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The number $\pi$ has infinite decimals whom appear to be randomly distributed.

If we had $\pi$ fingers, and would therefore use $\pi$ as base instead of ten, could I then count integers on my fingers?

Jyrki Lahtonen
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Per Eriksson
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  • What do you mean by exist other integers? They will always exist regardless of the base you assume. Do you mean would they be expressible with a finite "$\pi$-expansion"? – Milind Hegde Dec 01 '15 at 13:07
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    Of course, if Ana has $\pi$ fingers and Bob has $\pi$ fingers, together they have $2\pi$ fingers. – Antoine Dec 01 '15 at 13:07
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    What would be the "digits" in a base $\pi$ system? – coffeemath Dec 01 '15 at 13:09
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    And what does it mean to have $\pi$ fingers...? – Hans Lundmark Dec 01 '15 at 13:10
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    The most interesting non-integer base, I think, is the golden ratio $\phi$ ("phinary"). The digits are $0$ and $1$, and it has the odd characteristic that $100=11$. (Thus, in phinary, one does not usually allow two $1$s to be next to each other.) For example, the number $2$ is written as $10.01$ in phinary, since $2=\phi+\phi^{-2}$. However, I don't think that it makes sense to have $\phi$ fingers or $\pi$ fingers… – Akiva Weinberger Dec 01 '15 at 13:15
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    An interesting question. – orangeskid Dec 01 '15 at 15:33
  • Being an integer or not has nothing to do with the base in which a number is written. – Alex M. Dec 01 '15 at 16:00
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    @Akiva Are there similar representations for other rational numbers ? – Peter Dec 01 '15 at 18:27
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    @Peter Yes; check the Wikipedia page on phinary. For example, I believe — though I'm not sure — that $\frac12$ is $0.01\overline{001}$ in phinary. – Akiva Weinberger Dec 01 '15 at 18:31
  • Well I'm a software engineer, not a mathematician really. But I sometimes wonder about these types of questions and this time I was very happy to realize that I could ask it here on Stack. Some liked the question, some did not. I hope, at least, that no one was offended. – Per Eriksson Dec 01 '15 at 19:35
  • I think many of you gave good answers in different ways. How can I edit the question to put it off hold but still keep the philosophical quirk to it? – Per Eriksson Dec 01 '15 at 19:42
  • Integers do not have a finite representation in base pi. This is equivalent to the fact that pi is transcendental. – Justpassingby Dec 02 '15 at 06:32
  • @HansLundmark By this time it's reasonably clear what is being asked. – Justpassingby Dec 02 '15 at 06:34
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    So you wanna have $3.141592....$ fingers. – Kushal Bhuyan Dec 02 '15 at 06:35
  • @Justpassingby: Not to me! – Hans Lundmark Dec 02 '15 at 07:50
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    @HansLundmark: the question is if, and how, integers can be expressed as sums (finite or possibly infinite) of positive and negative powers of $\pi.$ That is what the base of a number system does: it expresses integers, fractions and irrationals as progressively more complicated sums of powers of the base. – Justpassingby Dec 02 '15 at 07:55
  • If that's the question, it should be asked that way instead of talking about counting on an irrational number of fingers, which is just nonsense in my opinion. – Hans Lundmark Dec 02 '15 at 09:20
  • Your question is more or less answered at http://math.stackexchange.com/questions/1465617/can-pi-be-rational-in-some-base-radix – Mark S. Jan 14 '16 at 00:52

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