something up. It’s highly unlikely it would make up the same thing.
That is what I did, and as the answer was the same I felt confident enough to link it. But I cannot prove it.
BTW: My birthday appears twice in the first 200 million digits in european notation (DD/MM/YYYY) but none in american notation (MM/DD/YYYY).
Suppose you enter a pi reciting contest, and you’ve memorized the first million digits. You’ve got this! No one can possibly beat you. And then you forget to say “Three point” at the beginning. Would they disqualify you?
What a bummer!
In fact, not only is pi believed to be normal, and it is known that most numbers are irrational, it’s even known that most numbers are normal.
Which isn’t really as significant as you’d think, because it’s also known that, given any notation system, most numbers can’t even be expressed at all in that notation. Not even if your notation system includes every form of notation human mathematicians ever have or ever will come up with.
In other words, every number that we can express is, in some sense, special. And so it’s possible that, for almost all of those cases, that specialness is somehow such that it prevents the number from being normal. Hence, both why we suspect that pi (and e, and sqrt(2), and so on) is normal, and why we’re not actually sure.
Thanks for that link. My grandson turned 11 on November 11, 2011 so I looked up 11/11/11 (that’s 11/11/11 for you non-Americans). It comes up 191 times in the first 200 million places.
That is just plain damn hospitable and considerate of you. Ta muchly. 
It would seem to me that if your grandson turned 11, you should have looked for 11111111. Which comes up three times. Lucky number! For your grandson, anyway.
Glad I could help 
Looking for Pi in the emojis I only found a Japanese variant 
and a musical one 
. Strange omission.
So somewhere it say “Help, I’m trapped in a universe-creating factory!”
Yep, it’s guaranteed to be in there, but we just don’t know where the index is. It could be past 100 trillion digits, past Tree(3) digits, or even past a Loader’s number of digits. The point is, when compared to infinity, even these massive numbers are nothing.
If the universe could be simulated by 1s and 0s, it will also have a simulation of the entire history of the observable universe (since that is finite) from the moment of the Big Bang.
Somewhere it says “Please disregard all of the following digits.”