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Comment by pdpi | Hacker News Reader

It's worth noting that the article slightly confuses the definition of normal number: A number isn't "normal with respect to a given base". It's either normal, or it's not. The definition is in terms of a given base, but it must hold true for all bases.

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I think the "confusion" is justified. It's a good thing(tm) to start with the concept of "normal with respect to a base" and then move on to say that "normal" or "absolutely normal" means "normal for every base (greater than 1)."

Wikipedia does this, adding the extra term "simply normal" for the case where we are talking about relative to a single base:

    We say that x is simply normal in base b if
    the sequence S(x,b) is simply normal, and
    that x is normal in base b if the sequence
    S(x,b) is normal.  The number x is called a
    normal number (or sometimes an absolutely
    normal number) if it is normal in base b
    for every integer b greater than 1.

For the purposes of explanation, especially to non-technical audiences, I think the approach taken is fully justified.

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

Are there any cases where a number is normal in one base but not another?

If you look at the definition on Wikipedia[1], you can see that a normal number, per definition, is base agnostic.

[1]: https://en.wikipedia.org/wiki/Normal_number

I think the question was whether it is possible for a number to be normal in one base but not in another. Intuitively I would say that randomness should transcend the base and therefore a number that is normal in one base should be normal in any other base and therefore absolutely normal. But if that were the case the distinction between normal in a specific base and absolutely normal would be unnecessary. So I don't know but would also be interested in an example in case one exists and is known.

EDIT: It is easily possible for simply normal numbers, i.e. if you only consider single digit frequencies but not frequencies of digit pairs, triples and so on. 0.(0123456789) is simply normal in base 10 because every digit occurs with frequency one tenth but it is not normal in base 10 because only ten out of one hundred digit pairs occur. But it is also not simply normal in base 10¹⁰ because it then consists only of the single digit representing 0123456789 repeated indefinitely.

Ah, yes.

Brought up here as well: https://news.ycombinator.com/item?id=10966819

This doesn't prove that the number in the edited example isn't simply normal in a base b for some b.

I am not sure if I get your point. The number is simply normal in base 10 but is not simply normal in base 10¹⁰. And that is what Retr0spectrum asked with simply normal replaced with just normal, i.e. for a number that is normal in at least one base but not normal in at least one other base.

That number is rational, and all rational numbers are not normal.

But they can be simply normal. It's a much weaker property.

It is known that Stoneham's constant alpha_2,3 is normal to base 2 but not to base 6.[1]

[1]http://crd-legacy.lbl.gov/~dhbailey/dhbtalks/dhb-nonnormalit...

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