# Is π normal in base π?

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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5
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

1
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

## 1 Answer

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.

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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
Thank for explaining it. – BanksySan Oct 22 '12 at 11:45
Kind of like how there would be no prime numbers without the restriction to integer divisors... – Alistair Buxton Nov 4 '12 at 10:54
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