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.