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I struggle to understand why disjunctive numbers are necessarily transcendental.

A rich number (or disjunctive number) is a real number whose expansion, in a given base $b$ is a disjunctive sequence over the alphabet $\{0, ..., b  −  1\}$, i.e. it contains any finite sequence of digits, at least once, expressible in that base (in other words, it contains the whole “universe” of finite integers, at least once, expressible in base $b$)

I read this in the OEIS Wiki : "[...] Thus, disjunctive numbers (in base $b$) contain, at least once, all $n$-digits approximations (ignoring the fractional point) for all real numbers (including at least a second occurrence of all $n$-digits approximations of itself or not?), which means that disjunctive numbers are transcendental numbers and constitute an uncountable subset of the real numbers."

But I don't understand the argument.

Why the fact that containing at least once all n-digits approximations of every real numbers means that the number cannot be the root of a non-zero polynomial of finite degree with rational coefficients ?

(I apologize for the potential english mistakes I'm still learning it)

LexLarn
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    Not sure I understand the claim. Surely "normal" implies "disjunctive" and it is at least widely conjectured that all the quadratic irrationals (like $\sqrt 2$) are normal. – lulu Dec 17 '22 at 03:45
  • To be fair: I'm not entirely clear on the meaning of "disjunctive". I thought that "normal" was the same, only you also insist that each string of length $n$ occur with the appropriate frequency. If I have that right, then, indeed, normal implies disjunctive. – lulu Dec 17 '22 at 03:47
  • Rich numbers (with the definition that they only must contain all finite strings , not necessarily with the correct frequency) must be irrational, but there is no reason that they must be transcendental. In fact, the algebraic irrationals are conjectured to be normal which is a stronger property. – Peter Dec 17 '22 at 08:13
  • So the argument is probably wrong, but we have to admit that we know almost nothing about normality. – Peter Dec 17 '22 at 08:16

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