It's approximately the second occurrence of 9:26 in my timezone, so I'm making a thread, mostly so I can rant about misunderstandings about appearances of finite sequences in the digits of pi. http://en.wikipedia.org/wiki/Pi So approximately every 2pi radians around the sun or so, we come across the meme that pi contains every single finite sequence of decimal digits somewhere in its expansion. This may be true! But it's a mathematical statement, and thus is subject to a test that is, if not stronger, perhaps perpendicular to truth: provability. Mathematicians aren't supposed to accept statements unless they're proved or are specifically stated to be unfounded assumptions for the purpose of hypotheticals or counterfactuals. So let's examine what we can say about the digits of pi. Pi is an irrational number, which means that there is no finite string of digits such that pi is eventually just that string repeated over and over again. Some people take this to mean that pi is "random", but that's not the case. Pi is a constant. We can compute what its digits are, so it's certainly not unpredictable in any way once we know that we're dealing with pi. Furthermore, merely being irrational doesn't mean its patternless. Champernowne's number, 0.1234567891011121314... is irrational, but I'm pretty sure we can all guess what comes next. http://en.wikipedia.org/wiki/Champernowne_constant Champernowne's number also has the property that, given any finite string of decimal digits, you're going to find it eventually somewhere in the expansion. This property is called being a Disjunctive number. http://en.wikipedia.org/wiki/Disjunctive_sequence Puzzle 1: suppose the string starts with a bunch of 0s. Champernowne's number is built out of integers that don't start with a bunch of 0s, but the string appears anyway. Where? It is also not the case that pi, by virtue of being irrational, is necessarily disjunctive. The Liouville constants, which are some nonzero digit in the (k!)-th digit and 0 everywhere else, are irrational, but if all of those nonzero digits are 1 then we certainly don't get every finite string. http://en.wikipedia.org/wiki/Liouville_number Before anyone asks, pi is also transcendental, but that also doesn't tell us anything. Champernowne's number and the Liouville constants are all transcendental. Actually, we expect pi to have a property that's stronger than being disjunctive: we expect pi to be normal. A normal number is one where for any fixed k, the strings of length k occur equally often. This is impossible for finite expansions, because you can pick k to be the exact length of the expansion and that's the only string of that length that can occur, so a normal number must have an infinitely long expansion. http://en.wikipedia.org/wiki/Normal_number So what do we know about pi? Very little! We have methods for computing the digits of pi out to any finite accuracy we want, but we don't know much about the infinitely long tail that remains uncomputed. For instance, we don't even know that, say, the digit 7 appears an infinite number of times. http://www.maa.org/sites/default/files/pdf/pubs/BaileyBorweinPiDay.pdf So what, one may ask, if there are only, say, 10106 7s or whatever? Well, it doesn't matter much to most people, but to a mathematician, the rebuttal is that almost all* finite strings of decimal digits contain at least 10106+1 7s in them, so if there are only 10106 7s, then pi is missing almost all finite strings! That's certainly not disjunctive at all! Speaking of almost all, almost all** real numbers are normal. This tells us absolutely nothing about pi, because pi is not a randomly chosen real number, anymore than 1 is, and 1 is certainly not normal. Contrast with the fact that almost all real numbers are noncomputable, but we have several dozen methods for computing pi. So any property of "almost all real numbers" could hit or miss pi. There have been statistical tests performed on the known digits of pi, and they are consistent with being disjunctive, even consistent with the stronger statement of normality. But this is a statistical test, and while that level of accuracy might be good enough for scientists, science is not math, and the tests that work for scientists don't work for mathematicians. Other numbers expected to be but not known to be normal: e, log(2), sqrt(2), Euler's constant, or indeed almost any real number not specifically created to be normal. See also: http://mathoverflow.net/questions/23547/does-pi-contain-1000-consecutive-zeroes-in-base-10 *Almost all in the following sense: consider the proportion of strings of length at most k that contain 10106 7s out of all of the strings of length at most k. As k goes to infinity, the proportion goes to 1. **Almost all in the Lebesgue sense, i.e. the complement has measure 0. A perhaps more intuitive picture: if you have a function f that is 0 at normal numbers and 1 at nonnormal numbers, and you integrated f from -infinity to +infinity, you'd get 0.