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/
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