[BruceZ]

It's quite simple really. When two sets cannot be put into one-to-one correspondence with each other, they are said to have different cardinalities or "sizes". This extends the notion of the size of a set to sets with an infinite number of elements. If a set can be put into one-to-one correspondence with the integers, it is said to have a "countable infinity" of elements. The set of integers has a countable infinity of elements. So do just the odds, so do just the evens, so do just the primes, so do the rationals. It may seem odd that a subset of a set can have the same size as the set, but that's what happens when you deal with transfinite sets. Infinity + 1 is still infinity, and it is the same size infinity. However, it is easy to show that the set of irrationals cannot be put into one-to-one correpondence with the set of integers, hence it has a higher infinity of elements, an "uncountable infinity". So does the set of all real numbers. In fact, there are an infinite number of different infinities. The set of all subsets of a set (called the "power set" of a set) always has a greater size than the set itself.

Yet even though the irrationals cannot be put in one-to-one correpondence with the rationals, there is still a rational as close as you please to any irrational, and there is an irrational as close as you please to any rational. In fact, there are an infinte number of each as close as you please to each other.

If you pick a truely random number between 0 and 1, the proability is 0 that it will be rational and 1 that it will be irrational. That doesn't mean that it must be irrational and can't be rational. Probability of 1 and dead certainty are two different things.