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This question states that π is normal:

Does Pi contain all possible number combinations?

My understanding of this is that it means that the statistically, the distribution of every number is equal across the infinite range.

If the numbering system is base π, wouldn't the number just be 1, so not normal, or does the definition only mean the bases that the number would be infinately non-repeating?

e.g π, in base π is 1 and not infinately non-repeating π in base 10 is infinately non-repeating

Let me summerise, does π base π = 1 mean that π isn't normal, or is π base π excluded from the definition because it in not infinately non-repeating?

Cheers

Dave

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BanksySan
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    What does base $\pi$ mean after all? *Usually* one talks about bases $b$ only if $b$ is an integer $\ge 2$. – Hagen von Eitzen Oct 19 '12 at 14:58
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    You have to define what it means to be in a base that is not an integer. In particular, with integer base, "almost" every real number is uniquely represented in the base. I'm not sure this is true for non-integer bases. – Thomas Andrews Oct 19 '12 at 15:00
  • Base phi (golden ratio) is a thing. (It's sometimes called "phinary.") I think there's a Wikipedia article on it. – Akiva Weinberger Jul 16 '15 at 05:48
  • @AkivaWeinberger indeed, see https://en.wikipedia.org/wiki/Golden_ratio_base – Stéphane Gourichon Jan 06 '16 at 19:55
  • @StéphaneGourichon Yup! By the way, since $L_n=\phi^n+(-\phi)^{-n}$ (where $L_n$ is the $n$th Lucas number), we have:$$L_{2n}=\phi^{2n}+\phi^{-2n}=10\dots00.0\dots01_\phi$$(where the last thing is written in phinary). This means that, to find the phinary representation of a large number, a useful idea is to write it as a sum of (even-indexed) Lucas numbers, convert to phinary, and simplify. – Akiva Weinberger Jan 06 '16 at 20:17
  • @AkivaWeinberger Mathematicians in my former university studied bases with almost any number – irrational, negative, complex... It's all possible, to some degree. – The Vee Nov 17 '16 at 13:22
  • @TheVee Complex? Really? I haven't seen that before. (Nor have I seen negative bases, actually.) – Akiva Weinberger Nov 17 '16 at 13:24
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    @AkivaWeinberger That also turns out to have a Wikipedia page, [here](https://en.wikipedia.org/wiki/Complex-base_system), and more generally listed under [Non-standard positional numeral systems](https://en.wikipedia.org/wiki/Non-standard_positional_numeral_systems). – The Vee Nov 17 '16 at 13:30

2 Answers2

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When we say that $x$ is normal, what we mean is that it's normal to base $b$ for every integer $b\ge2$. Base $\pi$ does not enter into the discussion.

Gerry Myerson
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    Both of you have mentioned that bases can only be integers... I can't see why this should be at all, surely what values are integers and what are not ate nearly a product of your choice of base. However, my original question was if pi as excluded from the rules for a normal number definition in base pi... which it seems is the case. In think I'll start a new question about integers being defined by your choose of base. Thanks all. – BanksySan Oct 21 '12 at 20:49
  • Having reconsidered... I see the argument. I was on the wrong track. Thanks all for answering though. – BanksySan Oct 21 '12 at 21:29
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    Careful --- I didn't say "bases can only be integers" --- I said that *when it comes to deciding whether something is normal* bases can only be integers. The reason for this is clear; if you don't restrict the bases, there aren't any normal numbers, since no number $x$ is normal to base $x$. Bases don't have to be integers, and there are many papers devoted to the properties of expansions to non-integer bases. It's just in the context of normality that we can't get anywhere without the restriction. – Gerry Myerson Oct 21 '12 at 22:51
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    Kind of like how there would be no prime numbers without the restriction to integer divisors... – Alistair Buxton Nov 04 '12 at 10:54
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In base pi, pi is represented as 10. The same way that 10 is represented as 10 in base 10. If you are asking if pi is normal in base pi then the answer is no because it only has the digits 1 and 0 in it.

Neil
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  • Good point. Also, kudos for reviving a question over two years old. – BanksySan Apr 13 '15 at 11:29
  • If you check the wikipedia page it says that a normal number is normal for all bases >1. It doesn't say that you aren't allowed to discuss the normality of a number in a specific base. In fact in that same page they discuss numbers that are normal in some bases but might not be in others. Your intuition was right. Also I don't avoid putting in the right answer if something is over a certain age. In fact my first accepted answer on English stack exchange was on a 3 year old question. Also someone put a delete vote on me for this answer, but I'm happy you were able to see it before removal. – Neil Apr 13 '15 at 12:38
  • Gerry Meyerson is right in saying that any x in base x is 10 and so is not normal in that base. But having 1 base where something isn't normal is fine because the Lebesgue measure is still 0. It's when you have more than the single base where it isn't normal that the Lebesgue measure is not 0. – Neil Apr 13 '15 at 12:41
  • I suspect, Neil, that if $x$ is real and not zero, then the set of real bases to which $x$ is not normal has Lebesgue measure zero. – Gerry Myerson Aug 26 '15 at 22:58
  • Don't forget that in base pi, pi is also represented as 3.01102111... in exactly the same way as 10 is also represented in base 10 as 9.9999.... – Leo B. Jul 30 '20 at 00:11