random numbers

What are the qualities of random numbers?  As I remember from math class, a truly random number is one without any pattern. 

Is it safe to read into that and say that in approach of the infinity-ith digit of that random number, every possible sequence of digits will have presented itself in that number?  If this were not the case, would there not be different types of random numbers, each presenting a different type of set of numbers out to the infinity-ith digit?  Could this be possible without a pattern?

If every possible sequence of numbers is present in any random number, is not every other random number present in that random number?  Is that random number not present within itself?  Is that not a pattern? 

There's something the fsck wrong here.  Either there is no infinity or there is no random.  Either that or I'm looking at this the wrong way. 

For any finite sequence of numbers, and with sufficient bribery, a crazy mathematician will find some pattern in it.  True randomness requires some degree of infinity, which most minds are incapable of comprehending in a consistent manner.

I'm going to assume you are referring to the digits of an arbitrary irrational number.  In such a case, it is not necessary that every possible (finite!) sequence of digits will present itself.  Those that do have this property are referred to as uniformly distributed.  It is speculated (but unproven) that pi and e have this property.

Let A, B be non equal uniformly distributed numbers.  By definition, for any finite N, the 1st N digits of A will appear in B infinitely often (and vice versa).  However, it is almost certain that the entire A sequence (infinite!) will not appear in B.

>>2
which most minds are incapable of comprehending in a consistent manner.
Middle school bullshit trying to sound profound. F-

What you're trying to argue is that an infinite series must contain every other infinite series in order for it to be truly random. That's not right.

>>2
Yah, a uniformly distributed arbritrary (UDA) number.  Pi is what came to mind.

I was thinking that different UDA numbers (pi and e, perhaps) could be the same, ie: a pattern could exist in a bit by bit comparison between the two UDA numbers, or any UDA numbers.

But... I'll have to brush up on the math and matlab skills to go anywhere with that. 

hah... http://www.everything2.net/index.pl?node_id=1302963


Peace.

>>3
I did not intend that to be profound.  It was a simple, and probably obvious statement.

>>5
Doubtful.

You're going to get pwnt by matlab due to the limits of double precision.  You might need to implement your own standard (or use rationals or something).

A random number is a map from a field equipped with a sigma algebra and a sigma-finite measure to the real lines.

A definition that might line up better with your intuition is that a "random" sequence, with random being taken in the conventional sense, is a sequence such that the conditional expectation of each successive digit given the entire previous sequence is exactly the same as the unconditional expectation of the next digit, and each digit is drawn from some nondegenerate distribution.

Of course, to prove definitively that a sequence is completely random forever does require an infinite number of digits, but to decide whether or not a finite-length realization of that sequence is random generally only requires a few hundred realizations of the sequence of that length to do some testing.

Remember, you can never prove something random, all you can do is fail to prove that it's not random.

Pi's digits have not been proven to occur randomly.

i proved it but i fucking forgot to save the doc
aint that some shit