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Suppose number$R$ expands in decimal,and $\lim_{n\rightarrow\infty}\frac{C_{n}(d)}{n}=\frac{1}{10}$ where $d$ is one of the ten digit,and ${C_{n}(d)}$ the counting numbers of $d$ from first digit to $n$ digit.We call such a number as normal number.

Now, could any one give any irrational algebraic number $x$ which is a normal number?.

MJD
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XL _At_Here_There
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    Possible duplicate of Normal Number - Intuition – Nij Aug 20 '17 at 03:27
  • This property is called normality in numbers. There are already a huge number of questions about it. One in particular gives that almost all numbers are normal, so any irrational algebraic number is almost certain to also be normal. But it is not yet shown that a specific given number is normal for sure. – Nij Aug 20 '17 at 03:31
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    @Nij, am I missing something? I don't see anything in the link about the question of whether there exists an irrational algebraic number which is normal. Also your second comment might be confusing to someone unfamiliar with the subject (since we do know that some "specific given numbers" are normal.) – Daniel McLaury Aug 20 '17 at 03:33
  • @DanielMcLaury I think Nij is confused, and I suspect the question is open. I guess there is not irrational algebraic number which is normal. – XL _At_Here_There Aug 20 '17 at 03:37
  • @XL_at_China: The question is open according to Wikipedia, but presumably every irrational algebraic number is normal. It would be very surprising if that weren't the case. – Daniel McLaury Aug 20 '17 at 03:52
  • @Nij could you give any example? Why do you think it is solved? and actually, I do not know the terminology and basic knowledge,sorry. – XL _At_Here_There Aug 20 '17 at 03:56
  • @DanielMcLaury in the context of irrational algebraics, we have no known examples, though we expect (almost) all of them to be normal. But I agree that my original phrasing could cause confusion. – Nij Aug 20 '17 at 03:57
  • @XL_at_China that's the point, it is not* solved*, though we have good reason to expect the result in previous comments vis-à-vis "(almost) all irrational algebraics are normal". – Nij Aug 20 '17 at 04:02
  • @DanielMcLaury I suspect there is no irrational algebraic number which is normal when I study another problem, so I ask the question – XL _At_Here_There Aug 20 '17 at 04:04
  • We can easily constuct normal rational number of base 10, and construct normal transcendental number of 10 easily, But we have not found or constructed any normal irrational algebraic number. – XL _At_Here_There Aug 20 '17 at 04:08

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According to Wikipedia this is an open question:

It has been conjectured that every irrational algebraic number is normal; while no counterexamples are known, there also exists no [irrational] algebraic number that has been proven to be normal in any base.

https://en.wikipedia.org/wiki/Normal_number

Daniel McLaury
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