Pi and normal numbers

The Internet seems to be an ideal breeding place for urban legends and questionable "facts". Just think about the claim that only on the equinoxes it's possible to stand an egg on its end . It's not true, but the story seems hard to eradicate; it pops back up around 21 March in many newspapers and magazines.

But while there are claims which are easy to refute, for others it's not that easy. Take for instance the following claim which can be found here and there on the Internet:

Pi is an infinite, nonrepeating 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.

This sure sounds exciting, but we're not sure if it's true. Being infinite and nonrepeating (which pi is) isn't enough. For instance, the ASCII sequence "01001000100001…" is infinite and nonrepeating, but doesn't contain any name at all.

Normal numbers

What's required is that pi is normal. A normal number is an irrational number for which any finite pattern of numbers occurs with the expected limiting frequency in the expansion in a given base (or all bases). For example, for a normal decimal number, each digit 0-9 would be expected to occur 1/10 of the time, each pair of digits 00-99 would be expected to occur 1/100 of the time, etc.

The problem is that we don't know if pi is normal. It's paradoxal that we know that most irrational numbers are normal, but we only know a couple normal numbers. Nevertheless it is conjectured that pi is normal in all bases, and analysis of the known digits of pi seem to confirm it, but there's no proof.

Some thoughts

If pi is normal, it will contain "the answers to all the great questions of the universe", but not only that. It will contain just any text, true or false. It will contain "pi is not normal", and "the square root of two is rational" for instance. So all great questions will have an infinite false answers to them in the sequence next to the right one.

Also, don't forget the word "infinite". Any finite sequence, no matter how long, only can contain a finite amount of information, and the information you want may not be in it. There's no way to represent all digits of pi; there's simply not enough matter in the (finite) universe. So, while statistically improbable, it's possible that your name does not appear in the ASCII representation of the first gazillion digits of pi. Improbable doesn't mean impossible.