Numbers don’t terminate or repeat (or fail to do either). Number representations do. Similarly, no representation is either rational or irrational, but every number is.
Understanding the distinction between a number and its representations is key to this sort of discussion. I think part of the problem is that we math types just get that they’re not the same thing, and don’t think to explain that to the questioners.
Actually, I was wrong - they are directly representable.
2 = (pi + pi )/ pi.
3 = (pi + pi + pi) / pi
etc.
So, integers are always directly representable in any number system, no matter the base.
Not quite. There is not a one-to-one correspondence, because there are multiple ways to represent some numbers. pi, for instance, could be written as 10, but it could also be written as 3.something . But then again, plain ordinary integer bases also don’t have a one-to-one correspondence between the real numbers and the representations, since you have things like 0.999… = 1.
Back to the OP, if you get down to the nitty-gritty, most fundamental level, numbers in mathematical proofs often are (or can be) expressed in forms like “The number after the number after the number after the number after 1”. So you can make statements like “The number after the number after the number after the number after 1 is prime”. That statement remains true regardless of whether you express that number as “5[sub]dec[/sub]”, or “101[sub]bin[/sub]”, or “12[sub]trin[/sub]”.
Why do you have to go and derail my grand mathematical arguments with your facts all the time?
So yes, there are multiple representations. What we need from a representation system is this: 1) given a representation, we can algorithmically calculate the number it represents; and 2) given a number, we can algorithmically pick a representation for it. As long as base pi satisfies those properties, it’s fine.
This thread got hijacked to discussion of bases like 3.5 (ugh) and pi (ugh,ugh) for which unique representation is not readily possible. A particularly nice base is 3 where you use -1,0,and 1 as the three symbols (or -,0,+, if you like). If you have ever solved that puzzle of weighing everything from 0 to 40 on a pan balance with only 4 weights, you will see the utility of such a base.
Less obvious is that you don’t have to use the same base for every digit. At the beginning of the 20th century, British coinage was based on a system in which the first (that is, least significant) digit was in base 4 (ha’pennies and farthings), the next was in base 12 (expressed using decimal digits, to be sure), the next in base 20 (ditto) and finally segueing into ordinary base 10.
Even less obvious is that you don’t even need a fixed base in each place. It would be possible to create a counting system in which the first place was in a base that varied from 28 to 31, depending in a complicated way on the values in the remaining places (and using 1 – n, not 0 – n-1 as “digits”), the second in base 12 and the remaining places in ordinary base10. Not only is it possible, but we are sufficiently perverse that we use such a system on a daily basis.
It’s probably worth pointing out what is probably obvious – that some results in number theory that depend upon the representation of the number (and therefore the base used) do depend upon the base used.
One of the more intriguing results of number theory is the Benford Probabilities. If you make up a chart showing the number of times the number 1 is the first digit in a table of, say, the lengths of rivers (or populations of cities, or any other statistic that we would expect to be randomly distributed) and compare it with the likelihood that the other digits appear as the first digit, we find that the probabilitiers are not equal. One would naively expect each digit to show up 1/9 of the time (since you never take “0” as the first digit). (I know I’d expect that, anyway). But it turns out that “1” is the first digit almost 1/3 of the time, with “2” as the first digit slightly less often than that, and so on down the line to “8” and “9”, which are the rarest ones. (It doesn’t matter, by the way, what units you measure the lengths of those trivers in – miles or kilonmeters or furlongs or Smoots. Which itself is a powerful clue as to what the distribution must be.)
the likelihood that the first digit is “N” turns out to be (log(N+1)) - (log(N)), where we’re taking the common logarithm.
in different bases, wee have to take the appropriate base logarithm. In base 2, the first digit is always “1”, of course. But in base 3, the first digit is “1” about 2/3 of the time.
Interesting. I can see how that’s true, but it’s equally true that you can do arithmetic in base 1.
1111 + 11 = 111111
111111111 / 111 = 111
Obviously, on paper, you have the extra symbols and the whitespace to convey information.
Could that possibly be aleviated with some simple rules about always writing numbers in a certain form?
Or, it lends support to the idea that the symbols should not correspond to integral values 0, 1, 2, 3… etc, but to fractions of pi. You could then do base pi with only two symbols, 0 and 1, which would just be base 2 multiplied by pi, or with n symbols, for each pi/nth part, which would be base n multiplied by pi.
astro, do you consider your question in the OP to be answered? We’ve taken this thread into directions that you didn’t anticipate, I expect. Despite all the explanations that we’ve given on how it’s theoretically possible to use negative, fractional, and every other sort of numbers for the base for a number system, I can’t imagine a civilization that would naturally use such bases, even if you contemplated a world where humans had more or less than ten fingers.
No, in base 1, there would only be one symbol, representing 0. The guy said base one is well defined, which may be correct, but still, you can’t ever actually represent any numeral other than 0 using such a base.
-Kris
I’m trying to figure out how a base pi system would work.
You’d have a “ones place,” a “pi’s place” and a “pi-squared’s place,” and so on.
So 123 should represent pi-squared plus 2*pi plus 3.
Okay, so far so good.
Now let’s count.
pi.
pi+1
pi+2
pi+3
2pi
2pi+1
2pi+2
2pi+3
3pi
3pi+1
3pi+2
3pi+3
pi^2
I am bothered by the following:
Well, who says you have to be able to do arithmetic with a numbering system? Their primary function is to represent numbers, I guess, and being able to do arithmetic with them is just a nice bonus. 
Still, I wonder if something counting as a kind of “base pi” system can be developed which does allow for easy arithmetic.
-FrL-
You can’t, strictly speaking, have a base 1, but there is a number system known as unary that can be used for integer arithmetic. If you restrict your Turing machines to unary (with appropriate other symbols for demarking the boundaries between numbers), you can show that P = NP.
The Benford distribution isn’t really a matter of number theory, though–it’s more descriptive statistics. Still, it’s an interesting result.
And that’s why we like to use integer bases. The definition of a basis representation is as I posted above. If you restrict your bases to certain nice algebraic structures (say Z, rationals with absolute value greater than 1, the algrebraic numbers), then the nice properties that you’re used to will hold. If not, they don’t.
I don’t know what Z or algebraic numbers are, but I do know what a rational with absolute value greater than 1 is. I don’t think those work. Take 1.5, for example. You’ll “count” as follows (representation in 1.5 first, then represented numeral as expressed in base 10 after the colon):
0: 0
1: 1
10: 1.5
11: 2.5
100: 2.25
and so on
As you can see, this sequence would bother me in the same way the pi sequence bothered me.
Without having thought through it completely, it’s looking to me like every non-integer base will have these “bothersome” properties. Do you think I’m wrong, though?
-Kris
Good point. Anyone who has used the unit circle in trigonometry has used base pi a bit. I hadn’t thought of it in that way.
Pretty clumsily, as seen.
Yes, I think an algorithm for doing even simple addition on two arbitrary base-pi numbers would be mighty ugly. Many simple sums would result in an infinite sequence of digits, even when the inputs are finite. And, you might need to “carry” in both directions, left and right from each column. And, the carry value coming out of each column could itself have infinite digits. Finite precision would save you from total insanity, but only just.
Or perhaps iterating on the columns of digits, as you do in integer bases, isn’t even the right approach to begin with. This is just how I’m trying to think it through.
It’s possible someone has worked out the details of base pi arithmetic already. (Base phi is workable for example, but that number has special properties pi doesn’t have.) Whether the details have been nailed down or not though, I think we can safely say that base-pi computers aren’t in our future.
Consider how you would transmit your equation. You might say that you’d transmit 2 volts as a 1, and use some number of 1s as +. The whitespace you mention would be 0 volts. But, you’ve just created a base 2 system - 2 volts and 0 volts! In a real base 1 system, you are allowed only one voltage. Since the value of the voltage is immaterial, you can set it at 0 volts, which kind of shows you intuitively that no energy is required for base 1 computations - and thus no information is transmitted.
I came up with this idea my first term of grad school where three different classes felt compelled to review binary arithmetic.