Is every possible sequence of digits somewhere is the number pi?

Question

Pi is an irrational number so it has an infinite number of digits. So is it absolutely certain that every sequence of digits occurs somewhere in pi?

Is there a string of one hundred straight 5's somewhere?
Without computing pi to trillions of digits, can it be proven that it contains this sequence of digits?

Answer 1

I think that such a possibility would defeat the purpose of irrationality. If it could be proven when, where, or if a sequence exists, then pi becomes more predictable. With greater analysis over a larger string of digits, patterns of some sort show up. This cannot be if pi is irrational. Its digits are supposed to be unpredictable, by definition of irrationality. Right? A truly irrational number might as well be an infinite string of randomly generated digits.

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I know what irrationality means... thank you.

The point of what I was trying to say was that a pattern cannot be located or otherwise proven to exist. It can only be reasoned to exist... assumed... "somewhere". But not definitely proven.

Pi, like other irrational numbers in nature and not ones artificially constructed, have no discernible pattern or predictability from one digit to the next. We know no equation to solve for the xth digit of pi or the xth +1 digit of pi. We dont know all digits and never will. How might we ever prove that a certain specific sequence exists? We cant!

It stands to reason that they all do. Given an infinite number of digits. But once sequences are proven... once formulas allow us to provide solid proof... then the seemingly randomness diminishes. Eventually, pi's irrationality would be as trivial as .122333444455555666666. Therefore, pi could be reduced to a formula.

There is no formula as pi isnt just irrational.. its transcendental. What function could be constructed to not only show digits from a given point on... but actually find a predetermined sequence?

Answer 2

"So is it absolutely certain that every sequence of digits occurs somewhere in pi? Is there a string of one hundred straight 5's somewhere?"

Most mathematicians believe the answer is "yes," but to date, this has not been proven. We don't know.

"Without computing pi to trillions of digits, can it be proven that it contains this sequence of digits?"

We don't know. No one has proven that pi has to contain all possible sequences of digits (though many believe that does); nor has anyone demonstrated the existence of a sequence of numbers which cannot be in pi.

It is unknown whether pi contains all possible sequences of digits at some point.

It is also unknown whether pi contains sequences of digits with roughly the same frequency one would expect by picking digits randomly.

In mathematical terms, the question you are asking is closely related to whether pi is a "normal number." (A normal number is a number in which each sequence of digits appears about the same number of times one would expect if the digits were selected randomly.)

See http://en.wikipedia.org/wiki/Normal_number or http://mathworld.wolfram.com/NormalNumber.html .

Answer 3

It seems probable. I seem to remember some web page where you could enter in a string of numbers, and if it found it in the decimal expansion of pi, then it would show you where.

But, unless I am slightly out of date with my math news aggregator, the property has not been proven nor disproven. Not only that, I don't personally know if it can. As tempting as it is to say that it is true, we don't know every digit of pi, and we never will.

There are infinite digits in the expansion of pi, and there are infinite finite digit sequences of digits, and since the only way we can show that any finite digit sequence exists in the decimal expansion of pi is to actually show you where it occurs, we can never verify it is true for every digit sequence.

Answer 4

in an infinte string of seminly random numbers every number should appear but since infinty is not really a number more of a concept I would say that all the numbers 1-infinity don't appear.

Answer 5

I don't know the proof, but I'm guessing yes, for any finite string you'd care to name, it's in there somewhere.

This definitely isn't true for irrational numbers in general. I can make a number like 0.1223334444555556666667777777... that isn't rational, but still has a pattern that would exclude most arbitrary strings.

Edit--the reasoning in the answer below is incorrect. Irrational doesn't mean "lacking in pattern," it just means not rational--can't be expressed as a ratio of integers. If there really were no pattern whatsoever--if you just had one truly random number after another, than that WOULD guarantee that any string would show up. The whole idea that monkeys banging on keyboards for long enough will eventually write anything (like this answer).

Answer 6

Yes, simply because it is infinite, every sequence would eventually occur if you mapped it far enough.