Does $\pi$ contain all possible number combinations?

Question

$\pi$ Pi

Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time and manner of your death, and the answers to all the great questions of the universe.

Is this true? Does it make any sense ?

Answer 1

It is not true that an infinite, non-repeating decimal must contain ‘every possible number combination’. The decimal $0.011000111100000111111\dots$ is an easy counterexample. However, if the decimal expansion of $\pi$ contains every possible finite string of digits, which seems quite likely, then the rest of the statement is indeed correct. Of course, in that case it also contains numerical equivalents of every book that will never be written, among other things.

Answer 2

It is widely believed that $\pi$ is a normal number. This (or even the weaker property of being disjunctive) implies that every possible string occurs somewhere in its expansion.

So yes, it has the story of your life -- but it also has many false stories, many subtly wrong statements, and lots of gibberish.

Answer 3

This is an open question. It is not yet known if $\pi$ is a normal number.

http://mathworld.wolfram.com/NormalNumber.html

Answer 4

In general it it not true that an "infinite non-repeating decimal" contains any sequence in it. Consider for example the number $0.01001000100001000001000000100000001...$.

However, it is not known if $\pi$ does contain every sequence.

Answer 5

This is False. Claim: Infinite and Non-Repeating, therefore must have EVERY combination.

Counterexample: $0.01001100011100001111\ldots$
This is infinite and non-repeating yet does not have every combination.

Just because something is infinite and non-repeating doesn't mean it has every combination.

Pi may indeed have every combination but you cant use this claim to say that it does.

Answer 6

According to Mathematica, when $\pi$ is expressed in base 128 (whose digits can therefore be interpreted as ASCII characters),

• "NO" appears at position 702;

• "Yes" appears at position 303351.

Given (following Feynman in his Lectures on Physics) that any question $A$ with possible answer $A'$ (correct or not) can be re-expressed in the form "Is $A'$ a correct answer to $A$?", and that such questions have either "no" or "yes" answers, this proves the second sentence of the claim--and shows just how empty an assertion it is. (As others have remarked, the first sentence--depending on its interpretation--is either wrong or has unknown truth value.)

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Code

pNO = FromCharacterCode[RealDigits[\[Pi], 128, 710]];
pYes = FromCharacterCode[RealDigits[\[Pi], 128, 303400]];
{StringPosition[pNO, "NO"], StringPosition[pYes, "Yes"]}

{{{{702, 703}}, {}}, {{{303351, 303353}}, {}}}

Answer 7

I believe the statement could be worded more accurately. Given the reasonable assumption that PI is infinitely non repeating, it doesn't follow that it would actually incude any particular sequence.

Take this thought experiment as an analogy. Imagine you had to sit in a room for all eternity sayings words, without every ever uttering the same word twice. You would very soon find yourself saying very long words. But there's no logical reason why you should have to use up all the possible short words first. In fact you could systematically exclude the words "yes" or every word containing the letter "y", or any other arbitrary subset of the infinite set of possible words.

Same goes for digit sequences in PI. It's highly probably that any conceivable sequence can be found in PI if you calculate for long enough, but it's not guaranteed by the prescribed conditions.

Answer 8

Whether or not it's true, it's absolutely useless.

Imagine finding your life story: a copiously documented and flawless recounting of every day of your life... right up until yesterday where it states that you died and abruptly reverts back to gibberish. If pi truly contains every possible string, then that story is in there, too. Now, imagine if it said you die tomorrow. Would you believe it, or keep searching for the next copy of your life story?

The problem is that there is no structure to the information. It would take a herculean effort to process all of that data to get to the "correct" section, and immense wisdom to recognize it as correct. So if you were thinking of using pi as an oracle to determine these things, you might as well count every single atom that comprises planet Earth. That should serve as a nice warm up.

Answer 9

Let me summarize the things that have been said which are true and add one more thing.

1. $\pi$ is not known to have this property, but it is expected to be true.
2. This property does not follow from the fact that the decimal expansion of $\pi$ is infinite and does not repeat.

The one more thing is the following. The assertion that the answer to every question you could possibly want to ask is contained somewhere in the digits of $\pi$ may be true, but it's useless. Here is a string which may make this point clearer: just string together every possible sentence in English, first by length and then by alphabetical order. The resulting string contains the answer to every question you could possibly want to ask, but

• most of what it contains is garbage,
• you have no way of knowing what is and isn't garbage a priori, and
• the only way to refer to a part of the string that isn't garbage is to describe its position in the string, and the bits required to do this themselves constitute a (terrible) encoding of the string. So finding this location is exactly as hard as finding the string itself (that is, finding the answer to whatever question you wanted to ask).

In other words, a string which contains everything contains nothing. Useful communication is useful because of what it does not contain.

You should keep all of the above in mind and then read Jorge Luis Borges' The Library of Babel. (A library which contains every book contains no books.)

Answer 10

Challenge accepted. In the following file are the first 1,048,576 digits (1 Megabyte) of pi (including the leading 3) converted to ANSI (with assistance from the algorithm described in https://stackoverflow.com/questions/12991606/):

https://docs.google.com/file/d/0B9plORbvSu2ra1Atc0QwOGhYZms/edit