Re: Pi as the Mother Number
- From: "Jim Black" <tramspap@xxxxxxxxx>
- Date: 11 Feb 2006 20:00:59 -0800
dougwedel@xxxxxxxxxxxxx wrote:
It has occurred to me during idle moments that pi, if it is normal and never
repeats, contains within itself all possible finite integer strings. Is
this interesting? It could be in the sense that pi therefore may contain
ALL RANDOM NUMBERS. What does it mean to say that pi contains all random
numbers? It means that you could generate any 100-digit number with your
best random number generator and if you searched long enough through the
expansion of pi you would find your 100-digit number. Once we find the
supposedly random number in pi it doesn't seem so random anymore. In fact,
in some cases, when we find the number in pi, we will have found a way to
actually "compress" the digit string and thus lower its Kolmogorov
complexity. For example if we find our 100-digit string starting at the
37th digit of pi, we can just say, go to the 37th digit of pi and print the
next 100 digits. But as a very astute mathematician pointed out to me, the
problem with using this principle to compress numbers is the problem of the
index, i.e. the index into pi. Typically it becomes larger than the random
number we are trying to compress. I have been working on half-baked notions
for how to compress indices into pi (eg. go to the 10^24 + 37th digit of pi
and print the next 100 digits) so as to increase the universe of
compressible, formerly "random" numbers. I think what I am doing is
attacking what I intuit are shaky intellectual foundations of the
mathematical treatment of randomness ;-) I wonder if these speculations
have any resonance in the "real world" of mathematics or if anybody has any
ideas on how to compress indexes into pi?
Let's say you have a string of up to 100 digits, chosen at random.
There are 10^100+10^99+10^98+... = 1.111....*10^100 possible strings.
Now suppose you run them through some sort of compression algorithm.
The compression algorithm outputs a string of digits which you could
uncompress if you wanted the original string of digits back. There
must therefore be at least 1.111...*10^100 possible output strings, or
the compression would not be reversible. Since we want to make the
output strings as short as possible, we wouldn't use a 101-digit string
as a possible output if there was an string of 100 digits or fewer that
wasn't being used to represent something else. Thus, we will end up
using all 1.111...*10^100 possible 100-digit-or-fewer strings as
outputs. So the average length of a string after compression will be
the same as the average length of a string before compression.
The reason that compression works, despite this argument, is that not
all inputs are equally probable. For example, for a program designed
to compress text, the input "Have a nice day" is much more probable
than the input "dkjrc4jg5ghd3bv" is. But if the input is random, i.e.,
all inputs are equally likely, then compression is impossible.
.
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