D’oh! Would you believe, twenty-seventh (for “twenty-seventy” above).
Here are the rules for digits in base n:[ul]
[li]If |n| > 1, the symbols are the integers in the range [0, |n|) and everything works exactly like you’d expect.[/li][li]If 1 > |n| > 0, the symbols are the integers in the range [0, |1/n|). Number representations may extend infinitely to the left, but will terminate to the right. For instance, (1/3)[sub]10[/sub] = …333.[sub]1/10[/sub].[/li][li]If |n| = 1 or |n| = 0, it’s just not going to work.[/li][li]If n < 0, you don’t need a sign marker (a negative sign, a bit, etc.) because that information is already contained in the digits.[/li][/ul]As Exapno Mapcase mentions, you can use any symbols you like, but since they have to behave exactly like integers with respect to addition and ordering, what’s the benefit of using something odd?
Wikipedia has articles on non-standard positional numeral systems, some of which are not base n for any n.
Just to make it clear, you need three symbols, for example 0, 1 and 2.
Pi would be 10.
Right… I’ll come in again.
You need four symbols, 0, 1, 2, and 3.
I will put them into bold-face type to make them really pop. 0, 1, 2, 3. Ok, so what do these symbols represent? Well,
0 = 0
1 = Pi/4
2 = Pi/2
3 = 3Pi/4
10 = Pi
11 = 5Pi/4
et cetera.
At least, this is how I would guess it would work. If I remember I’ll come back tonight and see how this thread is going.
sinjin
For the record, I have most of a PhD in mathematics and at no point in my mathematical career has this ever been an issue. I’ll try to decide just what it means to have irrational bases and if I can write it up nicely without using .pdf files. 
Correct. (Cite: ultrafilter’s post.)
No, those symbols represent exactly what they would in any base: the integers zero, one, two, and three.
But how can that work? If 1=1, 2=2, 3=3 and 10=pi, how can you do arithmetic?
What would 2+2 equal? It can’t equal 10. What does 11 equal? Pi+1? But 100 has to eqal pi^2, right? What does 20 equal, 2pi?
If you have a non-integer base, then I can’t see how any integer in that counting system could be a decimal integer.
I’m still not getting this.
Wouldn’t 3.3 represent 3 + 3/[symbol]p[/symbol], which is about 3.955[sub]10[/sub]? But that is greater than 10[sub][symbol]p[/symbol][/sub]. So you can define a number base such that 3.3 > 10?
In base n, a[sub]k[/sub]…a[sub]0[/sub].a[sub]-1[/sub]…a[sub]-m[/sub] represents sum( a[sub]i[/sub]n[sup]i[/sup], -m < i < k ). The second three only represents 3/[symbol]p[/symbol] because of its position.
Back up a minute there.
In base pi, what does 2+2 or 3+3 equal? How would you write that?
1=1
2=2
3=3
10=pi
11=pi+1
12=pi+2
13=pi+3
20=2pi
21=2pi+1
22=2pi+2
23=2pi+3
30=3pi
31=3pi+1
32=3pi+2
33=3pi+3
100=pi^2.
It seems to me that 2 and 3 are pretty useless characters. How do you write 4pi? How do you write pi/2? I mean, in base pi, shouldn’t our standard base 10 integers be irrational numbers?
You’re right…I thought I understood his post but I had it garbled.
Never mind…
No, but they do have non-terminating and non-repeating representations.
perhaps the source of confusion here is that we are conflating the integers 1, 2, and 3 with the symbols 1,2, and 3 in sinjin’s numbering system. Perhaps if we used p, q, r and s it might be clearer. In a system of base pi, you would not be able to exactly represent a standard integer. We can keep 0
Actually, I’d think the number of symbols between 0 and “10” (pi) would be arbitrary. You’d have just as good a system if you used
p = pi / 8
q = pi / 4
etc. In general, if you choose n symbols between 0 and pi, the ith would be i pi / n.
The one glitch I can see is how you prove things using real integers when real integers are not well defined in the system. You would pretty much need to define integers anyhow independently.
On preview, Lemur866 brings up a good point. You need integers anyhow to express what
pi+pi is in a positional numbering system. The problem is that 1 is fundamental as the multiplicative identity, so I’m not sure how you’d express this in a system where 1 is not directly and exactly expressible. Or could we just assign a symbol to it, like we do for, say, pi and e, which are not expressible in our standard number systems.
Base 0 is not well defined. Base 1, on the other hand is. I’ve written some parody papers on base 1 arithmetic, and you can prove, using information theory, that base 1 machines need no power. (They also produce no information. )
I call the basic unit of base 1 a nit, so a base 1 engineer is a nit picker, and a genius at base 1 arithmetic is a nitwit.
BTW, I missed Exapno’s post on the arbitrariness of symbols when I did mine, but at least I applied it to the base pi problem.
We agree completely on that but it doesn’t address my question. I am responding to the post that says the digits in base pi are the decimal digits 0-3. Then by definition (as you have detailed) 3.3 > 10. This seems as nonsensical as saying 9.9 > 10 if you’re in base 10.
Dammit, Tripler!!
::Searching for something to gouge my mind’s eye out. Perhaps a rusty spoon?::
With all the other things that behave oddly in base [symbol]p[/symbol], I wouldn’t be surprised if our intuition about magnitudes is off. It doesn’t matter, though–as long as there’s a one-to-one correspondence between representations and real numbers, every other property is up for grabs.
True, but unless you do it as I have, it’s not base [symbol]p[/symbol].
The problem is that the coefficients are not directly representable in base p, where p is irrational! But since you need an integer number of symbols, traditional integers factor into things anyhow. So, you can’t represent integers exactly in the base selected, but they are used as if represented exactly in arithmetic in this base. This shows, if nothing else, you can’t reason as simply about irrational bases as you can about integer bases.
OK, I think I’ve had an intuition.
In base pi, the symbols 1, 2, and 3 are equivalent to e and pi and phi. They are interesting numbers, but you can’t do much with them, except 1.
So you can write 2+2, but there isn’t a simple number equivalent to our base 10 4, just like there’s no simple way to write pi + pi or pi**e* in base 10. You just have to leave those pi’s in.
So…
0=0
1=pi^0, or 1
10=pi^1, or pi
100=pi^2
1000=pi^3
etc.
The only reason 1 doesn’t equal pi is that it’s so much easier if 1=1, that way you can still do arithmetic. And you’ve got some special characters 2 and 3 but using them could cause problems, so 3.3 is analgous to writing e.e in base 10.
One more thing. What’s the difference between a nonterminating, nonrepeating number and an irrational number?
The coefficients aren’t written in base pi. They’re written as whatever set of symbols you care to use.
Sure you can. It just requires an infinite number of digits. Think like a mathematician, not a computer scientist.
If it’s any consolation, 10[sub]pi[/sub] can be rewritten as 3.01102111[sub]pi[/sub] . . ., which makes it look nice and small again compared to 3.3[sub]pi[/sub], like it’s “supposed” to.
But in general, all numbers except zero are going to have multiple representations in base pi, and that’s going to obscure their relative ordering — at least to the human eye.