Borel (1909) shows that most numbers are normal. In fact, the set of non-normal numbers, while still quite large, has measure zero. It's just hard to determine whether or not a number is normal. Certain numbers are suspicious because so far we've observed a random distribution of digits as far as we've checked. But it's risky to go ahead with that, especially knowing the other most familiar numbers are from these few classes of non-normal numbers.

You should look into websites like Mathworld, PlanetMath or Wikipedia, which often cite sources for information, allowing you to sidestep any unclear language and hear what people have claimed from the horse's mouth. Although papers on normal numbers can be a little hard to handle, that tells you why we've found so few examples.

conjecture that all the irrational algebraic numbers are normal? (2) Regarding the last sentence of your post, note that Liouville's numbers could "start" the conjecture that transcendental numbers are NOT normal, but hardly the conjecture you state. – Did Jan 27 '12 at 6:57