A normal number [1] is pretty close to what you're describing. Although there are a lot of normal numbers, theoretically, they are very difficult to construct, and equally difficult to prove normality. Unfortunately, proving non-normality is also difficult, although the wikipedia article has some examples of base-10 normal numbers and a couple of examples of irrational non-normal numbers.
It's been 20 years since I majored in math, so maybe I did encounter these and just don't remember. Looks like there is a proof that almost all real numbers are normal, but where it comes to specific numbers pi, e, sqrt(2), they are thought to be normal but no proof.
No, of course not. You just gave an example of a number which was real, not rational, and easily described with finite information 0.001012012301234...
So... does this mean that this statement in the abstract:
"Moreover, a better terminology for the so-called real numbers is "random numbers", as their series of bits are truly random"
is demonstrably false? That there exist real, non-rational numbers with a series of bits that are not truly random?
Well, thinking about this after reading other posts in this thread, perhaps the statement really means that "almost every" real, non-rational number has a series of bits that are truly random, so this isn't a counterexample.
Are the bits of a real, non-rational number random.
I suppose another way to think of it would be
Are the bits of a real, non-rational number expressed as a decimal random integers between 0 and 9?
I'm out of my depth here, but it seems that is a statement that could be proven.