Are All Digit Strings in Pi?

Date: 08/06/99 at 15:23:50
From: Ilan Halfi
Subject: Repeating Digits in Pi?

Hi there,

I was just wondering if you could help solve an argument. My two 
friends have a theory that since Pi is infinite, all possible number 
combinations have to, and will, be used. For example: somewhere down 
the line of digits, there may be a thousand twos in a row, or even a 
million twos without a single other digit interrupting those twos. My 
defense was that this is extremely improbable; why should the 
uniformly random distribution of numbers in the first 51 billion 
digits of Pi be altered so greatly the further down you go? If it is 
infinite, shouldn't it be absolutely random throughout? Just because 
something is infinite, does that mean there is some 'law' that says 
all possible number combinations will be implemented?

Thanks so much for your help!
-Ilan

Date: 08/06/99 at 20:04:56
From: Doctor Rob
Subject: Re: Repeating Digits in Pi?

Thanks for writing to Ask Dr. Math.

There are irrational numbers with no repeat that don't contain any 
occurrences of certain multiple-digit strings. An example would be

0.10010000100000010000000010000000000100...,

where all the digits are zero except for those in positions whose 
number is a perfect square:  1, 4, 9, 16, 25, 36, and so on, that are 
all 1's. This contains no 2's, 3's, ... 9's. It doesn't even contain 
11. It also is not periodic.

On the other hand, it is known that most irrational numbers are 
"normal," that is, every k-digit string appears "equally often," for 
every positive integer k. (To make this rigorous is possible, but not 
quite obvious.)

From these examples, it is clear that not every irrational number is 
normal.

It is not known whether or not Pi is normal, although most 
mathematicians probably believe this. It is extremely hard to prove 
that any particular irrational number is normal. If Pi is normal, then 
your friends are right, and somewhere down the line, there will be a 
million 2's in a row. In fact, this will occur at infinitely many 
different places in the decimal expansion of Pi. If Pi is not normal, 
then a million 2's may or may not ever occur.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/