4 ms·

Here's the question I have: 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-rat

• ouid 8y ago
  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...
  • geebee 8y ago
    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.
    • gowld 8y ago
      There are some non "random" reals. But almost all of them are "random" in this sense. The exceptions are an infinitesimal fraction.
• andrewla 8y ago
  I think generally speaking, the answer is no. 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. [1] https://en.wikipedia.org/wiki/Normal_number https://en.wikipedia.org/wiki/Normal_number
  • geebee 8y ago
    Wow, yes, that's exactly it. 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. Well that was informative, thanks for the link!