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what has led to conjecture that all the irrational algebraic numbers are normal? is there some kind of evidence somewhere?

i know that there exists non-normal trascendental numbers like liouville's number. is this the only thing that "started" the conjecture?

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closed as off-topic by Did, Ramiro de la Vega, Benjamin Steinberg, Chris Godsil, Yemon Choi Jul 6 '13 at 19:58

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Benjamin Steinberg, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

(1) Do you have a reference for the 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
This question appears to be founded on an incorrect premise – Yemon Choi Jul 6 '13 at 19:58
In On the Random Character of Fundamental Constant Expansions (by David H. Bailey and Richard E. Crandall), it can be read : "one could further conjecture that every irrational algebraic number is absolutely normal". See also Algebraic irrational binary numbers cannot be fixed points of non-trivial constant-length or primitive morphisms (by J.-P. Allouche and L. Q. Zamboni). – Watson Apr 1 at 21:18
up vote 1 down vote accepted

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.

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