Hey! You there, average Internet reader. I bet you’ve heard something along the lines of this before.

Did you know, that since the digits of pi never repeat, you can find everything ever written in the digits somewhere?

There are two things wrong with such a claim.

• The reasoning is completely false.
• The conclusion is thought to be true, however it is still an open question in mathematics.

Let’s first think about the reason given. The statement that the digits never repeat, is that you can’t take a sequence of digits and then that same sequence just repeats indefinitely after a certain point in the expansion. For example, one eleventh ($\frac{1}{11}$) has a decimal expansion $\frac{1}{11}=0.09090909...$ – the sequence ‘$09$‘ keeps repeating forever. The fact that the digits of $\pi$ don’t ever repeat, is simply the statement that $\pi$ is an irrational number. So to suggest this statement was true for that reason, would say that it’s true of all irrational numbers.
To show this is clearly false reasoning, simply remove every occurrence of the numeral 7 from the expansion of $\pi$; this new number clearly doesn’t contain any sequences with 7s.

The special thing about $\pi$ that makes people believe every possible string of digits can be found somewhere, is that the digits of $\pi$ are thought to be evenly distributed. Personally, I think that it’s amazing that there are simple things about numbers that we don’t have an answer to yet!
Check out my friend’s statistics blog for a discussion on the distribution of the digits of $\pi$ at Everything is Bayesian.

Now if the answer is in the affirmative – or if we consider some number which is known to be normal – then that’s when we can start stating some cool/crazy stuff, like what you’ve probably heard before. If you convert the digits to text by some means, then every book ever written can be found among the digits. Even the full velociraptor genome could be found in there! Of course, every piece of nonsense you can imagine is in there too; there’s definitely no useful information to be extracted from it.

You can use this website to find strings of digits in the first 200000000 digits of $\pi$ – my birthday occurs around the 334,000th digit.

From → Math

I did not realise “almost all” real numbers are normal, that’s super cool. I assume though that the set of normal numbers is smaller than the irrationals? I guess your example shows that it must be, since I can’t imagine how extracting all the 7’s from a normal number could possibly render it rational. Still, strictly speaking, a number doesn’t have to be normal to contain all possible sequences, does it? Surely other distributions than uniform also work, so long as all the digits do indeed have a non-zero probability of occurring at any decimal place (in a Bayesian sense I guess). P.S. Lol thanks for the link :p I should do a new post sometime…

• As soon as I started posting about it, I remembered you had calculated that stuff on your blog 😛
I would avoid saying that the set is smaller since they both have the same cardinality. The complement of the normal numbers is also uncountable though, so perhaps it’s smaller in some sense.
I think normal is sufficient, but not necessary, to contain all finite sequences. I’m not sure exactly what you call the numbers that have that property, but I figured normal was something I’d heard of before so I ran with that as the example.